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Intermediate Algebra/Solving Absolute Value Equations

Absolute Values
Absolute Values represented using two vertical bars ($$\vert$$) are common in Algebra. They are meant to signify the number's distance from 0 on a number line. If the number is negative, it becomes positive. And if the number was positive, it remains positive:

$$\left\vert4\right\vert = 4 \,$$ $$\left\vert-4\right\vert = 4 \,$$

For a formal definition:

If $$x\ge0$$, then $$\left\vert x\right\vert= x$$ If $$x<0$$, then $$\left\vert x\right\vert=-x$$

The formal definition is simply a declaration of what the function represents at certain restrictions of the $$x$$-value. For any $$x<0$$, the output of the graph of the function on the $$xy$$ plane is that of $$y=-x$$.

Please note that the opposite (the negative, -) of a negative number is a positive. For example, the opposite of $$-1$$ is $$1$$.

Practice Problems
 {For all of these problems, a = -2 and b = 3. Evaluate the following expressions.}

{ $$|a|=$${ 2_1 }
 * type="{}"}

{ $$|b|=$${ 3_1 }
 * type="{}"}

{ $$|b + a|=$${ 1_1 }
 * type="{}"}

Absolute Value Equations
Now, let's say that we're given the equation $$\left\vert k\right\vert=8$$ and we are asked to solve for $$k$$. What number would satisfy the equation of $$\left\vert k\right\vert=8$$? 8 would work, but -8 would also work. That's why there can be two solutions to one equation (and later, even more solutions). (Answer this: why?)

Recall what the absolute value represents: it is the distance of that number to the left or right of the starting point, zero. This means that whatever the inside value represents, it must be either $$8$$ or $$-8$$. As such, $$2k + 6 = 8\quad\text{OR}\quad2k + 6 = -8$$. All that is left to do is to solve the two equations for $$k$$:
 * $$2k+6=8\qquad 2k+6=-8$$
 * $$\Leftrightarrow 2k=2\qquad 2k=-14$$
 * $$\Leftrightarrow k=1\qquad k=-7$$

A basic principle of solving these absolute value equations is the need to keep the absolute value by itself. This should be enough for most people to understand, yet this phrasing can be a little ambiguous to some students. As such, a lot of practice problems may be in order here.

We will show you two ways to solve this equation. The first is the standard way, the second will show you something incredible.

Standard way: Multiply the constant multiple by its inverse.

We'd have to divide both sides by $$3$$ to get the absolute value by itself. We would set up the two different equations using similar reasoning as in the first example: $$2k + 6 = 4\quad\text{OR}\quad2k + 6 = -4$$. Then, we'd solve, by subtracting the 6 from both sides and dividing both sides by 2 to get the $$k$$ by itself, resulting in $$k = -5, -1$$. We will leave the solving part as an exercise to the reader.

Other way: "Distribute" the three into the absolute value.

Play close attention to the steps and reasoning laid out herein, for the reasoning for why this works is just as important as the person using the trick, if not moreso. Let us first generalize the problem. Let there be a positive, non-zero constant multiple $$c$$ multiplied to the absolute value equation $$|2k+6|$$: $$c\cdot|2k+6|=|c|\cdot|2k+6|\quad\text{OR}\quad c\cdot|2k + 6|=|-c|\cdot|2k+6|$$. Let us assume both are true. If both statements are true, then you are allowed to distribute the positive constant $$c$$ inside the absolute value. Otherwise, this method is invalid!
 * $$\begin{align}

|c|\cdot|2k+6|&=|c(2k+6)|&\qquad|-c|\cdot|2k+6|&=|-c(2k+6)|\\ &=|2ck+6c|&\qquad &=|-2ck-6c|=|-(2ck+6c)|\\ &=|1|\cdot|2ck+6c|={\color{red}1\cdot|2ck+6c|}&\qquad &=|-1|\cdot|2ck+6c|={\color{red}1\cdot|2ck+6c|} \end{align}$$ Notice the two equations have the same highlighted answer in red, meaning so long as the value of the constant multiple $$c$$ is positive, you are allowed to distribute the $$c$$ inside the absolute value bars. However, this "distributive property" needed the property that multiplying two absolute values is the same as the absolute value of the product. We need to prove this is true first before one can use this in their proof. For the student that spotted this mistake, you may have a good logical mind on one's shoulder, or a good eye for detail.

We will first look at the  simplest case : the two values $$b\text{ and }c$$ are constants. We know the following properties are true: From this, we can conclude that $$|b|\cdot|c|=bc$$ for any $$b,c\in\mathbb{R}$$. We also know the following is true: Because we multiplied two absolute values, the product is either positive or zero. We can therefore use the second and third bullet point to conclude that
 * $$|-b|=|b|=b$$
 * $$|-c|=|c|=c$$
 * $$bc=d<0\Leftrightarrow |d|=-d>0$$. This simply means that for some product $$bc$$ that equals a negative number $$d$$, the absolute value of that is $$-d$$, or the distance from zero. Because $$d<0$$, multiplying the two sides by $$-1$$ will change the less than to a greater than, or $$d<0\Leftrightarrow-d>0$$.
 * $$bc=d=0\Leftrightarrow |d|=d=0$$. For some product $$bc$$ that equals a number $$d=0$$, the absolute value of that is $$0$$.
 * $$bc=d>0\Leftrightarrow |d|=d>0$$. For some product $$bc$$ that equals a positive number $$d$$, the absolute of the product is $$d$$.
 * $$|b|\cdot|c|=bc\ge0\Leftrightarrow |d|=d=bc=|bc|\ge0$$.

We can use all five bullet points to show that for any constants $$b,c\in\mathbb{R}$$, $$|b|\cdot|c|=|bc|\blacksquare$$. This was the simplest case. However, we have already proven the hardest case, where both $$b\text{ and }c$$ are variables. The five bullet points we have is enough to demonstrate this fact. As a result, this proof will be left as a trivial exercise for the reader. By confirming the general case, we may be employ this trick when we see it again. Let us apply this property to the original problem (this gives us the green result below):
 * $$3|2k+6|={\color{green}|6k+18|=12}$$

This all implies that $$6k+18=12\quad\text{OR}\quad 6k+18=-12$$. From there, a simple use of algebra will show that the answer to the original problem is again $$k = -5, -1$$.

Let us change the previous problem a little so that the constant multiple is now negative. Without changing much else, what will be true as a result? Let us find out.

We will attempt to the problem in two different ways: the standard way and the other way, which we will explain later.

Standard way: Multiply the constant multiple by its inverse.

Divide like the previous problem, so the equation would look like this: $$|2k + 6| = -2$$. Recall what the absolute value represents: it is the distance of that number to the left or right of the starting point, zero. With this, do you notice anything strange? When you evaluate an absolute value, you will always get a positive number because the distance must always be positive. Because this is means a logically impossible situation, there are no real solutions. Notice how we specifically mentioned "real" solutions. This is because we are certain that the solutions in the real set, $$\mathbb{R}$$, do not exist. However, there might be some set out there which would have solutions for this type of equation. Because of this posibility, we need to be mathematically rigorous and specifically state "no real solutions."

Other way: "Distribute" the constant multiple into the absolute value.

Here, we notice that the constant multiple $$c<0$$. The problem with that is there is no $$g$$ such that $$|g|<0$$. The only way this would be true is for $$-|g|<0$$ because
 * $$-|g|<0\qquad\text{Divide both sides by }-1$$
 * $$|g|>0$$

With this property, we may therefore only distribute the constant multiple as $$|c|$$ with a negative $$-1$$ as a factor outside the absolute value. As such,
 * $$-4|2k+6|=-|8k+24|=8\qquad\text{Divide both sides by }-1$$
 * $$|8k+24|=-8$$

It seems the other way has us multiply a constant by its inverse to both sides. Either way, this "other method" still gave us the same answer: there is no real solution.

The problem this time will be a little different. Keep in mind the principle we had in mind throughout all the examples so far, and be careful because a trap is set in this problem.

There are many we ways can attempt to find solutions to this problem. We will do this the standard and allow any student to do it however they so desire.
 * $$|3x-3|-3=2x-10\qquad\text{Add the }3\text{ to both sides.}$$
 * $$|3x-3|=2x-7$$

Because the absolute value is isolated, we can begin with our generalized procedure. Assuming $$2x-7>0$$, we may begin by denoting these two equations: $$ $$ These are only true if $$2x-7>0$$. For now, assume this condition is true. Let us solve for $$x$$ with each respective equation:

Equation
 * $$3x-3=2x-7\qquad\text{Add }3\text{ and subtract }2x\text{ on both sides.}$$
 * $$x=-4$$

Equation
 * $$3x-3=-(2x-7)\qquad\text{Distribute }-1\text{.}$$
 * $$3x-3=-2x+7\qquad\quad\text{Add }3\text{ and add }2x\text{ on both sides.}$$
 * $$5x=10\qquad\qquad\qquad\quad\text{Divide }5\text{ on both sides.}$$
 * $$x=2$$

We have two potential solutions to the equation. Try to answer why we said potential here based on what you know so far about this problem.

Because of this, we have to verify the solutions to this equation exist. Therefore, let us substitute those values into the equation:
 * $$|3(-4)-3|=2(-4)-7$$. Notice that the right-hand side is negative. Also, the left-hand side and the right-hand side are not equivalent. Therefore, this is not a solution.
 * $$|3(2)-3|=2(2)-7$$. Notice the right-hand side is negative, again. Also, the left-hand side and the right-hand side are not equivalent. Therefore, this cannot be a solution.

This equations has no real solutions. More specifically, it has two extraneous solutions (i.e. the solutions we found do not satisfy the equality property when we substitute them back in).

Despite doing the procedure outlined since the first problem, you obtain two extraneous solutions. This is not the fault of the procedure but a simple result of the equation itself. Because the left-hand side must always be positive, it means the right-hand side must be positive as well. Along with that restriction is the fact that the two sides may not equal the other for the values whereby only positive values are given. This is all a matter of properties of functions.

All the properties learned will be needed here, so let us hope you did not skip anything here. It will certainly make our lives easier if we know the properties we are about to employ in this problem.
 * $$6\left\vert5\frac{a}{6}+\frac{1}{12}\right\vert=\frac{3}{5}|15a+15|\qquad\text{Distribute, so to speak, the constant terms.}$$
 * $$\left\vert5a+\frac{1}{2}\right\vert=|9a+9|$$

Looking at the second equation might be the first declaration of absurdity. However, an application of the fundamental properties of absolute values is enough to do this problem. $$ $$ Peel the problem one layer at a time. For this one, we will categorize equations based on where they come from; this should hopefully explain the dashes: 3-1 is first equation formulated from, for example. $$ $$ $$ $$ We can demonstrate that some equations are equivalents of the other. For example, and  are equivalent, since dividing both sides of  by $$-1$$ gives. After determining all the equations that are equivalent, distribute $$-1$$ to the corresponding parentheses. $$ $$ $$ Now all that is left to do is solve the equations. We will leave this step as an exercise for the reader. You will find that two of three potential solutions are identical, so that means there are two potential solutions: $$a=-\frac{19}{28},-\frac{17}{8}$$. All that is left to do is verify that the equation in the question is true when looking at these specific values of $$a$$:
 * $$a=-\frac{19}{28}$$
 * $$\left\vert5\left(-\frac{19}{28}\right)+\frac{1}{2}\right\vert=\left\vert9\left(-\frac{19}{28}\right)+9\right\vert$$ is true. The two sides give the same value: $$\frac{81}{8}=10.125$$.
 * $$a=-\frac{17}{8}$$
 * $$\left\vert5\left(-\frac{17}{8}\right)+\frac{1}{2}\right\vert=\left\vert9\left(-\frac{17}{8}\right)+9\right\vert$$ is true. The two sides give the same value: $$\frac{81}{28}\approx2.893$$.

Because both solutions are true, the two solutions are $$a=-\frac{19}{28},-\frac{17}{8}\blacksquare$$.

Practice Problems
 {$$|k + 6| = 2k$$ $$k=$${ 6 _2 }
 * type="{}"}

{$$|7 + 3a| = 11 - a$$ $$a\in\{$${ -9 _2 },{ 1 _2 }$$\}$$
 * type="{}" coef="2"}

{$$|2k + 6| + 6 = 0$$ How many solutions? { 0 _1 }
 * type="{}"}

Lesson Review
An absolute value (represented with |'s) stands for the number's distance from 0 on the number line. This essentially makes a negative number positive although a positive number remains the same. To solve an equation involving absolute values, you must get the absolute value by itself on one side and set it equal to the positive and negative version of the other side, because those are the two solutions the absolute value can output. However, check the solutions you get in the end; some might produce negative numbers on the right side, which are impossible because all outputs of an absolute value symbol are positive!

Lesson Quiz
 {Evaluate each expression.}

{ $$|-4|=$${ 4 _5 }
 * type="{}"}

{ $$|6-8|=$${ 2 _5 }
 * type="{}"}

{Solve for $$a$$. Type NS (with capitalization) into either both fields or the right field for equations with no solutions. Any solutions that are extraneous (don't work when substituted into the equation) should be typed with XS on either the right field or both. Order the solutions from least to greatest.}

{$$|3a-4|=5$$ $$a\in\{$${ -1/3 _5 }$$,$${ 3 _5 }$$\}$$
 * type="{}" coef="2"}

{$$5|2a+3|=15$$ $$a\in\{$${ -3 _5 }$$,$${ 0 _5 }$$\}$$
 * type="{}" coef="2"}

{$$3|4a-2|-12=-3$$ $$a\in\{$${ -1/4|-0.25 _5 }$$,$${ 5/4|1 1/4|1.25 _5 }$$\}$$
 * type="{}" coef="2"}

{$$|a+1|-18=a-15$$ $$a\in\{$${ -2 _5 }$$,$${ NS _5 }$$\}$$
 * type="{}" coef="4"}

{$$2\left\vert\frac{a}{2}-1\right\vert-2a=-4a$$ $$a\in\{$${ -2 _5 }$$,$${ XS _5 }$$\}$$
 * type="{}" coef="4"}

{Read the situations provided below. Then, answer the prompt or question given. Type NS (with capitalization) into either both fields or the right field for equations that have no solutions. Any solutions that are extraneous should be typed with XS on either the right field or both. Order the solutions from least to greatest.}

{The speed of the current of a nearby river deviates $$1.5\tfrac{\text{m}}{\text{s}}$$ from the average speed $$20\tfrac{\text{m}}{\text{s}}$$. Let $$s$$ represent the speed of the river. Select all possible equations that could describe the situation. - $$|s-1.5|=20$$ - $$|s+1.5|=20$$ + $$|20-s|=1.5$$ + $$|s-20|=1.5$$ - $$|s+20|=1.5$$ - $$|1.5-s|=20$$
 * type="[]" coef="3"}
 * At least one of the current speeds for this equation would be negative, which is not representative of the speed of the river.
 * At least one of the current speeds for this equation would be negative, which is not representative of the speed of the river.
 * Good job! although all the magnitude of the other answers are correct, the actual direction (negative versus positive values) are incorrect, and therefore not representative of the situation.
 * Good job! although all the magnitude of the other answers are correct, the actual direction (negative versus positive values) are incorrect, and therefore not representative of the situation.
 * At least one of the current speeds for this equation would be negative, which is not representative of the speed of the river.
 * At least one of the current speeds for this equation would be negative, which is not representative of the speed of the river.

{A horizontal artificial river has an average velocity of $$-4\tfrac{\text{m}}{\text{s}}$$. The velocity increases proportionally to the mass of the rocks, $$r$$, in kilograms, blocking the path of the current. Assume the river's velocity for the day deviates a maximum of $$6\tfrac{\text{m}}{\text{s}}$$. If the proportionality constant is $$k=\frac{2}{5}$$ meters per kilograms-seconds, what is the maximum mass of the rocks in the river for that day? - $$5$$ kilograms. + $$15$$ kilograms. - $$25$$ kilograms. - $$35$$ kilograms.
 * type="" coef="4"}


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Diplomacy/Negotiation

Negotiation
Negotiation is the most important aspect of Diplomacy, but is the hardest to master. Great negotiation skill comes from experience.

Before opening negotiations, be sure to have a plan for yourself, as well as a plan for the power or powers with whom you are negotiating. Without a plan, negotiation does nothing except damage your reputation when you are unable to live up to the promises you made.

Negotiation is divided into these categories:

Attacking any power requires assistance. Show the advantages, and make sure that you divide pretty equally between the country's supply centers. It's pretty safe to be generous with another power's supply centers. However, be generous to a degree - be too generous, and you soon will see yourself as everyone's ally, creating suspicion.
 * 1. Attack a power

If two powers have an unhelpful stalemate line, they can sign a non-aggression pact to clear that stalemate line or not to attack each other when either has an enemy to deal with. Beware though, somebody can still stab you seeing that you are empty.
 * 2. Non-aggression pact

If playing Turkey, the last thing that it needs is a strong Austro-Russian alliance, as it could crush Turkey's chance. If playing Austria, you should do everything to stop at least two countries to attack you in the first few years. As England, a Franco-German alliance would be disaster. This way, preventing certain alliances is necessary to live - even alliances on the other side of the board can hurt later in the game. A common method is to create distrust between two countries that seem to have an alliance. If Austria wants to break the alliance between Russia and Turkey, then Austria could tell Russia what Turkey might do to stab Russia, and vice versa. After enough of this, Turkey and Russia will not be so willing to help each other, which will be better for Austria.
 * 3. Prevent certain alliances

If you have a very loyal ally, you can negotiate so he completely empties of units the supply centers closest to your territory, while you prepare to take them. This means that a short time span you can take his supply centers and cripple him before he can react.
 * 4. Prepare for a stab

Furthermore, try to make it so your stab has immediate results. You should be able to take at least a supply center or two in the next fall. Unless he is much stronger then you are, this gives you the advantage over him, and means you might be able to avoid a counter-attack.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Diplomacy/Negotiation


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Talk:Intermediate Algebra/Chapter1Test

MATHOS: Don't forget to have question from 1-4 concerning Jelly Beans Guessing Contest to practice both Word Problems and Absolute Value Equations!

title: Talk:Intermediate Algebra/Chapter1Test


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Diplomacy/Software

There are both commercial software and freeware for playing Diplomacy.

Freeware
Freeware is software that is free to the public to use. Freeware software allows people to use and access the programs free of charge and may be shared with other individuals, if permitted. The right to freeware is often held by the owner, meaning no one is able to alter, change, or sell the created software. A common example of freeware is Preezo (similar to Microsoft PowerPoint), which is a presentation software that is free via the Internet. Like Preezo, freeware software is very commonly shared on the Internet for public use.

There are several freeware products for Diplomacy. The most used is Realpolitik and only for PC/Windows RealWorld.

webDiplomacy is free software that lets you play online via your web browser.

jDip is a very commonly used Diplomacy mapping/ajudication application, with many features and variants.

title: Diplomacy/Software


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Basic Algebra/Exponential Functions/Multiplication of Exponents

Example Problems
Ex.1

(c+3)(c-2)-(c+2)(c-3)=

(c²-2c+3c-6)-(c²-3c+2c+6)=

c²-2c+3c-6-c²+3c-2c-6=

Answer: 2c-12

Ex.2

3(2s+4)(s-3)-2(s+3)(2s-4)=

(6s+12)(s-3)-(2s+6)(2s-4)=

(6s²-18s+12s-36)-(4s²-8s+12s-24)=

(6s²-6s-36)-(4s²+4s-24)=

6s²-6s-36-4s²-4s+24=

Answer: 2s²-10s-12

Practice Problems
Use  for exponentiation  {Easy}

{ (x+2)(x-4)={ x^2 - 2x - 8 (i)|x^2-2x-8 (i) _23 }
 * type="{}" coef="1"}

{ x(x+2)+2(x-4)={ x^2 + 4x - 8 (i)|x^2+4x-8 (i) _23 }
 * type="{}" coef="1"}

{ (3a+4)(2a-2)={ 6a^2 + 2a - 8 (i)|6a^2+2a-8 (i) _23 }
 * type="{}" coef="1"}

{ 4s(s²+s)-4s²(s-1)={ 8s^2 (i)|8s^2 (i) _23 }
 * type="{}" coef="1"}

{ Medium}

{ 2(2x²+4)(2x+3)+(2x+5)={ 8x^3 + 12x^2 + 18x + 29 (i)|8x^3+12x^2+18x+29 (i) _23 }
 * type="{}" coef="2"}

{ 5(x+2)(3-4)-2(x+2)={ -7x - 14 (i)|-7x-14 (i) _23 }
 * type="{}" coef="2"}

{ Hard}

{ (a+2)²-3(a+2)+10-(a+2)²+3a={ -3a + 4 (i)|-3a+4 (i) _23 }
 * type="{}" coef="3"}

{ v(v+1)-v(v+2)+v(v+5)-v(v+4)={ 0_23 }
 * type="{}" coef="3"}

{ (2d+2)(3d-2)-(3d+2)(2d-3)={ 7d + 2 (i)|7d+2 (i) _23 }
 * type="{}" coef="3"}


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Basic Algebra/Exponential Functions/Zero and Negative Exponents

Lesson
There are two very important things you need to know when working with zero or negative exponents. The rules of them.

First, any number the zero power always equals one. For example -5000 = 1

Second, any number to the negative power is the opposite fraction of the exponent. For example 2-2 = 1/22 = 1/4 because 22 = 4 and when you have 2-2 it becomes its reciprocal to get 1/4.

This is a review of section 2 of the polynomials chapter.

Practice Problems
Easy

Medium

Hard


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Basic Algebra/Rational Expressions and Equations/Multiplying and Dividing Rational Expressions

Practice Problems
 {Easy}

{ -20(s^2-t^2) / 5(s-t) = { -4(s + t) (i)|-4(s+t) (i) _9 }
 * type="{}"}

{ Medium}

{ Hard}


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Basic Algebra/Rational Expressions and Equations/Adding and Subtracting with Like Denominators

Lesson
"Write" the original expression

5/2x + x-5/2x

"Add" the numerators

5x + (x-5)/2x

"Combine" like terms

x/2x

"Simplify" the expression

1/2

Practice Problems
Easy

Medium

Hard


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Basic Algebra/Rational Expressions and Equations/Adding and Subtracting When the Denominators are the Same

Lesson
You have to leave the same denominator and add the nominators:

x  +   3x = x+1     x+1

x + 3x = x+1

4x x+1

Practice Problems
Easy

Medium

Hard


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Basic Algebra/Rational Expressions and Equations/Adding and Subtracting When the Denominators are Different

Lesson
If you have to add two rational fractions with different denominators, as the first step, you have to find the LCM:

3  +   3   x+1     x-1

LCM = (x+1)(x-1)

Now divide the LCM by both denominators and multiply by their respectives numerators:

(x+1)(x-1) / (x+1) = (x-1). (3) = 3x-3 (x+1)(x-1) / (x-1) = (x+1). (3) = 3x+3

The sum of the two results would be the new nominator:

3x -3 +3x +3 = (x+1)(x-1)

6x (x+1)(x-1)

This is another example:

6x  +   9x 2x-6   x2-6x+9

We factorize both denominators and find the LCM

2x-6 = 2(x-3) x2-6x+9 = (x-3)2 LCM = 2(x-3)2

Now we divide and multiply:

2(x-3)2 / 2(x-3) = 2x2-12x+18 / 2x-6 = x-3 (x-3). 6x = 6x2-18x

2(x-3)2 / (x-3)2 = 2x2-12x+18 / x2-6x+9 = 2 (2) . (9x) = 18x

We add the results to obtain the nominator; the denominator is the LCM:

6x2 -18x +18x = 2(x-3)2

6x2 2(x-3)2

We can factorize the nominator to simplify the result:

2 (3x2) = 2 (x-3)2

3x2 (x-3)2

Practice Problems
Easy

Medium

Hard


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Basic Algebra/Rational Expressions and Equations/Simplifying Rational Expressions

Lesson
To simplify rational expressions, first we need to know what they are. They are fractions that have polynomials on both their numerators and denominators.

 * $$\frac{2x^2+8x+8}{6x-2}$$

You just can simplify factors, not terms. That means you have to factorize the nominator and the denominator:

 * $$\frac{2x^2+8x+8}{6x-2}$$

 * $$= \frac{2(x+2)^2}{2(3x-1)}$$

Now, both are divisible by 2. So just divide them by 2, the result would be:

 * $$= \frac{(x+2)^2}{3x-1}$$

Practice Problems
Easy

Medium

Hard


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Basic Algebra/Rational Expressions and Equations/Chapter Review

Lesson 1. Direct and Inverse Variation Lesson 2. Simplifying Rational Expressions

Lesson 3. Multiplying and Dividing Rational Expressions

Lesson 4. Adding and Subtracting with Like Denominators

Lesson 5. Adding and Subtracting When the Denominators are the Same

Lesson 6. Adding and Subtracting When the Denominators are Different

Lesson 7. Solving Rational Equations


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Basic Algebra/Exponential Functions/Chapter Review

Lesson 1. Multiplication of Exponential Expressions Lesson 2. Zero and Negative Exponents

Lesson 3. Graphs of Exponential Functions

Lesson 4. Division of Exponential Expressions

Lesson 5. Scientific Notation

Lesson 6. Growth and Decay Functions


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Basic Algebra/Quadratic Functions/Chapter Review

Lesson 1. Graphing a Parabola

Lesson 2. Using the Discrimnant to Help Graph a Parabola

Lesson 3. Finding the Axis of Symmetry of a Parabola

Lesson 4. Finding the Vertex of a Parabola

Lesson 5. Graphing Quadratic Inequalities


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Basic Algebra/Solving Quadratic Equations/Chapter Review

Lesson 1. The Zero Product Property

Lesson 2. Solving Quadratic Equations by Factoring

Lesson 3. Solving Quadratic Equations By Taking The Square Root of Both Sides Lesson 4. Solving Quadratic Equations By Completing The Square

Lesson 5. Solving Quadratic Equations By Using The Quadratic Formula

Lesson 6. The Discriminant

Lesson 7. Quadratic Equation Word Problems


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Basic Algebra/Factoring/Chapter Review

Lesson 1. Factors of Integers

Lesson 2. Dividing Monomials

Lesson 3. Monomial Factors of Polynomials

Lesson 4. Multiplying Binomials

Lesson 5. Squares of Binomials

Lesson 6. Factoring x^2 + bx + c

Lesson 7. Factoring x^2 - bx + c

Lesson 8. Factoring x^2 - bx - c and x^2 + bx - c

Lesson 9. Factoring by Grouping

Lesson 10. Factoring Cubic Polynomials

title: Basic Algebra/Factoring/Chapter Review


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Basic Algebra/Polynomials/Chapter Review

Lesson 1. Exponents

Lesson 2. Zero and Negative Exponents

Lesson 3. Adding and Subtracting Polynomials

Lesson 4. Multiplying Monomials

Lesson 5. Powers of Monomials

Lesson 6. Multiplying Monomials and Polynomials

Lesson 7. Multiplying Polynomials


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Basic Algebra/Systems of Linear Equations/Chapter Review

Lesson 1. Solving Linear Systems by Graphing

Lesson 2. Solving Linear Systems by Substitution

Lesson 3. Solving Linear Systems by Linear Combinations

Lesson 4. Using Matricies to Solve Linear Systems


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Basic Algebra/Linear Inequalities/Chapter Review

Lesson 1. Solving Inequalities

Lesson 2. Solving "AND" Compound Inequalities

Lesson 3. Solving "OR" Compound Inequalities

Lesson 4. Solving Absolute Value Inequalities

Lesson 5. Graphing Linear Inequalities


text:
Basic Algebra/Lines (Linear Functions)/Chapter Review

Lesson 1. Graphing Points

Lesson 2. Equations in Two Variables

Lesson 3. Graphing Lines(Linear Functions)

Lesson 4. Slope of a Line

Lesson 5. Slope-Intercept Form of a Line

Lesson 6. Find Parallel and Perpendicular Lines

Lesson 7. Find the Equation of the Line Using Two Points

Lesson 8. Functions and Relations


text:
Basic Algebra/Proportions and Proportional Reasoning/Chapter Review

Lesson 1. Proportions

Lesson 2. Percents

Lesson 3. Percent of Change

Lesson 4. Weighted Averages

Lesson 5. Direct and Inverse Variation

Lesson 6. Solving Similar Triangles using Proportions


text:
Basic Algebra/Solving Equations/Chapter Review

Lesson 1. Solving Equations with Addition and Subtraction

Lesson 2. Solving Equations with Multiplication and Division

Lesson 3. Solving Multi-Step Equations

Lesson 4. Solving Equations with Variables on Both Sides of the Equation

Lesson 5. Equations with More than One Variable

Lesson 6. Formulas


text:
Basic Algebra/Working with Numbers/Chapter Review

=Lesson 1. Integers and the Number Line=

=Lesson 2. Absolute Value=

=Lesson 3. Adding Rational Numbers=

=Lesson 4. Subtracting Rational Numbers=

=Lesson 5. Multiplying Rational Numbers=

=Lesson 6. Distributive Property=

=Lesson 7. Combining Like Terms=

=Lesson 8. Dividing Rational Numbers=

=Lesson 9. Formulas=

Games and Activities for Review

 * http://www.quia.com/cm/19994.html (30 second integer practice - add, subtract, multiply and divide)
 * http://www.quia.com/cb/25168.html (add and subtract positive and negative numbers)


text:
Novial/Finni numres

Pri li numral sisteme del finni lingue.

On dikte ofte ke li finni lingue es tre desfasili. In serteni maneres disu es veri, ma in altris finnum es plu fasil kam multi altri lingues. Un tal manere es li sisteme de numres, kel es li teme de dis kurti esaye. Altri es li sisteme de ortografia e pronuntiatione. Por ke li lektere pove pronuntia korektim li finni numres, li sisteme de ortografia bli brevim deskripte.

Vokales:

a	kom fransi a

e	kom fransi é

i	kom fransi qui

o	kom sud-angli hot

u	simil ye germani unter

y	simil ye fransi u

ä	kom sud-angli bat

ö	kom germani schön

Omni disus es kurt vokales. Longi vokales kun li sami suones es skriptet kom duopli vokales. Exemplim, ii es li longi i kun sami suone ma plu longi. Es anke kelki diftonges ma disis non es besonat in dis esaye.

Konsonantes:

b	kom angli bat (in finnum nur in vordes pruntat fro stranjeri lingues)

c	kelkitem kom angli cap, kelkitem kom angli sap (nur in stranjeri nomes)

d	kom angli dip

f	kom angli for (nur in vordes pruntat fro stranjeri lingues)

g	kom angli go

h	kom angli hat (anke talim pronuntiat devan konsonante)

j	kom angli yes

k	kom fransi kilomètre

l	kom angli let

m	kom angli man

n	kom angli now

ng kom angli sing, ma li suone es duopli inter du vokales

p	kom fransi pied

q	nur in “qu“, kom germani Quelle (kv)

r	kom spani pero

s	kom angli sit

t	kom fransi terre

v	kom angli van

w	kom v (nur in stranjeri nomes)

x	kom ks (nur in stranjeri vordes)

z	kom ts (nur in vordes pruntat fro stranjeri lingues)

Finni vordes preske sempre bli pronuntia kom skripte segun li ortografi regles. Duopli vokales mus bli pronuntia kom longi e samiman duopli konsonates mus bli pronuntia kom du konsonantes. Li strese de simpli, non-komposit vorde es sempre sur li unesmi silabe. Komposit vordes bli pronuntia quasi separat vordes.

Li unesmi dek numres es:

1.	yksi

2.	kaksi

3.	kolme

4.	neljä

5.	viisi

6.	kuusi

7.	seitsemän

8.	kahdeksan

9.	yhdeksän

10.	kymmenen

Interesanti trete del finni gramatike es li forme del vorde after numre. Li gramatikal kasu es li partitivi singulare, exept after yksi kand lu es li nominativi singulare. Tamen, por li eme de dis esaye, on pove regarda li partitivi singulare kom plurale.

Exemplim:

yksi talo (un hause)

kaksi taloa (du hauses)

kymmenen taloa (dek hauses)

e seteru.

Disu es importanti por komprenda qualim li sistema funktiona. Li sekuenti dek numres es:

11.	yksitoista

12.	kaksitoista

13.	kolmetoista

14.	neljätoista

15.	viisitoista

16.	kuusitoista

17.	seitsemäntoista

18.	kahdeksantoista

19.	'''yhdeksäntoista '''

20.	kaksikymmentä

Li numres fro dek-un a dek-nin bli forma per adi li sufixe -toista a li apti numre sub dek (faktim li vorde toista signifika ”altri” o ”duesmi”). Duanti, kaksikymmentä es literalim ”du dekos”, kymmentä esent li partitivi singulare de kymmenen. Numres plu grandi kam duanti have li forme ”du dekos un”, ”tri dekos set” e seteru:

21.	kaksikymmentäyski	(”du dekos un”)

22.	kaksikymmentäkaksi	(”du dekos du”)

23.	kaksikymmentäkolme	(”du dekos tri”)

24.	kaskikymmentäneljä	(”du dekos quar”)

25.	kaksikymmentäviisi	(”du dekos sink”)

26.	kaksikymmentäkuusi	(”du dekos six”)

27.	kaksikymmentäseitseman	(”du dekos set”)

28.	kaksikymmentäkahdeksan	(”du dekos ot”)

29.	kaksikymmentäyhdeksän	(”du dekos nin”)

30.	kolmekymmentä		(”tri dekos”)

40.	neljäkymmentä		(”quar dekos”)

50. viisikymmentä		(”sink dekos”) 60. kuusikymmentä		(”six dekos”)

70. seitsemänkymmentä	(”set dekos”)

80. kahdeksankymmentä	(”ot dekos”)

90. yhdeksänkymmentä	(”nin dekos”)

100. sata

Sentos bli forma samiman kam dekos. Li partitivi singulare de sata (sent) es sataa (sentos).

200. kaksisataa			(”du sentos”)

300. kolmesataa			(”tri sentos”)

301. kolmesataayksi			(”tri sentos un”)

423. neljäsataakaksikymmentäkolme		(”quar sentos du dekos tri”)

859. kahdeksansataaviisikymmentäyhdeksän	(”ot sentos sink dekos nin”)

1000. tuhat

Li partitivi singulare de tuhat (mil) es tuhatta (milos).

2000. kaksituhatta			(”du milos”)

2016. kaksituhattakuusitoista		(”du milos six-altri”)

5325. viisituhattakolmesataakaksikymmentäviisi	(”sink milos tri sentos du dekos sink”)

Vida quam longi kelki numres apari!

title: Novial/Finni numres


text:
Talk:Hebrew/Tips for Contributing

Transliteration
I agree that the Hebrew spelling (characters) are primary and should always be in the lesson. But I disagree with the comment in the tips that we shoud avoid transliterations. I think we should encourage transliterations for the elementary lessons. In some cases there may be multiple transliterations, and that's ok. Also in elementary lessons, I think we should have the vowels and other diacritic marks. (Although I think it's good to have some exercises practicing reading without them.) --CocoaZen 02:22, 17 July 2006 (UTC)

Course?
To be honest, despite using words as 'lesson', I miss the actual 'tutoring' towards the student. I get the feeling of 'an information dump', instead of teaching Hebrew...

Hebrew diacritics (nikud) with Windows 10
This article needs updating to include instructions on how to include nikud with Windows 10. 238-Gdn (discuss • contribs) 17:38, 21 June 2017 (UTC)

title: Talk:Hebrew/Tips for Contributing


text:
User:Mindspillage

See w:User:Mindspillage for my page on en.wikipedia.

title: User:Mindspillage


text:
Hebrew/Greetings

Description: This lesson contains some basic greetings. There are approximately 10 vocabulary words and 5 phrases. This lesson has no prerequisite lessons, although the Aleph-Bet would be useful.

Adjectives

 * Can be used with appropriate time of day (morning, afternoon, evening, night) as a greeting.

Phrases

 * לְהִתְרָאוֹת : See you (Lehitra-ot)

 * ?מָה שְלוֹמְךָ : How are you? (to a man; Ma sh'lomkha)
 * ?מָה שְלוֹמֵך : How are you? (to a woman; Ma sh'lomekh)

 * : What is your name? (Ma Shimkha?) מה שמך?

 * : My name is X. (Shmi hu X) שמי הוא X

-- good night

Example Dialog
?מָה שְלוֹמְךָ בְּסֵדֶר Translation: How are you (to a man)? Ok.

שָׁלוֹם לילה טוֹב ?מָה שְלוֹמֵך טוֹב Translation: Hello. Good night. How are you (to a woman)? Good.

Exercises
Translate the following conversation into Hebrew:

"Hello!" "How are you?"(Translate to a man, then a woman.) "Good." "Goodbye, and good night." "Goodbye."

Related lessons
Hebrew/Basic 1

For a more advanced lesson on greetings, see Hebrew/Greetings2 (add there: I'm happy to/It's good to meet you.     I'm fine.  She is sick.  )

title: Hebrew/Greetings


text:
Diplomacy/Geography

Divisions of the Diplomacy Gameboard
The countries of the diplomacy board are most importantly organized along the St. Petersburg-Switzerland-North Africa stalemate line. The three countries north of the line are called the Western Triangle while those south are called the Eastern Rectangle. Each half of the board holds 17 centers, so a power must be able to cross the stalemate line in force to achieve victory.

Western Triangle
Composed of England, France, Germany and St. Petersburg, the Western Triangle is more geographically spread-out than its eastern counterpart. This provides difficulties for France when trying to gain a solo victory. Closer centers, like Tunis, Italy, and Warsaw require crossing an easily-held stalemate line, while Scandinavia and St. Petersburg are very far away. Thus France has a good record of strong draws, but has only a middling record of wins because it is simply too far away from anything else. England suffers from the same problem—Scandinavia is closer, but it is difficult for England to deploy armies to Central Europe to conquer MUN, and TUN is far away and across a stalemate line. Germany has the best winning record of the three due to his ability to make up for centers that are difficult for him to conquer, for example SPA, POR, and MAR, with closer centers across the stalemate line, usually VIE, TRI, VEN, and SEV.

Eastern Rectangle
Composed of Italy, Turkey, Austria, Russia except St. Petersburg, and Tunis, The Eastern Rectangle is geographically more compact than the Western Triangle, but has four powers vying for five neutrals, instead of three powers (and Russia's northern fleet) battling over seven neutral SC's. The extra competition often turns the East into a complete quagmire, allowing one or two Western Powers to clean up and then turn to attack Italy and Russia. Turkey sometimes gains strong draws out of these games, although often the Western power will rush to a quick solo victory while the four Eastern powers battle each other for dominance. Except for Turkey, tucked in its own corner of the board, the powers of the Eastern Rectangle have an easier time breaking through stalemate lines than their Western counterparts. Italy is more prepared to push into the Western Med than France is to break into the east. Austria's home centers are slightly closer to the stalemate line than Germany's, and it is much easier for Austria to take MUN than Germany to take VIE. Russia can ignore stalemate lines completely so long as he holds onto St. Petersburg or maintains fleets in Northern waters.

Italy as an Independent
Other theories of board division contain a Western and an Eastern Triangle and treat Italy as an independent variable. This is based on the effect that a northwardly aggressive Italy can have on the Western Triangle. While attacks into Germany or France rarely yield worthwhile results for Italy, its actions can decide which power is ultimately victorious in the west.

Stalemate Lines
Stalemate Lines are lines that cross the board that have more spaces on one side than the other that are able to be defended by fewer units than the number of SC's they protect. They are quite controversial, especially in the design of variants, and are considered by most to be a weakness of the game because they do not parallel reality. Whereas real war is won by concentration of force in proper areas, the rules of Diplomacy allow for only one unit to occupy one province.

The most important stalemate line runs from St. Petersburg to North Africa, running between the non-supply center territories and Germany until Switzerland, then between PIE and MAR, GOL and SPA, and the Western Mediterranean Sea. This line separates the Western Triangle from the Eastern Rectangle, and it is around this line that most victories are based. To win, most powers conquer everything on their side of the line, and reach for one center on the other side. This is easier the closer the power is to a stalemate line.

Abbreviations
These are the standard three-letter abbreviations for provinces, that will be used for the rest of the book. Provinces are sorted first by region (separated into land and sea areas), then alphabetically by name. If there are multiple province abbreviations available, they are listed in order of frequency.

 * Impassable areas are spaces which are marked on most maps, but not a part of gameplay.
 * Tunisia is misnamed on the map. The space is called "Tunis", despite the naming rule that only powers' SCs would be named with cities.
 * Although the game begins in "Spring 1901", Sweden and Norway are separate provinces for gameplay purposes. This is because the map nevertheless reflects the 1914 borders, in Scandinavia as well as the Balkans.
 * "Livonia" and "Liverpool" conflict, so neither is abbreviated "LIV". "Romania" and "Rome" would conflict, but the former was called "Rumania" instead by Allan Calhammer.
 * "North Atlantic Ocean", "North Sea", "Norway", "North Africa", and "Norwegian Sea" all begin with "Nor", so none of them use this abbreviation.
 * Neither "Gulf of Bothnia" or "Gulf of Lyons" are abbreviated "GUL"; see chart.

title: Diplomacy/Geography


text:
Talk:Diplomacy/Introduction

Moving Rules information to Rules Section
I'll move the rules covered in this section to the Rules section.

Diplomacy is not a zero sum game. Spain and balkans can be taken by players without the other players losing anything.

title: Talk:Diplomacy/Introduction


text: Diplomacy/Books

title: Diplomacy/Books


text:
Principles of Economics/Preface/To The Student

Dear Student
''This text is designed to provide you with a very comprehensive and easy-to-understand introduction to economics. This text will assume no background in economics. All you really need is an open mind and an interest in learning!''

Thank you,

Endless melee - Initiating Author

Why study economics?
Those who don't study economics will say that it's boring. Those who do will say that it's dismal. The former say it's boring because they can't see the relationships so crucial to the discipline. The latter say it's dismal because many of the conclusions reached reveal saddening facts. It's not knowledge that's easy to show off, either, nor is it particularly useful in day-to-day interactions.

So why would you ever want to study economics?

Study it because it interests you. Study it because monetary policy intrigues you, because you want to know the causes of inflation, because you wonder how the market works so flawlessly (or not), because you wish to comprehend socially applied concepts of efficiency, because the concepts of supply and demand are beautiful to you, because you find the analytical tools and theories of economics graceful, because you hope to learn more about the market and the "invisible hand", because...

-- Yunzhong Hou - Primary Author and Illustrator

Why economics?
The world is made up of billions of people, each with their own motivations and beliefs. To try to fully understand each individual would be a hopeless exercise in futility. Instead, the job of the economist is to simplify the world to the point where we can say something interesting about it. When building a model, it is always my hope that the interesting bit of information I walk away with will, by some miracle, still be useful when applied to the real world, which is anything but simple.

Note that I haven't said anything about an economy, money, or trade yet. You'll read quite a bit about those topics in this book, but they are not as essential to economics as you might think. Economics is the study of people and choice. It's a methodology. It's a way of thinking about the world. If the authors have done their job, you'll walk away from this text with a new perspective on the world and a deeper understanding of how choices are made.

Good luck!

~ BU Rob13 - Contributing author


text:
Principles of Economics/List Of Contributors

 * Endless melee
 * Yunzhong Hou
 * BU Rob13


text:
Massage - Synergistic Techniques

This book is created to explain and summarize a style of massage incorporating many aspects of international massages into a massage that aims to relieve all types muscle tension. This style is designed to cater to athletes or individuals with significant muscle tension. This book is designed to provide massages that relieve a specific type of muscle tension and should not be confused with massages that try to "provide a general feeling of relief".

(To readers: Please note this is my first book, and I am trying to create a concise outline of various topics to be covered. I realize the tone of voice may be very forceful and I aim to revise this in future edits. This forceful tone is aimed to keep me on track to follow my vision of what concepts need to be covered properly.)

This book aims to be a technical howto massage manual (readable for beginners), with referencing of specific groups of muscles (with technical names), by providing a specific set of solutions to relieve muscles pain. Ideally this book will try to provide a set of massage solutions that only the best physio-therapists can currently provide. Solutions will try to never be vague or provide "new-age" remedies (without explicit noting of unknowns).

Introduction
It is my belief that there is no style of massage that is better than other styles of massages. This style aims to increase the effectiveness of massage in general by combining my first hand experience with different massage techniques including; fingertip, pressure point/reflexology, chiropractor, and stretching.

General Massage Theory
Massage with your fingertips. The human fingertips have the most nerves in the human hand allowing a greater feel for the muscle tension in your massage subject. This is not as easy as it sounds as the pressure of the massage must be from the inside of the finger away from the fingernail. The fingernail should touch the subject as little as possible.

As such, I find rock-climbers and basketball players have the best hands for this massage as fingertip strength is used extensively in their sport. Professions who grip bars or handles for a living will find proper massage technique incredibly difficult (they tend to want to dig their fingernails in and lack fingertip strength vs grip strength).

The human body has many different types of muscles that serve to extend, retract, and rotate limbs. Muscles and tendons are often laid in a way that makes massage impossible on the inner muscles until the outer layers of muscle are relaxed.

Injuring muscles is a common side effect of deep-tissue massage and massaging very stiff muscles. Knowing which techniques cause the least damage allows the massuese to give an effective massage while minimizing the negative effects. This is especially important for athletes that want to compete and retain their balance/speed/strength.

 * Stroking technique
 * Often done with oil, this involves stroking the muscle along the length of the muscle. This technique works best with muscles located on the surface uncovered by other muscles.
 * Cross-stroking refers to the stroking of muscles across its axis to make the muscle "snap". This is much like plucking the string of a guitar on an instrument and should be avoided as much as possible. Extensive cross-stroking of muscles leads causes unnecessary wear and tear compared to other techniques. Note: this technique does often lead to a quicker relief of pressure but often leaves the muscle severely weakened. My experience has shown that the "pure pressure" technique is preferrable to Cross-stroking.
 * Cross-stroking should not be used as a massage technique so much as to locate tendons/muscles that are stiff.

 * "Pure pressure" techinique (Chiropractor based)
 * Many muscles, including most large muscles and many muscles located deep under layers of other muscles respond best to this technique. Stroking deep muscles often leads to the "cross-stroking" of larger muscles on top which damages the muscle ontop.
 * Simply done, the "pure pressure" is commonly done using the knuckles or elbow. The masseuse leans on and compresses the muscle at pressure points for 30+ seconds. Often this may take minutes per muscle (especially for larger muscles). Eventually the muscle will soften and releasing the pressure will give the subject a feeling that blood and relief is gushing into the muscle. (there should be only limited numbness associated with this technique, you should NOT be cutting off the blood supply)
 * "Pure Pressure" should be applied at an angle that generates some cross-stroking. That is, as the muscle relaxes, you should be able to feel the muscle gently slide by your elbow/knuckle. If it snaps past your elbow/knuckle, your angle is not correct.

 * "Tapping"
 * Basically just shaking the muscle by various means; chopping at the area, mechanical (ultrasound or consumer vibration devices), jacuzzi water-jets/bubbles. Of these, water-jets seem the most effective (possible due to the use of warm water often associated with jacuzzi's). Overuse of tapping techniques will definitely result in noticeable muscle damage including bruising. This technique should be used lightly and only as a supplement to other techniques.
 * Novices without enough fingertip strength to massage larger muscles will often revert to this technique. Note: Larger muscles often can take more abuse than smaller muscles and Tapping is a very effective technique if used sparringly.

 * Stretching
 * Although not strictly a massage. This is often the most effective way to relieve stress in large muscles or deep muscles (eg. rotator cuff). Effective stretching should last 30+ seconds and may last minutes depending on circulation and stiffness.
 * Do not bounce when stretching (this may result in injury and is not as effective). Constant pressure should be applied when stretching. Pressure will need to be periodically increased as the muscle relaxes during the stetch.

 * Promoting circulation (Heat/Ice Contrast baths)
 * Note: Although related, this is not a guide to sports injury and will not go extensively into RICE (Rest Ice Compress and Elevate) techniques that are used to treat sports injury.
 * I find my experience with Ice and Constrast baths (alternating 5min hot and cold water baths) to be a very effective technique to help deal with swelling and injuries. However, for general muscle stiffness, this does not help enough to be considered a technique.
 * Assuming your source of muscle tension is not an injury that is swollen and may worsen with heat, heat is a masseuses best friend. If you are very stiff, the subject should take a long 15min+ hot bath/jacuzzi to loosen up the muscles. This will make the massage less painful, less damaging, and more effective overall.
 * After a very physical massage which has released a lot of stress, another hot bath will help circulation and help clear the lactic acid from your muscles.

Shoulder stiffness - Rotator Cuff
The most common muscle tension state that baffles subjects and massagers. This is caused by either of the rotator cuff muscles (lookup name) located deep within the shoulder blade. This deep location means it is impossible to massage these muscles effectively without first relaxing the very large shoulder and back muscles that cover it up.

The pressure points for the shoulder and back muscles are quite far from the source of pain caused by the stiff rotator cuff. This causes the subject and massager to be often be misguided by the pain and stiffness which they are unable to release.

Locating the problem
Subject will often be seen moving their shoulder blade and swinging their upper arm(lookup technical name) trying to find relief.

Exact location: Betweent the lats (back muscle) and the rear deltoids (shoulder muscle) just below and behind the armpit.

Solution

 * 1) Relax the shoulder muscle (lookup name of specific group of muscles). The rotator cuff will to be totally inaccessible to massage if the shoulder or back is stiff.
 * 2) Stretching - Often the best way to provide relax the rotator cuff is to stretch it. The shoulder rotator cuff serves to rotate your forearm forward/backward.
 * 3) ; Stretch forward rotator cuff (lookup techinical name)
 * Find a pole (at least body height, need a diagram); with your right hand hold pole with palm facing forward, make a ~90degree bend in an "L" shape with your arm so you forearm is parallel with the pole. Place your elbow infront of the pole and lean forward with your body as if you wanted to snap off your forearm (don't actually do it, but it illustrates what this is trying to stretch).
 * Feel free to fidget and alter to angle of the stretch as it stretches different parts of the rotator cuff. Changing the angle of your elbow from 45degrees to 135 degrees also helps stretch all parts of the rotator cuff.
 * (This seems to be by far the most common baffling source of muscle tension in sports and in the workplace. The reverse rotator cuff doesn't seem to be as common or major a problem.)
 * 1) ; Stretch rear rotator cuff
 * Totally different stretch from the forward rotator cuff. Find a long flat table, place your right arm on top of the table in an "L" shape with your palm facing down and pointing to the left. Hold the edge of table with your right hand and, keeping your elbow on top of the table (and near the edge of the table), lean down and try to take your shoulders and hand as far down to the ground as you can.
 * 1) "Pure pressure" and Stroking''' - After stretching, either of these techniques can help fully relax the muscle (stretching usually helps the most). Applying these techniques, massage the rotator cuff using the 2 mentioned techniques (ideally with the subject lying face down). You may find the rotator cuff hard to locate due to a layer of stiff larger muscles on top. Go to step 1 to relax related muscles.

Common Problems
You may find the rotator cuff is hard to locate due to the following:

Small size of muscle - In non-muscular/athletic people, the rotator cuff muscle often feels more like a thin stiff tendon only a couple mm wide. It will be impossible to locate if the latizoids or deltoids are too stiff.

Lack of response to massage techniques - This muscle is very deep and will not respond to tapping techniques well. You must relax surrounding muscles first. As surrounding muscles may be difficult to fully relax in one massage session, always remember to stretch the subject as stretching will always provide a large amount of relief for this condition.

Prevention
Regular person: This stiffness is usually caused by bad posture in the workplace.

Athlete person: The rotator cuff is extremely useful in many contact sports (eg. basketball, rugby, wrestling, judo...) and sports involving the upper body. My personal experience has shown, even with years of extensive training, the rotator cuff is almost never strong enough and can always stand from having extra strength. Please refer to a sports trainer for specific exercises on improving the strength of this muscle.

My experience has also shown athletes not involved in contact sports will not readily understand how to provide training of the intensity necessary to create a really strong rotator cuff. Basic rotator cuff training should suffice for several months but finding a athelete with contact sport experience will help to create a rotator cuff strong enough to minimize future injury.

Areas for expansion

 * Neck stiffness - left/right tilt
 * Neck stiffness - rotating
 * Neck stiffness - forward/back tilt
 * Shoulder blade (trapezoids) stiffness - pain shugging shoulders rearward
 * Shoulder blade stiffness - pain shugging shoulders forward
 * Knee pain - below kneecap - usually stiff tibia
 * Knee pain - above kneecap - usually stiff quads
 * Lower back pain - spinal massage
 * Outer shoulder stiffness (deltoids)
 * Chest stiffness (pectorals)
 * Foot pain
 * Forearm pain
 * Hand massage

title: Massage - Synergistic Techniques


text:
Talk:Massage - Synergistic Techniques

Is this formatted properly? I don't think I have enough content to split up the book into pages like in the "naming policy - chess" article yet...

I'll keep working on it when I get the time and lookup all the terminology I can't remember off the top of my head.

How can I change the title of the page from "Massage - Synergistic Techniques" to "Massage - A Synergistic Approach"? I think this better reflects that fact that I want this to focus on techniques that work well together. --Rawb 09:01, 17 July 2006 (UTC)

title: Talk:Massage - Synergistic Techniques


text:
User:Little jonny jewel

'ello 'ello, I love this thing we call knowledge

title: User:Little jonny jewel


text:
Special Relativity/Introduction

Introduction
The Special Theory of Relativity is a theory of classical physics that was developed at the end of the nineteenth century and the beginning of the twentieth century. It changed our understanding of older physical theories such as Newtonian Physics and led to early Quantum Theory and later the Theory of General Relativity. Special Relativity is one of the foundation blocks of physics.

This book will introduce the reader to, perhaps, the most profound discovery of the twentieth century and the modern world: the universe has at least four dimensions.

Historical Development
Special Relativity is not a theory about light, it is a theory about space and time, but it was the strange behaviour of light that first alerted scientists to the possibility that the universe had an unexpected geometry. The short history of Special Relativity given here will start with light but will end with the discovery that the behaviour of light is related to the geometry of the universe.

In the nineteenth century it was widely accepted that light travelled as waves in a substance called the “aether”. Light was thought to travel in this aether in a similar way to how other types of waves travel in material substances; for example, sound waves travel in air (and other substances). Light would travel to our eyes as waves through the aether like sound travels to our ears as waves in the air.

The nature of the aether was unknown but a possible link between the aether and electrical and magnetic fields became apparent during the first half of the nineteenth century. Faraday demonstrated that the polarisation of light was affected by magnetic fields and Weber showed that electrical effects could be transmitted across non-conducting materials so there was a strong suggestion that light might be some sort of electromagnetic effect.

In 1865 the Scottish physicist James Clerk Maxwell drew together the various experiments on electricity and magnetism into an electromagnetic theory of light based on the idea of an aether. He identified that electricity and magnetism, previously considered to be separate forces, were two aspects of one force. He was able to calculate the speed at which electromagnetic waves moved as a simple ratio between the strength of the electric field and the strength of the magnetic field. From this, one of his key observations was that electromagnetic effects seemed to propagate at nearly light speed. He wrote of the velocity of electrical interactions that:

“This velocity is so nearly that of light that it seems we have strong reason to conclude that light itself (including radiant heat and other radiations, if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.”

Maxwell's theory explained radio, heat radiation, light and many other phenomena as electromagnetic waves travelling in an aether. The velocity of these waves depended upon the properties of the aether itself. Someone who was stationary within the aether would measure the speed of light to be constant as a result of the constant properties of the aether. A light ray going from one stationary observer to another in the aether would take the same amount of time to make the journey no matter which stationary observer measured it. However, although stationary observers would all observe the same velocity for light, moving observers would measure the velocity of light as the sum of their velocity relative to the aether and the velocity of light in the aether.

If space were indeed full of an aether then the motion of objects through this aether should be detectable by measuring the velocity of light rays. In practice it is difficult to measure the velocity of light with sufficient precision. Maxwell suggested that an instrument called an "interferometer" would provide the required accuracy. He proposed that if an interferometer were moved through the aether the addition of the velocity of the equipment to the velocity of the light in the aether would cause a distinctive interference pattern. Maxwell's idea was submitted as a letter to Nature in 1879 (posthumously).

Albert Michelson read Maxwell's paper and in 1887 Michelson and Morley performed an 'interferometer' experiment to test whether the observed velocity of light is indeed the sum of the speed of light in the aether and the velocity of the observer. Michelson and Morley discovered that the measured velocity of light did not change with the velocity of the observer. To everyone's surprise the experiment showed that the speed of light was independent of the speed of the destination or source of the light in the proposed aether. How might this "null result" of the interferometer experiment be explained? How could the speed of light in a vacuum be constant for all observers no matter how they are moving themselves? It was possible that Maxwell's theory was correct but the theory about the way that velocities add together (known as Galilean Relativity) was wrong. Alternatively it was possible that Maxwell's theory was wrong and Galilean Relativity was correct. However, the most popular interpretation at the time was that both Maxwell and Galileo were correct and something was happening to the measuring equipment. Perhaps the instrument was being squeezed in some way by the aether or some other material effect was occurring.

Various physicists attempted to explain the Michelson and Morley experiment. George Fitzgerald (1889) and Hendrik Lorentz (1895) suggested that objects tend to contract along the direction of motion relative to the aether and Joseph Larmor (1897) and Hendrik Lorentz (1899) proposed that moving objects are contracted and that moving clocks run slow as a result of motion in the aether. Fitzgerald, Larmor and Lorentz's contributions to the analysis of light propagation are of huge importance because they produced the "Lorentz Transformation Equations". The Lorentz Transformation Equations were developed to describe how physical effects would need to change the length of the interferometer arms and the rate of clocks to account for the lack of change in interference fringes in the interferometer experiment. It took the rebellious streak in Einstein to realise that the equations could also be applied to changes in space and time itself.

By the late nineteenth century it was becoming clear that aether theories of light propagation were problematic. Any aether would have properties such as being massless, incompressible, entirely transparent, continuous, devoid of viscosity and nearly infinitely rigid. In 1905 Albert Einstein realised that Maxwell's equations did not require an aether. On the basis of Maxwell's equations he showed that the Lorentz Transformation was sufficient to explain that length contraction occurs and clocks appear to go slow provided that the old Galilean concept of how velocities add together was abandoned. Einstein's remarkable achievement was to be the first physicist to propose that Galilean relativity might only be an approximation to reality. He came to this conclusion by being guided by the Lorentz Transformation Equations themselves and noticing that these equations only contain relationships between space and time without any references to the properties of an aether.

In 1905 Einstein was on the edge of the idea that made relativity special. It remained for the mathematician Hermann Minkowski to provide the full explanation of why an aether was entirely superfluous. He announced the modern form of Special Relativity theory in an address delivered at the 80th Assembly of German Natural Scientists and Physicians on September 21, 1908. The consequences of the new theory were radical, as Minkowski put it:

"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

What Minkowski had spotted was that Einstein's theory was actually related to the theories in differential geometry that had been developed by mathematicians during the nineteenth century. Initially Minkowski's discovery was unpopular with many physicists including Poincaré, Lorentz and even Einstein. Physicists had become used to a thoroughly materialist approach to nature in which lumps of matter were thought to bounce off each other and the only events of any importance were those occurring at some universal, instantaneous, present moment. The possibility that the geometry of the world might include time as well as space was an alien idea. The possibility that phenomena such as length contraction could be due to the physical effects of spacetime geometry rather than the increase or decrease of forces between objects was as unexpected for physicists in 1908 as it is for the modern high school student. Einstein rapidly assimilated these new ideas and went on to develop General Relativity as a theory based on differential geometry but many of the earlier generation of physicists were unable to accept the new way of looking at the world.

The adoption of differential geometry as one of the foundations of relativity theory has been traced by Walter (1999). Walter's study shows that by the 1920's modern differential geometry had become the principal theoretical approach to relativity, replacing Einstein's original electrodynamic approach.

It has become popular to credit Henri Poincaré with the discovery of the theory of Special Relativity, but Poincaré got many of the right answers for some of the wrong reasons. He even came up with a version of $$E=mc^2$$. In 1904 Poincaré had gone as far as to enunciate the "principle of relativity" in which "The laws of physical phenomena must be the same, whether for a fixed observer, as also for one dragged in a motion of uniform translation, so that we do not and cannot have any means to discern whether or not we are dragged in a such motion." Furthermore, in 1905 Poincaré coined the term "Lorentz Transformation" for the equation that explained the null result of the Michelson Morley experiment. Although Poincaré derived equations to explain the null result of the Michelson Morley experiment, his assumptions were still based upon an aether. It remained for Einstein to show that an aether was unnecessary.

It is also popular to claim that Special Relativity and aether theories such as those due to Poincaré and Lorentz are equivalent and only separated by Occam's Razor. This is not strictly true. Occam's Razor is used to separate a complex theory from a simple theory, the two theories being different. In the case of Poincare's and Lorentz's aether theories both contain the Lorentz Transformation which is already sufficient to explain the Michelson and Morley Experiment, length contraction, time dilation etc. without an aether. The aether theorists simply failed to notice that this is a possibility because they rejected spacetime as a concept for reasons of philosophy or prejudice. In Poincaré's case he rejected spacetime because of philosophical objections to the idea of spatial or temporal extension (see note 1).

It is curious that Einstein actually returned to thinking based on an aether for philosophical reasons similar to those that haunted Poincaré (See Granek 2001). The geometrical form of Special Relativity as formalised by Minkowski does not forbid action at a distance and this was considered to be dubious philosophically. This led Einstein, in 1920, to reintroduce some of Poincaré's ideas into the theory of General Relativity. Whether an aether of the type proposed by Einstein is truly required for physical theory is still an active question in physics. However, such an aether leaves the spacetime of Special Relativity almost intact and is a complex merger of the material and geometrical that would be unrecognised by 19th century theorists.

Intended Audience
This book presents special relativity (SR) from first principles and logically arrives at the conclusions. There will be simple diagrams and some thought experiments. Although the final form of the theory came to use Minkowski spaces and metric tensors, it is possible to discuss SR using nothing more than high school algebra. That is the method used here in the first half of the book. That being said, the subject is open to a wide range of readers. All that is really required is a genuine interest.

For a more mathematically sophisticated treatment of the subject, please refer to the Advanced Text in Wikibooks.

The book is designed to confront the way students fail to understand the relativity of simultaneity. This problem is well documented and described in depth in: Student understanding of time in special relativity: simultaneity and reference frames by Scherr et al.

What's so special?
The special theory was suggested in 1905 in Einstein's article "On the Electrodynamics of Moving Bodies", and is so called because it applies in the absence of non-uniform gravitational fields.

In search of a more complete theory, Einstein developed the general theory of relativity published in 1915. General relativity (GR), a more mathematically demanding subject, describes physics in the presence of gravitational fields.

The conceptual difference between the two is the model of spacetime used. Special relativity makes use of a Euclidean-like (flat) spacetime. GR lives in a spacetime that is generally not flat but curved, and it is this curvature which represents gravity. The domain of applicability for SR is not so limited, however. Spacetime can often be approximated as flat, and there are techniques to deal with accelerating special relativistic objects.

Common Pitfalls in Relativity
Here is a collection of common misunderstandings and misconceptions about SR. If you are unfamiliar with SR then you can safely skip this section and come back to it later. If you are an instructor, perhaps this can help you divert some problems before they start by bringing up these points during your presentation when appropriate.

Beginners often believe that special relativity is only about objects that are moving at high velocities. Strictly speaking, this is a mistake. Special relativity applies at all velocities but at low velocity the predictions of special relativity are almost identical to those of the Newtonian empirical formulae. As an object increases its velocity the predictions of relativity gradually diverge from Newtonian Mechanics.

There is sometimes a problem differentiating between the two different concepts "relativity of simultaneity" and "signal latency/delay." This book text differs from some other presentations because it deals with the geometry of spacetime directly and avoids the treatment of delays due to light propagation. This approach is taken because students would not be taught Euclid's geometry using continuous references to the equipment and methods used to measure lengths and angles. Continuous reference to the measurement process obscures the underlying geometrical theory whether the geometry is three dimensional or four dimensional.

If students do not grasp that, from the outset, modern Special Relativity proposes that the universe is four dimensional, then, like Poincaré, they will consider that the constancy of the speed of light is just an event awaiting a mechanical explanation and waste their time by pondering the sorts of mechanical or electrical effects that could adjust the velocity of light to be compatible with observation.

A Word about Wiki
This is a wikibook. That means it has great potential for improvement and enhancement. The improvement can be in the form of refined language, clear mathematics, simple diagrams, and better practice problems and answers. The enhancement can be in the form of artwork, historical context of SR, anything. Feel free to improve and enhance Special Relativity and other wikibooks as you see necessary.

title: Special Relativity/Introduction


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Hungarian/Exercises

LESSON 2 (Solutions) LESSON 3 (Solutions) LESSON 4 (Solutions)

title: Hungarian/Exercises


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User:EvanR~enwikibooks

you reached EvanR 's user page on wikibooks. For some odd reason wikibooks and wikipedia arent connected! so... check out my page on wikipedia

[EvanR on en.wikipedia.org]

Approaches to Special Relativity

title: User:EvanR~enwikibooks


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Cookbook:Falafel

Falafel is a fritter popular in Mediterranean cuisine. It is typically fried, but may also be baked. It always consists of one or more types of beans; usually fava beans or chickpeas. In some regional varieties parsley is added too.

All other ingredients may vary, depending on flavor preferences or preparation technique.

Ingredients

 * 2 cups (480 g) chickpeas, drained
 * 4 cloves of garlic, peeled and sliced
 * 4 tablespoon parsley, chopped
 * 1 onion, cut into chunks
 * 1 teaspoon (0.25 g) ground coriander
 * 1 teaspoon (0.75 g) ground cumin
 * 1 teaspoon (0.46 g) cayenne pepper
 * 1 teaspoon baking powder
 * 6 tablespoon flour
 * ½ cup vegetable oil (sunflower or corn oil) for frying

Procedure

 * 1) The night before, place the chickpeas in a bowl and cover with water (they will expand, so cover with at least a couple of inches). Drain, rinse, and dry before using. This will be about 4 cups' worth, all of which gets used.
 * 2) In a food processor or blender purée chickpeas. To the chickpeas add the other ingredients (except oil); process until well mixed.
 * 3) Shape into balls, using about 1 tablespoon for each.
 * 4) Heat oil in a medium skillet over medium-high heat. Add balls, and brown on all sides.
 * 5) Serve with pita.

title: Cookbook:Falafel


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Cookbook:Falafel Seitan Bratwurst

| Vegan cuisine

Falafel seitan bratwurst is a vegetarian bratwurst substitute that combines chickpea falafel and seitan (gluten).

Ingredients

 * Gluten flour
 * Spiced falafel mix (the chickpea variant, not the fava bean variant)
 * Water
 * Soy sauce
 * Vegetable oil
 * Chili powder (optional)

Procedure

 * 1) Mix gluten flour and falafel mix in a ratio of 30:70 by volume.
 * 2) Take 25% more water by volume than falafel-gluten mixture. Mix water with vegetable oil and soy sauce.
 * 3) Form a dough from the dry mixture and the liquid mixture.
 * 4) Roll the dough into sausages and fry them in an oiled non-stick pan.

Notes, tips, and variations

 * If the dough is too runny you can form round sausages by squeezing from both sides with two spatulas and frying the sausages on four sides. When you turn the sausages on the side you may also have to squeeze from above to give them an approximately round profile.

title: Cookbook:Falafel Seitan Bratwurst


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Talk:Blender 3D: Noob to Pro/Model a Low Poly Head

Confusing
The second part - the subdivision - is very confusing. It is not clear which edges are supposed to be selected and how often they should be subdivided. Even with the help of the good, numerous screenshots I couldn't reproduce the model. 205.201.100.227

Very bad page layout. small screen-shots, and staggered between left and right side of the screen. I can't even tell what I'm supposed to be doing. 68.82.83.197

Note from new user: This tutorial is definitely one of the more badly written ones, if only the writer had kept it simple. I reproduced the model, albeit in a rather exasperated state.

really confusing and doesn't really teach some thing new... I suggest to re-arrange it and put it in an earlier section of this unit...

I agree! I've done what I can for it. Pearts (talk) 16:26, 1 August 2010 (UTC)

When I subdivide and set smooth my nose is highlighted by a dark ring. Any reason why?Rupoe (talk) 16:46, 26 August 2010 (UTC)

Nevermind... I guess it's not a big deal. It's not there when I render.Rupoe (talk) 16:49, 26 August 2010 (UTC)

Poly Count
It is a good practice to keep the number of triangles to a minimum. I think this tutorial should be tweaked so that it doesn't teach people bad habits. ie. it use edge loops and no tris - the pictures show a lot of good effort though. Venar303

Steps to do this need to be added, who knows how? Pearts (talk) 16:25, 1 August 2010 (UTC)

You need to use proper topology, etc. blendercookie has a very good explanation of this: http://www.blendercookie.com/2010/01/28/topology-overview-the-head/ Untitled2 (discuss • contribs) 22:44, 29 August 2011 (UTC)

Images
It would be easier to look at and better to download for low-bandwidth and non-flatrate users, if the images showed only the necessary parts instead of the whole screen. You can render each viewport individually, and save the render as JPEG for example. DanMan

I've trimmed down all the images Pearts (talk) 16:24, 1 August 2010 (UTC)

Crowds
This might be great for rendering crowds at a stadium - a sea of faces. Perhaps wrap some sort of face image around each head.

If you have lots of heads (several thousand), and they were at a distance, I would even not bother sub-surfacing, to reduce render time.

Add a low-poly chest. And some legs on the front-row.

Necessary.
I just wonder, how this far in the tutorial, a simple project like this is included so late. To be honest so far in this tutorial and, I enjoy this tutorial more than any other blender tutorial out right now, I have learned all the basic functions and tools of blender to be able to easily create this face with my own knowledge. So what i am asking is this, Does anyone feel that this page should be placed a little earlier in the tutorial? Honestly in my opinion it was just a bump in my road to more advanced things,which I'm eager to learn now.Thebigba (talk) 05:30, 8 November 2010 (UTC)

I 100% agree with you. I also go through each tutorial with passion and after the tank tutorial this and the one before seem like a huge joke. Also the result from this tutorial is neither astonishing nor does it yield any new techniques.

Also I am quite confused why his surface at "Make the facial features" becomes subdivided, if he only subdivedes the edge (at least in v.2.67 only the edge becomes subdivided - as expected). Even though this seems to be the weakest tutorial so far, this whole book is still amazing.Adam.Elenktik (discuss • contribs) 19:16, 21 May 2013 (UTC)


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Talk:Special Relativity/Introduction

Light velocity in aether
It is stated that the observed velocity of light is the sum of the velocity of the light relative to the aether and the observer's velocity relative to the aether.

Would basic consideration of these two vectors not indicate that it is in fact the difference?

This is better. RobinH 10:47, 17 July 2006 (UTC)

Correct Isimangaliso Mohapi (discuss • contribs) 15:06, 25 June 2020 (UTC)

Better?
Hi RobinH, Perhaps next time you might like to discuss before you totally delete and rewrite an article next time. I had some understanding when I read this current (reverted) version, but I was totally lost by your rewrite about who proposed and theorised the concept of special relativity. Please discuss this, maybe I am wrong, please tell me so. I'm sure we are supposed to discuss major edits first, aren't we?--Read-write-services 22:32, 7 November 2006 (UTC)

 * Please accept my apologies. The problem with the existing introduction is that it does not embrace the scope of Special Relativity. Newtonian physics begins with a discussion of velocity and acceleration but no one would say that Newtonian physics is about relative velocities - it extends to force, energy and thermodynamics. Similarly, relativity begins by reconsidering the basis of Newtonian physics but it is actually a theory that entirely replaces everything that Newton and pre-relativistic physics had derived. The idea that modern physics is just a tweak of nineteenth century physics misses the entire point, modern physics is a total replacement for what went before.  Modern physics is deeply shocking, it proposes that the universe is a four dimensional manifold containing probability waves. Newtonian physics proposes that the world is a three dimensional manifold of no duration containing lumps.

 * Special relativity has a different ontology from Newtonian physics. Length contraction is due to the differing slices of the worldtube of an object that are observed by observers in relative motion, quantum physics originates in de Broglie waves which are due to the relativity of simultaneity, time dilation is due to the time gaps that result from four dimensional space-time etc. etc.. RobinH 12:54, 13 November 2006 (UTC)

 * This intro needs to say that SR is not just a tweak to the physics that went before. It also needs to point out that SR is not solely Einstein's work. Einstein got his nobel prize for photoelectric emission, people knew that Minkowski, Poincare, Lorentz etc. had all been involved in SR. 86.14.2.37 13:08, 13 November 2006 (UTC)

Hi, I didn't mean to upset you, I'm having problems just trying to understand it all. I thought I knew something about physics-until I came upon this. I consider myself reasonably educated. I edit, proofread and write articles for the general public, the photographic community and I have taught Medical Imaging, Radar, Communications electronics as well as aviation navigational aids. perhaps it may be presumtuous of me, however, if I cant understand it, (I think you lost me at four dimensional manifold containing probability waves.) then I think that a layperson reading it would be "blown-away". I certainly take your point about getting the facts right-for sure! Maybe we need to consider the potential audience that this is aimed at? anyway, thank you for your reply. Cheers!--Read-write-services 21:53, 13 November 2006 (UTC)

 * I agree that the introductory version of this text should be accessible to the layperson. However, it is incredibly important that we do not compromise the integrity of the theory.  Relativity is not just a clever way of calculating velocities.

 * The key difference between relativity and Newtonian physics is that Newtonian physics assumes that euclidean geometry, including the fifth postulate, describes the world. This difference is expressed in the laws of physics. The laws of physics are statements about the symmetries and corresponding invariants that exist in nature.

 * In a three dimensional euclidean space the conservation laws are conservation of momentum, conservation of angular momentum and conservation of energy. In space-time the conservation law that is of prime importance is conservation of four momentum which contains energy and momentum conservation in a single concept.

 * In a thoroughly modern treatment (not for laypersons) we would introduce noether invariants and the generalised concept of a "physical law" then point out that the observed world corresponds to a 4 dimensional manifold of signature (---+). This is an entirely different entity from a 3 dimensional manifold of signature (+++). For instance, the 4 D manifold contains the possibility of quantum physics (See: de Broglie, L. (1925) On the theory of quanta. A translation of : RECHERCHES SUR LA THEORIE DES QUANTA (Ann. de Phys., 10e s´erie, t. III (Janvier-F ´evrier 1925).by: A. F. Kracklauer. http://www.ensmp.fr/aflb/LDB-oeuvres/De_Broglie_Kracklauer.pdf ). It is also not generally appreciated that the schrodinger equation was almost immediately replaced by the Dirac Equation which is Lorentz invariant, introducing the relativistic 4 momentum and hence predicting quantum spin (See [Dirac equation http://en.wikipedia.org/wiki/Dirac_equation]).

 * It is a diabolical tragedy that popular science articles often describe QM and relativity as if they are alternative theories. QM is a prediction of SR and a logical consequence of the signature of spacetime.

 * Somehow we have got to transmit to the lay reader that relativity is a huge idea, it is not just a tweak to Newtonian physics. RobinH 17:41, 17 November 2006 (UTC)

 * I have tried to merge some of my approach with yours. RobinH 12:34, 19 November 2006 (UTC)

SP Relativity
speaking of einstein's development of SR; i believe his wife or wife to-be at that time (not sure) had a great deal to do with the r&d of this and many of the genius's credited publications and awards. Her contributions went without barely a peep towards her co-development of einsteins lifetime of great works. i do not often hear much of her regarding the contributions she made and am not entirely sure exactly how much she actually did, but i suspect given the time and the man, her assistance would barely rate a mention by the egomaniacs on top of the physics world during this rapid theoretical development period.

any comments

"same speed as the bus" is incorrect
From the introduction, "Bob, seeing the dog on the street move by, determines that the dog is moving at the same speed as the bus. Jim on the other hand, determines that the dog is not"

This seems inconsistent. Since Bob is ON the bus, Bob doesn't see that the bus is moving at all. But he does see the dog as moving. Thus, to say that Bob determines that the dog is moving at the "same speed as the bus" is incorrect. In a book like this, the distinction is important, obviously. Inconsistencies cause confusion.

Can't handle accelerating frames?
The first paragraph in the section "What's so Special?" is thoroughly incorrect: special relativity and newtonian mechanics can both handle accelerating frames of reference (and even rotating ones) perfectly fine; what makes special relativity "special" is that the principle of special relativity is restricted to inertial frames. --Marzojr (talk) 21:20, 5 May 2008 (UTC)

The text has now been amended RobinH (talk) 19:49, 16 November 2010 (UTC)

title: Talk:Special Relativity/Introduction


text:
Bodybuilding and Weight Training/Bench Presses

- The Bench press - Main Muscles Used:

Chest, Triceps

- Description: - The bench press is similar to a push up. however, the differences are that the performers body position is reversed, where the person is lying with their back on a bench like seat and pushing weights connected to two sides of a standard 45 pound metal bar instead of their body weight.

weights used are in common sizes of 45lbs, 35lbs, 25lbs, 10lbs, 5lbs, and 2.5 pound plates.

Starting position is the bar with, for example, two 45 pound weights on both sides (135 pounds) set on the starting point.

The lifter then positions themselves under the bar near the neck and positions their arms for the bench press. Arms held closer together puts more focus on the triceps, where as the opposite puts more focus on the chest. Find a comfortable grip (this varies by each lifter.)

The lifter then lifts the weights off the hooks holding the bar+weights, and positions it across their chest. This is followed by a lowering of the weights, then pushing it back up to the starting point. Some say one should lower the bar completely to your chest. However, this may cause injuries. Forcing the bar to your chest when it does not occur naturally can cause temporary to permanent injuries to your elbows. - Common Exercises: -

3 sets of 8 repetitions with stretching and a few minutes of rest in between each set. Weights lifted vary from person to person. however, it should be a weight that allows the last set to be completed, but at the same time, moderately intense/tiring.

Pyramid sets are also used as a high intensity workout. sets go from 8, 6, 4, 2, 4, 6, 8 with increasing weight, and then decreased as the sets increase again. example of weight increase Example: 8 x 95, 6 x 105, 4 x 125, 2 x 135, 4 x 125, 6x 105, 8x 95 - Safety: -

In general, it is always good to have a spotter when dealing with free weights, especially with a bench press. Failure to be able to lift the weight back onto the safety bar after dismounting it can result in severe injury and even death. A spotter is there to help you lift the weight back onto the safety hook in case you can't do it yourself.

Also, when bench pressing, avoid over-training to prevent injuries. be smart and lift an amount of weight where it can help you advance in your climb in weight lifting, but not injure you.

Make sure your equipment is in good condition and working order . "Hardware failure" with heavy weights involving you is not fun.

Good form prevents stupid injuries.

Most of all, Have fun!

- Variations of the Bench press -

- Incline Press - Decline Press

Each variation works a different area of the general chest area (the upper area and lower area) and are done on an inclining or declining bench.

--- Tips to Prevent Injury: ---

The bench press is notorious for causing injuries to the shoulder and sometimes the elbow joint. Great care should be taken when preparing to lift and during execution. Before you begin, ensure that you have the correct placement of hands on the bar. Next, activate shoulder stabilizing muscles and carefully lift the bar and move to the correct position over the chest (for basic bench press). Lower the bar slowly, without allowing it to free-fall, as too much speed can place excessive stress on joints. There is no real need to touch the bar to the chest at the bottom. This should be reserved for advanced lifters or football players who must perform this move for tests. In general, stop the bar a few inches off the chest, then push up. This will save your shoulder joint.


text:
Horticulture/Leaf Vegetables

Leaf vegetables and grown for their edible foliage. Most are picked young, so successive plantings are often used to extend the season.

Outline:

 * Growing methods
 * Growing in soil
 * Growing in flats
 * Harvesting
 * One crop harvesting
 * Cut-and-come-again harvesting
 * Cooling and storage

Leaf vegetable plants:

 * ../Lettuce/
 * ../Pac Choi/
 * ../Mustard Greens/

/Finder/Types/Fruits and Vegetables

title: Horticulture/Leaf Vegetables


text:
User talk:EvanR~enwikibooks

Hi. Your revised intro to Special Relativity was good. It is useful to have someone who can write english contributing (I tend to write in notes - I've read too many maths and physics books). RobinH 14:01, 17 July 2006 (UTC)

 * yeah we gotta make this book awesome, ie featured book. EvanR 20:37, 17 July 2006 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called EvanR. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name EvanR~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

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Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 04:57, 19 April 2015 (UTC)

title: User talk:EvanR~enwikibooks


text:
User talk:LOST fan

Jokebook
I'm sorry to say that the jokebook here on wikibooks has long since been deleted, as it was found to be in violation of wikibooks policy. Some of the material still exists in the deletion logs, but it would be far too big a pain to undelete the necessary pages, retrieve any information (if it still exists, and is even worth while), and then redelete the pages again. Sorry I can't be more help on this. Good luck with your other project. --Whiteknight (talk) (projects) 14:48, 17 July 2006 (UTC)
 * Okay. I'll look for other stuff. LOST fan 14:49, 17 July 2006 (UTC)

title: User talk:LOST fan


text:
User:The-thing~enwikibooks

Wassup?!?!

title: User:The-thing~enwikibooks


text:
User talk:The-thing~enwikibooks

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called The-thing. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name The-thing~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:46, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 05:35, 19 April 2015 (UTC)

title: User talk:The-thing~enwikibooks


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Analog and Digital Conversion/Sampling Theorem

Sampling Theorem
The Sampling Theorem is a mathematical construct that allows us to take a single sample of an input waveform at a time T. This is not the same as the star transform.

Definition
The sampling theorem states that if we convolute the input function with an impulse, centered at the sampling time T, the output will be the value of the input function at time T. More specifically, our output will be the sample at time T. Here is the definition:

 * $$\bar f(T) = f(t) * \delta(t - T)$$

Where $$ \bar f(T)$$ is the sampled value at time T.


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Analog and Digital Conversion/ADC Converters

ADC Converter Steps
An Analog to Digital Converter (ADC) takes an analog input signal and converts the input, through a mathematical function, into a digital output signal. While there are many ways of implementing an ADC, there are three conceptual steps that occur:

 * 1) The anti-aliasing filter blocks unwanted frequencies.
 * 2) The signal is sampled.
 * 3) The sampled signal is quantized.
 * 4) The quantized signal is digitally coded.

Flash Converters ADC
Flash ADC's are the fastest ADC as it takes only on clock pulse to produce digital output. It is a direct compare type architecture. It requires 2^n-1 comparator, 2^n Resistor and a thermometer to binary encoder.

Successive Approximation ADC
Successive approximation is one of the most widely used popular method due its efficiency The block diagram of SAR ADC s as shown: Working: Initially, let us set the MSB bit of SAR register i.e. d1 = 1. It is applied to 4-bit D to A converter i.e. as 1000. The D/A converter will generate its analog value and send to control logic The output of control logic is VR. Now at the comparator, there are two inputs VR (Reference voltage) and VA (Input analog). If VR > VA  output of comparator is low and bit d1 is reset correspondingly. If VR <VA  output of comparator is high and bit d1 remains high. The same procedure is repeated for all bits i e for d2, d3,…. dn, and output may be taken in serial or parallel manner. Advantage: The conversion time is fixed as it does not depend upon amplitude of analog input.

Sigma-Delta ADC
In systems that requires at least 16 bits of resolution, such as audio electronics, by far the most common type of ADC converter is the sigma-delta converter.

The sigma-delta converters, unlike other kinds of ADCs, never sample the signal at any particular point in time. Instead, the output digital number is a weighted average of the signal over the entire sampling time.


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Analog and Digital Conversion/DAC Converters

DAC Converter Steps
An Digital to Analog converter (DAC) takes a digital signal and converts it, through a mathematical function, into an analog signal. Again, the DAC may be implemented in a number of ways, but conceptually it contains two steps:

 * 1) Convert each time step of the digital signal into an "impulse" with the appropriate energy. In a real system, this could be accomplished by creating short pulses that have the same voltage, but whose total power is modified by changing the pulse length.  This pulse train produces a signal whose frequency response is periodic (and theoretically extends to infinity).
 * 2) Apply a low-pass filter or an appropriate holding circuit to the time sequence of impulses. This removes all of the high-frequency periodicities, leaving only the reconstructed signal.


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Set Theory/Ordinals

Standard representation of ordinal numbers
The definition of ordinal numbers offers little insight into their nature. In situations like this pure mathematicians create representations of the objects they wish to study. Such representations are built from familiar mathematical constructions and are equivalent to the obstruce objects. By manipulating the familiar objects in the representation, the pure mathematician may thus investigate the structure of the mysterious abstract entities.

The most common representation of the ordinal numbers, due to Von Neumann, is as follows. The ordinal 0 is defined to be the empty set $$\emptyset$$; the ordinal 1 is defined to be the set $$\{0\}$$, which is of course equal to $$\{\emptyset\}$$. Similarly, the ordinal 2 is the set $$\{0, 1\}$$; the ordinal 3 is the set $$\{0, 1, 2\}$$; the ordinal 4 is the set $$\{0, 1, 2, 3\}$$. Any finite ordinal $$n$$ is defined to be the set $$\{0, 1, 2, \ldots, n-1\}$$ (or, in a rigorous notation, the successor of an ordinal α is defined as the set $$s(\alpha) = \alpha \cup \{ \alpha \}\,$$).

In fact this definition extends naturally to transfinite ordinals. The ordinal $$\omega$$ is the set consisting of every finite ordinal $$\{0, 1, 2, 3, \ldots\}$$. Again, $$\omega + 1$$ is the set $$\{0, 1, 2, 3, \ldots, \omega\}$$; $$\omega + 2$$ is the set $$\{0, 1, 2, 3, \ldots, \omega, \omega + 1\}$$; and so on.

The ordinal $$\omega + \omega$$ (or $$\omega * 2$$) is the set consisting of all finite ordinals and ordinals of the form $$\omega + n$$, where $$n$$ is a finite ordinal. Thus $$\omega + \omega = \{0, 1, 2, \ldots; \omega, \omega + 1, \omega + 2, \ldots\}$$.

The ordinal $$\omega_1$$ is the first uncountable ordinal, and is the set of all countable ordinals.

Ordinal arithmetic
Teoria dos conjuntos/Números ordinais

title: Set Theory/Ordinals