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User talk:Tinukaushik~enwikibooks

Welcome!
Welcome, Tinukaushik!

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.--ЗAНИA talk 19:19, 11 April 2008 (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 Tinukaushik. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Tinukaushik~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 23:47, 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) 05:36, 19 April 2015 (UTC)

title: User talk:Tinukaushik~enwikibooks


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User talk:Aakanksha.singhal

Welcome!
Welcome, Aakanksha.singhal!

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.--ЗAНИA talk 19:20, 11 April 2008 (UTC)

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

title: User talk:Aakanksha.singhal


text:
SAT Study Guide/Part 4 - The Writing Section/Improving Sentences

 Improving Sentences  Just like Identifying Sentence Errors questions, the large majority of Improving Sentences questions related to a surprisingly limited number or grammatical issues. The main difference between the two question types is that while the errors in Identifying Sentence Errors questions consist of single words or short phrases, the errors in Improving Sentences questions generally involve the structure of the whole sentence. So improving Sentences questions can be little trickier and more involved. WHAT ARE IMPROVING SENTENCES QUESTIONS? Remember our "spot-the-mistake" strategy? We've got a similar strategy for Improving Sentences. In addition to finding the mistake in each sentence, you have to pick the answer choice that best corrects it. Like we said, a little harder. Here is an example: The Emancipation Edict freed the Russian serfs in 1861; that being four years before the Thirteenth Amendment abolished slavery in the United States. (A) in 1861; that being four years (B) in 1861 and is four years (C) in 1861 and this amounts to four years (D) in 1861, being four years (E) in 1861, four years


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Cookbook talk:Egg

Merge
I think that any content on the page Cookbook:Separating_eggs that does not already exist in the section on separating eggs on the egg page should be moved to the egg page, and then the Cookbook:Separating_eggs page deleted. I don't think there will every be enough content on separating eggs for it to warrant its own page.

Thoughts? Wikidsoup (talk) 21:36, 11 April 2008 (UTC)

i agree as well.

I also agree with this --Draco Advigilat (talk) 04:22, 25 July 2008 (UTC)

No, the articles should not be merged. Eggs are an ingredient, separation is a technique. Mike Hayes (talk) 14:40, 23 August 2008 (UTC)

Raw egg danger
So, what is it? What is the point of having a warning everywhere if there's no explanation for it? 203.214.115.130 06:17, 25 April 2006 (UTC)

 * It's dumb. There doesn't need to be a warning on every page including eggs. Kellen T 00:21, 1 May 2006 (UTC)

 * There is an explanation if you care to read the article. Raw eggs may contain salmonella which is present in most chickens.  However, whereas most chickens are asymptomatic carriers of the organism, humans are not as a rule (a famous exception was "typhoid Annie").  It is a very pathogenic organism in humans, typhoid being one of the more pathogenic strains of this species of bacterium. Mike Hayes (talk) 14:40, 23 August 2008 (UTC)

Requested recipe
Would someone create a page for huevos rancheros? 164.58.89.3 19:39, 7 August 2006 (UTC)

This module, or a version of it in the page history, uses text from the Wikipedia article Egg.

Some text was also adapted from the Project Gutenberg version of The Whitehouse Cookbook, by Mrs. F.L. Gillette.

The section Egg products was adapted from the public domain U.S. Department of Agriculture article Focus on Egg Products

This module was the Featured Ingredient on the main Cookbook page from April 7 2004 - April 18, 2004 Gentgeen

???
Are amphibian, monotreme, or invertebrate eggs ever eaten? --Whoop whoop pull up Bitching Betty 21:33, 17 January 2011 (UTC)

title: Cookbook talk:Egg


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User talk:129.21.42.51

--Jomegat (talk) 21:46, 11 April 2008 (UTC)

title: User talk:129.21.42.51


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User talk:68.45.92.51

Join Us!
-- – Mike.lifeguard  | talk 22:03, 11 April 2008 (UTC)

title: User talk:68.45.92.51


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RC Airplane/Basic controls

Ailerons control the Roll axis, keeping the wings level, or rolling one wing up, and one down. Ailerons are usually on the back side of the main wing, closer to the tip.

Elevators move the nose up or down, changing the Pitch axis, also known as Attitude. Elevators are usually part of the tail.

Rudder moves the nose right and left, changing the Yaw axis. The rudder is usually part of the tail.

Flaps change the shape of the wing to add lift and reduce speed. Flaps are often used during landing, and sometimes during take-off. Flaps are usually on the back edge of the main wing, closer to the center of the plane.

Spoilers reduce lift by spoiling the shape of the wing, like air brakes. Spoilers can be used while landing to give better traction for the tires on the runway. Spoilers can also be used during flight to descend quickly, if necessary. Spoilers are usually on the top of the wing.

title: RC Airplane/Basic controls


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User:RogerWickes

Hi! I'm a nice guy, really. I love to write, and express ideas, and find creative solutions to problems.

title: User:RogerWickes


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Real Analysis/Dedekind's construction

Historically Dedekind gave the first construction of the real numbers. Like Cantor's construction, Dedekind's method constructs the real numbers from the set rational numbers. Dedekind's construction gives a more geometric picture of the real numbers.

The idea of the construction is that every real number $$r$$ should cut the number line into two subsets, the numbers less than $$r$$ and the numbers greater than or equal to $$r$$. These two sets are different for every real number, and given these sets we could determine $$r$$. In fact, because each real number divides the number line, it also divides the rational numbers into two. And if we knew these two sets of rational numbers, then we could also determine $$r$$. Hence when looking for a way to define the real numbers, we could define them as the collection of ways to partition the rational numbers into two sets. Today when discussing Dedekind cuts one usually only keeps track of one of these two sets. Our definition of a cut could informally be thought of as the numbers less than $$r$$.

Definition
We say that $$\alpha\subset\Q$$ is a cut if and only if the following statements are true:
 * 1) The set $$\alpha$$ is nonempty. That is, there exists some a rational number $$p$$ such that $$p\in\alpha$$.
 * 2) If $$p\in\alpha$$ and $$q$$ is a rational number so that $$q<p$$, then $$q\in\alpha$$ . So for any element of the cut, every rational number smaller than it is also an element of the cut.
 * 3) There exists a rational number $$a$$ such that $$p<a$$ for all $$p\in\alpha$$ . So there's a rational number that's larger than any element in a particular cut.
 * 4) For every $$p\in\alpha$$, there exists an $$r\in\alpha$$ such that $$p<r$$ . So every element of the cut has yet another element larger than it. If that seems impossible, imagine a series of rational numbers that approximate the real number we're trying to define. For example, the cut $$x^2<2$$ contains a series of countably infinitely many rational numbers like 1, 1.2, 1.4, 1.41, 1.414, 1.4142, etc. each of which is larger than every previous one and a bit closer to $$\sqrt2$$.

The Construction
We now outline how to make the set of Dedekind cuts form an ordered field that satisfies the least upper bound axiom.

Order
First we discuss how to compare the relative size of two cuts.

Definition Given two cuts $$\alpha,\beta$$, we say $$\alpha=\beta$$ if they are the same subset of the rational numbers. That is if $$\alpha\sube\beta$$ and $$\beta\sube\alpha$$.

Exercise Show equality defined as in the previous definition is an equivalence relation.

Definition Given two cuts $$\alpha,\beta$$, we say $$\alpha<\beta$$ if $$\alpha\subset\beta$$.

Exercise Show the above order satisfies the order axioms and that $$\R$$ is totally ordered.

We now also assume the usual definitions to $$\ge,<,>$$, we leave this to the reader to supply the correct definitions.

Exercise Given two cuts $$\alpha,\beta$$, show that if $$\alpha < \beta$$ then there is a rational number $$p\in\beta$$ so that $$p\notin\alpha$$.

Exercise (Trichotomy) Show that with this order, every cut satisfies exactly one of the following three inequalities: $$\alpha<0,\alpha=0,\alpha>0$$.

Field Axioms
Let $$\alpha,\beta$$ be two cuts

Define addition as $$\alpha+\beta=\{p+q:p\in\alpha,q\in\beta\}$$.

Define negation as $$-\alpha=\{r\in\Q:r<-p\text{ for some }p\not\in\alpha\}$$

Let 0 be the cut defined by $$\mathbf0=\{r:r<0\}$$.

 * Exercise For two cuts α, β verify that α+β, −α, and 0 are cuts. Furthermore show that the set of cuts forms an abelian group with this definition of addition.  What is the identity element?

Defining multiplication requires a bit more care. We define
 * $$\alpha\cdot\beta=\begin{cases}\{p\cdot q\mid p\in \alpha, p>0, q\in\beta\}&\text{if }\alpha\geq\mathbf{0},\beta\geq\mathbf{0};\\

\{p\cdot q\mid p\notin \alpha, q\in\beta, q>0\}&\text{if }\alpha\geq\mathbf{0}, \beta<\mathbf{0};\\ \{p\cdot q\mid p\in \alpha, q\notin\beta, q>0\}&\text{if }\alpha<\mathbf{0}, \beta\geq\mathbf{0};\\ \{r\mid r\mathbf{0};\\

\{r\mid r<\frac{1}{p}, p\not\in \alpha,\text{ and }p<0\}&\text{if }\alpha<\mathbf{0}. \end{cases}$$

Exercise For two cuts α, β verify that α·β, α-1 (assuming α ≠ 0), and 1 are cuts. Show that with these definitions the set of Dedekind cuts forms an ordered field.

Least Upper Bound Property
We will now show the set of Dedekind cuts satisfies the least upper bound axiom.

Let A be a non-empty collection of cuts, and suppose that there is a cut β such that α<β for all α. In other words, A is a non-empty collection of cuts that has an upper bound given by β.

Now we define a subset of the rational numbers, δ, by taking the unions of all of the subsets given by the cuts in A. That is $$\delta=\displaystyle\bigcup_{\alpha\in A}\alpha$$

Now we wish to show that δ is a cut. Suppose p∈δ, if follows that p∈α0 for some α0 in A. Then if q<p then q∈α0 and hence q∈δ. This shows the first property. Also, there is some r∈α0 with r>p, but then r∈δ. Hence the third property holds. To see that the second property holds, notice that δ⊆β because each α⊆β. By the cut properties for β there is a rational number b so that b>p for all p∈β. Therefore, b>p for all p∈δ. Hence δ is a cut.

Now, by definition of the order, δ is an upper bound of A. If η any cut with η<δ, then there is a rational number p∈δ and p∉η. But p∈α0 for some α0 in A. Hence η is not greater then α0, and therefore η is not an upperbound for A. Therefore δ is the least upper bound. This shows that the set of Dedekind cuts satisfies the least upper bound axiom.

Reference

 * W. Rudin, Principles of Mathematical Analysis, McGraw-Hill International

title: Real Analysis/Dedekind's construction


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Real Analysis/Symbols

We begin with listing various sets of numbers that are important in mathematical analysis.

title: Real Analysis/Symbols


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A-level Physics (Advancing Physics)/Electron Behaviour as a Quantum Phenomenon

So far, we have considered how quantum physics applies to photons, the quanta of light. In reality, every other particle is also a quantum, but you only need to know about photons and electrons.

The image on the right shows what happens when you fire electrons through a pair of slits: it arrives in lumps, but you get fringes due to superposition as well. The electrons are behaving as both waves and particles. Actually, they are behaving as quanta. The equations describing quantum behaviour in electrons are similar to those describing it in photons.

Frequency and Kinetic Energy
We know that, for photons:

$$f = \frac{E}{h}$$

In suggesting that electrons are quanta, we assume that they must have a frequency at which the phasors representing them rotate. We also know that h is a constant; it does not change. So, when adapting the above equation to apply to electrons, all we need to adapt is E. In electrons, this energy is their kinetic energy. If the electron has some form of potential energy, this must first be subtracted from the kinetic energy, as this portion of the energy does not affect frequency. So:

$$f = \frac{E_{kinetic} - E_{potential}}{h}$$

De Broglie Wavelength
If electrons exhibit some wavelike properties, they must also have a 'wavelength', known as the de Broglie wavelength, after its discoverer. This is necessary in order to work out a probability distribution for the position of an electron, as this is the distance the electron travels for each phasor arrow rotation. The de Broglie wavelength λ is given by the equation:

$$\lambda = \frac{h}{p} = \frac{h}{mv}$$,

where h = Planck's constant, p = momentum, m = mass of electron = 9.1 x 10−31kg and v = velocity of electron.

Potential Difference and Kinetic Energy
Potential difference is what causes electrons to move. You already know how power is related to charge, voltage and time:

$$P = \frac{QV}{t}$$

Since power is the rate at which work is done:

$$W = QV\,$$

We know that the charge on an electron equals -1.6 x 10−19, and that work done is energy, so:

$$E_{kinetic} = 1.6 \times 10^{-19} \times V$$

Energy, in the SI system of units, is measured in Joules, but, sometimes it is measured in electronvolts, or eV. 1 eV is the kinetic energy of 1 electron accelerated by 1V of potential difference. So, 1 eV = 1.6 x 10−19 J.

Questions
1. An electron moves at 30,000 ms−1. What is its de Broglie wavelength?

2. What is its frequency?

3. What is its kinetic energy, in eV?

4. Given that it is travelling out of an electron gun, what was the potential difference between the anode and the cathode?

5. An electron is accelerated by a potential difference of 150V. What is its frequency?

/Worked Solutions/


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A-level Mathematics/MEI/C2

Core 2 Mathematics (MEI)
This section covers the entire MEI C2 Syllabus (Assessed by OCR). Within each section is the conventional formulae and methods as well as examples that are often featured in Exam Papers (As of 2004.) Knowledge of MEI C1 is assumed.

Contents

 * Sequences and Series
 * Differentiation
 * Integration
 * Trigonometry
 * Logarithms and Exponentials
 * Radians

title: A-level Mathematics/MEI/C2


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A-level Physics (Advancing Physics)/Electron Behaviour as a Quantum Phenomenon/Momentum

All moving objects have momentum. Momentum is the product of the velocity of an object and its mass, so:

$$p = mv\,$$

Momentum is usually denoted p, and is measured in kgms−1. Momentum is used when considering collisions of bodies as momentum, in a closed system, is constant. Force is the rate of change of momentum, so:

$$F = \frac{\Delta p}{t}$$


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A-level Physics (Advancing Physics)/Electron Behaviour as a Quantum Phenomenon/Worked Solutions

'''1. An electron moves at 30,000 ms−1. What is its de Broglie wavelength?'''

$$\lambda = \frac{h}{mv} = \frac{6.626 \times 10^{-34}}{9.1 \times 10^{-31} \times 30{,}000} = 2.43 \times 10^{-8}\,\mathrm{m}$$

2. What is its frequency?

$$f = \frac{E_{\mathrm{kinetic}}}{h} = \frac{\tfrac{1}{2} \, 9.1 \times 10^{-31} \times 30{,}000^2}{6.626 \times 10^{-34}} = 6.18 \times 10^{11}\,\mathrm{Hz}$$

3. What is its kinetic energy, in eV?

From the top half of the fraction in the previous question:

$$E_{\mathrm{kinetic}} = \tfrac{1}{2} \, 9.1 \times 10^{-31} \times 30{,}000^2 = 4.10 \times 10^{-22}\,\mathrm{J} = \frac{4.10 \times 10^{-22}}{1.6 \times 10^{-19}}\,\mathrm{eV} = 2.56\,\mathrm{meV}$$

4. Given that it is travelling out of an electron gun, what was the potential difference between the anode and the cathode?

2.56 mV – that's why we use eV!

5. An electron is accelerated by a potential difference of 150 V. What is its frequency?

$$E_{\mathrm{kinetic}} = 1.6 \times 10^{-19} \times 150 = 2.4 \times 10^{-17}\,\mathrm{J}$$

$$f = \frac{E_{\mathrm{kinetic}}}{h} = \frac{2.4 \times 10^{-17}}{6.626 \times 10^{-34}}\,\mathrm{Hz} = 3.62 \times 10^{16}\,\mathrm{Hz}$$


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Real Analysis/Exponential Function

Our aim in this chapter is to formally define the very interesting exponential and logarithmic functions for all real numbers. This will also highlight intriguing interpretations on how the field of mathematics operates, due to the odd nature of how and why mathematicians even defined exponentiation and logarithms to begin with. Simply put, these functions serve to make otherwise insurmountably complex hurdles easier. "What kind of hurdles?" one may ask. The ones where one might want to switch between addition and multiplication. Mathematically, they wanted to create some function ƒ and g such that

 * $$f(x + y) = f(x) \cdot f(y)$$ and $$g(x \cdot y) = g(x) + g(y)$$

[To summarize significant portions of this section, the function ƒ is exponentiation and the function g is logarithms]

As you can see, in some algebraic problems, such a function would be very desirable for some—and absolutely necessary for others. However, the guiding philosophy of mathematics dictates that there ought be a definable statement that composes these intriguing functions—it can't be arbitrary or else there may be some hidden contradiction somewhere! That drive is what this section will satiate.

Construction
We will begin constructing what the functions ƒ and g are through two streams. First, we will identify how we expect these functions to operate. Second, we build a definition of ƒ and g that happens to also match the behaviors of the function perfectly. How so? It's the definition, of course.

Behaviors
We will first identify something extremely important in terms of how day to day usage of the function ƒ will work. What if:

 * $$x = x \text{ and } y = -x \mid f(x + y) = f(x - x) = f(0)$$

Well, we can first observe that the function can be alternatively written as

 * $$f(x + y) = f(x) \cdot f(y) = f(x) \cdot f(-x) = f(0)$$

but still must equal to the earlier statement. Well, we can't make $f(0) = 0$ —the additive identity, since then that implies that x or negative x is 0, which makes the entire function ƒ worthless; it would necessarily mean that all inputs for ƒ equals 0. We could instead make $f(0) = 1$ —the multiplicative identity. This new definition avoids the pitfall of $f(0) = 0$ making the entire function worthless. However, it therefore necessitates that either $f(x)$ or $f(-x)$  be the multiplicative inverse. Since $f(x)$ for positive x is traditionally left alone (and also since—spoilers—it represents exponentiation of positive integers), we will assign that quality to $f(-x)$. Thus,

 * $$f(-x) = \frac{1}{f(x)}$$

If you noticed, we have inadvertently assumed that the variable x and y are at least integers by our addition of negation. Oops. Well, we can take it one step further by imagining them to be rational numbers. If we suppose that the function input can be rational too, we open up another kind of property to fulfill, namely

 * $$\underbrace{f\left(\frac{1}{q}\right) \times \cdots \times f\left(\frac{1}{q}\right)}_{q \text{ times}} = f\underbrace{\left(\frac{1}{q} + \cdots + \frac{1}{q}\right)}_{q \text{ times}} = f\left(\frac{q}{q}\right) = f(1)$$

and if we're multiplying p terms together instead of q terms,

 * $$\underbrace{f\left(\frac{1}{q}\right) \times \cdots \times f\left(\frac{1}{q}\right)}_{p \text{ times}} = f\underbrace{\left(\frac{1}{q} + \cdots + \frac{1}{q}\right)}_{p \text{ times}} = f\left(\frac{p}{q}\right)$$

which adds more to our plate. Luckily, this new definition does not break anything we have assumed before. Luckily.

Differentiation
We will first assume that ƒ is differentiable. When we do, we can write out the definition of ƒ being differentiable.

 * $$f' = \lim_{h \rightarrow 0}{\frac{f(x + h) - f(x)}{h}}$$

which, given the special relationship between addition and multiplication, can be applied here to give a special answer.

 * $$\begin{align}

f' &= \lim_{h \rightarrow 0}{\frac{f(x + h) - f(x)}{h}} \\ &= \lim_{h \rightarrow 0}{\frac{f(x)\cdot f(h) - f(x)}{h}} \\ &= \lim_{h \rightarrow 0}{\left(f(x) \cdot \frac{f(h) - 1}{h}\right)} \\ &= f(x) \cdot \lim_{h \rightarrow 0}\frac{f(h) - 1}{h} \end{align}$$

For now, let's do something irrational to our conceptions of mathematics. We suppose that the limit is equal to 1. If we do that—and hold on, we will show you how even this egregious disregard still keeps mathematics consistent (eventually), we have ourselves a new property for the function ƒ. The derivative of $$f'$$ is $$f$$. All in all, this exercise has led us to create a function immune to differentiation—so long as the addition of inputs for a function is equivalent to the multiplication of the function with inputs separated!

The Logarithm
We have made a lot of assumptions in that last sub-heading. A lot. We will, for all intents and purposes, spend the rest of the heading on justifying (i.e. ensuring a lack of contradiction to) these claims. This will also act the part of justifying those claims, but in lieu of defending the earlier claim of the function ƒ, it will bring to form a function wiith properties so intriguing and relationship to this exercise so destined, it is given a special name and notation in mathematics: the logarithm.

There's a big issue with the function ƒ, besides the claims we made earlier. Even with those claims we made, there's no obvious point to it. the function ƒ has yet to be defined—and thus suseptible to possible contradictions later on. Heck, with more assumptions laid on it it has even less likely of a chance to survive analysis. However, we still have a trick left to give the function form—the inverse function and its properties. If we forget about the function ƒ and focus on its inverse, we can do some cool things with it. Using the locally named Reciprocal Definition for inverse functions, we can give a definition for the derivative of the inverse function

 * $$\begin{align}

(f^{-1})'  &= \frac{1}{f' \circ f^{-1}} \\ &= \frac{1}{f \circ f^{-1}} \\ &= \frac{1}{x} \end{align}$$

That is one easy definition. With this, we will make another outrageous claim, albeit less intense. We can say that 1 over x is a primative for an integral. A special integral whose properties will give this inverse function some teeth. We will suppose (with a little more merit than the previous supposition) that

 * $$f^{-1} = \int_1^x{\frac{1}{t} \operatorname{d}\!t}$$

We're going to simply drop the function ƒ-1 notation now. This inverse function is the definition of the logarithm. This special logarithm, unlike the ones used in elementary mathematics, has no base. This is the mathematican's favorite version logarithm, and is either notated as simply "log" without any base, or "ln", which has a special significance (a defined base) that will be described later on. To summarize a key point for this section,

Note that in mathematics, you may see either $$\log$$ or $$\ln$$ (pronounced "lawn") used to refer to this special function. More often in fields where logarithms with bases (which will be covered in the next heading) are used, $$\ln$$ is preferred as it is clearly different from $$\log$$—which may appear as a mistake. In mathematics, logarithms with bases are often not used, so it would not be an issue in this discipline which one you use.

The Exponential
The exponential function is, unlike the logarithmic function, simply a single supposition. Essentially, the exponential function is the inverse of the logarithmic function. The purpose of this will be explored in the second heading of this page. All in all, the construction of the logarithmic and exponential functions is complete.

However, it should be emphasized that although the exponential function is similar in name to exponentiation as learned in elementary mathematics ($$10^x$$ as an example), they have a small, but significant difference that should not be easily glossed over in this heading.

Alternate Construction of Exponentiation
The usual approach of constructing exponentiation is by defining the logarithm as an integral and the exponent as its inverse (as done above). We will, in this appendix, follow the reverse approach by constructing—in the loose sense of the word—the exponential function that remains defined for the entire real numbers from that of rationals. (Unfortunately, it involves some tedious computations!)

We can now use what we know about continuity to construct rational powers of positive real numbers.

Continuity of x^n
We've already defined the integer powers as a series of multiplications, yet we haven't shown that it's continuous. Let's show they're continuous first.
 * $$f(x) = x^0 = 1$$ is continuous.

Given $$\epsilon>0$$, $$|f(x) - f(c)| = |1-1| = 0 < \epsilon$$. So, $$\forall \delta>0: |x-c|<\delta \implies |f(x)-f(c)| < \epsilon$$.

 * $$f(x) = x$$ is continuous.

Given $$\epsilon>0$$, let $$\delta = \epsilon$$. Then $$|x-c|<\delta \implies |x-c|<\epsilon \implies |f(x)-f(c)|<\epsilon$$.

 * $$f(x) = x^n $$ is continuous for all $$n \in \mathbb{N}$$ and all $$x \in \mathbb{R}$$.

We proceed by induction. We have already seen that $$f(x) = x^{1}$$ is continuous. Assuming $$f(x) = x^{n-1}$$ is continuous, we use the fact that continuity is preserved under algebraic operations to see that $$xf(x) = x^n$$ is continuous.

 * $$f(x) = x^{-n}$$ is continuous for all $$n \in \mathbb{N}$$ and all $$x \in \mathbb{R} \setminus {0}$$.

Since $$x^n$$ is continuous and nonzero on the set in question, $$\frac{1}{x^n} = x^{-n}$$ is continuous since continuity is preserved under division by a nonzero function.

We can now use the continuity of $$x^n$$ together with the intermediate value theorem to construct positive nth roots. As promised, this is much nicer than the construction of square roots in the first chapter:

Construction of nth roots
We begin with construction of rational powers of arbitrary positive reals.

Given $$c > 0$$, consider the function $$f(x) = x^n - c$$(it is clear that 0 has a unique nth root, so we do not consider this case). $$f(0) = -c < 0$$ and since $$1+c > 1$$, $$f(1+c) = (1+c)^n - c > (1+c)^n - (1+c) > 0$$. By the Intermediate Value Theorem, $$\exists x \in (0,1+c): f(x) = x^n - c = 0$$. Thus c has a positive nth root.

To prove uniqueness, let x and y be two nth roots of c. If $$x > y > 0$$, then $$x^n > y^n > 0$$. But then it would follow that $$c > c$$, a contradiction. Similarly we cannot have $$x < y$$, so it follows that $$x = y$$.

Definition and Properties of Rational Powers
Given $$n \in \mathbb{N}$$ we define $$x^{\frac{1}{n}} = \sqrt[n]{x}$$ to be the unique nonnegative nth root of x. We then define all rational powers as follows:

 * If $$r = \frac{p}{q}$$ is in lowest terms(i.e. p and q have no common factors and $$q>0$$), we define $$x^r=(\sqrt[q]{x})^p$$.

Our definition would work just as well if $$\frac{p}{q}$$ were not in lowest terms, as we'll see in a minute. First we must prove some basic facts:

 * $$\sqrt[m]{\sqrt[n]{x}} = \sqrt[mn]{x}$$

Note that $$(\sqrt[m]{\sqrt[n]{x}})^{mn} = ((\sqrt[m]{\sqrt[n]{x}})^m)^n =\sqrt[n]{x}^n = x$$. Thus $$\sqrt[m]{\sqrt[n]{x}}$$ is an mn-th root of x. The result follows immediately from uniqueness of positive roots.

 * $$x^{\frac{p}{q}} = \sqrt[q]{x^p} = \sqrt[q]{x}^p$$

Using what we know about integer powers we see that $$(x^{\frac{1}{q}})^p) = \sqrt[q]{(\sqrt[q]{x}^p)^q)} = \sqrt[q]{(\sqrt[q]{x}^q)^p)} = \sqrt[q]{x^p}$$

As promised, our definition does not depend on the fraction representing r:

 * If $$\frac{p}{q} = \frac{m}{n}$$, then $$x^{\frac{p}{q}} = \sqrt[n]{x^m}$$.

If $$\frac{p}{q} = \frac{m}{n}$$, then $$m = cp$$ and $$n = cq$$ for some $$c \in \mathbb{Z}$$. Thus $$\sqrt[n]{x^m} = (\sqrt[c]{\sqrt[q]{x}}^c)^p = \sqrt[q]{x^p} = x^{\frac{p}{q}}$$.

Now we'll prove the standard algebraic facts about rational powers:

 * If $$r,s \in \mathbb{Q}$$ and $$x>0$$, then $$ x^rx^s = x^{r+s}$$ and $$(x^r)^s = x^{rs}$$

Proof: Let $$r = \frac{a}{b}$$ and $$s = \frac{c}{d}$$. Then $$x^rx^s = x^{\frac{a}{b}}x^{\frac{c}{d}} = x^{\frac{ad}{bd}}x^{\frac{bc}{bd}} = (x^{\frac{1}{bd}})^{ad}(x^{\frac{1}{bd}})^{bd} = (x^{\frac{1}{bd}})^{ad + bc} = x^{\frac{ad+bc}{bd}} = x^{(r+s)} $$

Also, $$x^{rs}$$ $$= x^{\frac{a}{b}\frac{c}{d}}$$ $$= x^{\frac{ac}{bd}}$$ $$= (x^{ac})^{\frac{1}{bd}}$$ = $$(((x^{a})^{c})^{\frac{1}{b}})^{\frac{1}{d}}$$ $$= (((x^a)^{\frac{1}{b}})^c)^{\frac{1}{d}}$$$$ = (x^{\frac{a}{b}})^{\frac{c}{d}}$$

 * If $$r = \frac{a}{b} > 0 \in \mathbb{Q}$$ and $$y > x > 0$$, then $$y^r>x^r>0$$.

Proof: If $$y^{\frac{1}{b}} \leq x^{\frac{1}{b}}$$, then $$(y^{\frac{1}{b}})^b \leq (x^{\frac{1}{b}})^b$$, and $$ y \leq x$$, contradicting the assumption $$y > x > 0$$. So $$y^{\frac{1}{b}} > x^{\frac{1}{b}} > 0$$. Since a > 0, $$y^{\frac{a}{b}} > x^{\frac{a}{b}} > 0$$. Thus $$y^r>x^r>0$$

Continuity of rational powers
Now we'll use the preceding algebraic properties to prove continuity of all rational powers:

 * $$f(x) = x^{\frac{1}{n}} $$ is continuous for all $$n \in \mathbb{N}$$and $$x \geq 0$$.

Proof: Given $$\epsilon > 0$$, let $$\delta = |c^{\frac{n-1}{n}}|\epsilon$$. Then $$ |x-c|<\delta \implies $$

$$ |x^{\frac{1}{n}} - c^{\frac{1}{n}}| = \frac{|x^{\frac{1}{n}} - c^{\frac{1}{n}}||x^{\frac{n-1}{n}} + x^{\frac{n-2}{n}}c^{\frac{1}{n}} + \cdots + x^{\frac{1}{n}}c^{\frac{n-2}{n}} + c^{\frac{n-1}{n}}|}{|x^{\frac{n-1}{n}} + x^{\frac{n-2}{n}}c^{\frac{1}{n}} + \cdots + x^{\frac{1}{n}}c^{\frac{n-2}{n}} + c^{\frac{n-1}{n}}|}$$$$ = \frac{|x - c|}{|x^{\frac{n-1}{n}} + x^{\frac{n-2}{n}}c^{\frac{1}{n}} + \cdots + x^{\frac{1}{n}}c^{\frac{n-2}{n}} + c^{\frac{n-1}{n}}|}$$$$ \leq \frac{|x-c|}{|c^{\frac{n-1}{n}}|}< \frac{|c^{\frac{n-1}{n}}|\epsilon}{|c^{\frac{n-1}{n}}|} = \epsilon$$.

The preceding argument works for $$c \not= 0$$. If $$c = 0$$, then let $$\delta = \epsilon^{n}$$. Then:

$$|x-0|<\delta \implies |x|<\epsilon^n \implies |x|^{\frac{1}{n}} < \epsilon \implies |x^{\frac{1}{n}}-0|< \epsilon$$

So, $$x^{\frac{1}{n}}$$ is continuous for all $$x\geq0$$.

 * $$x^q$$ is continuous for all $$q \in \mathbb{Q}$$

Proof: If $$q = \frac{a}{b}$$, where a and b are integers and $$b>0$$, then $$x^q = (x^a)^{\frac{1}{b}}$$. Thus $$x^q$$ is the composition of continuous functions, and therefore is continuous itself.

Real Powers
We will define arbitrary real exponents as the supremum of the exponents of rational members of the "Cut" that corresponds to the given real number. But first, we need to establish that this operation indeed produces a unique real number.

Theorem
Let $$a,x\in\mathbb{R}$$ and let $$a>0$$

Let $$A=\{ a^q|q\in\mathbb{Q};q\leq x\}$$

Let $$B=\{ a^q|q\in\mathbb{Q};q\geq x\}$$

Let $$\alpha=\sup A$$ and $$\beta=\inf B$$

Then, $$\alpha=\beta$$

title: Real Analysis/Exponential Function


text:
User:Dan Polansky/monobook.css

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title: User:Dan Polansky/monobook.css


text: User talk:77.42.207.9

title: User talk:77.42.207.9


text:
Wikijunior:Europe/North Macedonia

North Macedonia (Република Северна Македонија in Macedonian) is a country located in south eastern Europe in the Balkan peninsula. Its neighbours are Serbia, Kosovo, Bulgaria, Greece and Albania. The capital of North Macedonia is Skopje. Other big cities in North Macedonia are Bitola and Kumanovo. The currency of North Macedonia is the Macedonian denar.

North Macedonia's History
The present country of North Macedonia has been inhabited for thousands of years and the country has been part of many empires including the Roman Empire, Byzantine empire, Bulgarian empire, Serbian empire, Ottoman Empire and, more recently, as a part of Yugoslavia. North Macedonia was one of the 6 republics of Yugoslavia.

North Macedonia became independent from Yugoslavia on September 8th, 1991. In the following years it faced disputes with Greece which also claims the Macedonia name (it's the name of a historical region which includes much of Greece as well as the Republic of Macedonia). North Macedonia has applied to join the European Union and has been a candidate country since 2005.

North Macedonia's Geography
North Macedonia is a mountainous landlocked (it has no coastline) country with many lakes and valleys. The highest mountain is Mount Korab at 2764 metres. Lake Ohrid (see the picture below) is the country's largest lake and the deepest lake in the Balkan peninsula. The country is regionally famous for its spa resorts which are popular with locals and visitors.

North Macedonia's People
People in North Macedonia are a mixture of many nationalities. The majority of people in the country are ethnic Macedonian but there are also large minorities of Serbs, Albanians and Bulgarians. The most widely spoken language is Macedonian which is very closely related to Bulgarian. There are large numbers of Serbian and Albanian speakers in North Macedonia.

Over 66% of North Macedonians are Orthodox Christians with the rest of the population mostly Muslim.

North Macedonia's Sights
700,000 people visit North Macedonia every year with the majority visiting the capital Skopje, the city of Dojran and Lake Ohrid. The country has 3 national parks which are popular with all people: Mavrovo, Galičica and Pelister. Bears, wolves and lynx can be found in the country making it a popular location for wildlife enthusiasts.

Festivals are common throughout the year including the Balkan Festival of Folk Songs and Dances and Galičnik Wedding Festival, where a chosen couple are allowed to marry in the traditional Galička-style wedding clothes during the festivities. North Macedonia's only UNESCO World Heritage Site is Ohrid Municipality, an area in the south-west of North Macedonia which includes Church of St. John at Kaneo (pictured above).

title: Wikijunior:Europe/North Macedonia


text:
User:Numediaweb

check : http://en.wikipedia.org/wiki/User:Numediaweb

title: User:Numediaweb


text:
Wikijunior:Europe/Georgia

Georgia is a small country in the Caucasus region of south-eastern Europe. Set on the coast of the Black Sea, it lies to the south of Russia and to the north of Turkey, Azerbaijan and Armenia. The capital city is Tbilisi and the currency used in Georgia is the Lari. Georgia was part of the Soviet Union until 1991. Since 1999, Georgia is a member of the Council of Europe and it is a candidate to join the European Union in the future.

Georgia's History
From the early 19th century Georgia was part of the Russian Empire. Georgia was made a part of the USSR, also known as the Soviet Union, in 1922. After Georgia gained independence from the USSR in 1991 it had a civil war which lasted for 3 years and was caused by a power struggle between different people who wanted to govern Georgia. Soon after further violence occurred in the Georgian regions of Abkhazia and South Ossetia which had large populations of Russians who either wanted to become independent or part of Russia. Since the 1990s both regions more-or-less govern themselves although they are not recognized as independent countries by most other countries.

In 2008 there was a major war between Georgia, Russia and fighters from Abkhazia and South Ossetia (2008 South Ossetia war). Russia bombed Georgia and invaded parts of the country before pulling out shortly after. Both regions continue to self-govern themselves though the Georgian government insists that they are part of Georgia.

Georgia's Geography
Georgia covers an area of about 69,700 km2 (about 26,900 sq mi). It is bounded by Russia on the north and by Azerbaijan, Armenia, and Turkey on the south. Rugged mountains make up about 85 percent of the total land area. The coastal area of Georgia has hot summers and cold winters while the mountain areas of Georgia have cool summers and rainy winters.

Georgia is a very mountainous country with many mountains over 4,000 metres high. The highest mountain in the country is Shkhara at 5,068 metres. Georgia is also home to many deep caves including Krubera Cave which is the world's deepest cave at over 2,000 metres deep. Georgia's Black Sea coast is full of sandy beaches and popular with tourists from the region and elsewhere.

Georgia's People
The population of Georgia is 4,646,003. Nearly 100 different ethnic groups make up Georgia's population. Georgians are the largest group, making up about 70 percent of the population, followed by Armenians, Russians, and Azerbaijanis. Georgian has been the country's official language since 1918. Russian is also a commonly spoken language. Education is free and required from the first grade to the eleventh grade. Georgia has an adult literacy rate of 99.5 percent. The majority of people are Christian.

Georgia's Sights
Each year Georgia is becoming more and more popular for international tourists. Ancient cathedrals and monasteries including Svetitskhoveli Cathedral, the architecture of Tbilisi's old town, traditional cuisine and fine wines are popular attractions for visitors.

Over 25% of Georgia's territory is designated as national park and forests cover 40% of the total area. Around 3 million foreign visitors enter the country every year. There are 4 UNESCO World Heritage Sites in Georgia which are: Bagrati Cathedral, monuments in Mtskheta (including the cathedral in the picture on the left) and the Upper Svaneti ruins.

title: Wikijunior:Europe/Georgia


text: User talk:206.255.68.23

title: User talk:206.255.68.23


text: User talk:88.107.101.184

title: User talk:88.107.101.184


text:
User:Cifuentes.mx~enwikibooks

Roberto Cifuentes
--Cifuentes.mx (talk) 15:05, 12 April 2008 (UTC)

title: User:Cifuentes.mx~enwikibooks


text:
User talk:Cifuentes.mx~enwikibooks

Welcome to Wikibooks, Cifuentes.mx!

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 Cifuentes.mx. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Cifuentes.mx~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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Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:02, 17 March 2015 (UTC)

Renamed
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title: User talk:Cifuentes.mx~enwikibooks


text:
User talk:David Trochos~enwikibooks

Welcome to Wikibooks, David Trochos!

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 David Trochos. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name David Trochos~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 23:08, 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) 04:52, 19 April 2015 (UTC)

title: User talk:David Trochos~enwikibooks


text:
User talk:JW159~enwikibooks

Νεοπτόλεμος ( talk | email | contribs ) 16:49, 12 April 2008 (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 JW159. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name JW159~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 23:19, 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) 05:03, 19 April 2015 (UTC)

title: User talk:JW159~enwikibooks


text:
Starting A Website/Understanding the Internet

In starting a website, one should understand the Internet. The Internet is a worldwide, publicly accessible series of interconnected computer networks that transmit data by packet switching using the standard Internet Protocol (IP). It is a "network of networks" that consists of millions of smaller domestic, academic, business, and government networks, which together carry various information and services, such as electronic mail, online chat, file transfer, and the interlinked web pages and other resources of the World Wide Web (WWW).

The Internet has also become a large market for companies; some of the biggest companies today have grown by taking advantage of the efficient nature of low-cost advertising and commerce through the Internet, also known as e-commerce. It is the fastest way to spread information to a vast number of people simultaneously. The Internet has also subsequently revolutionized shopping—for example; a person can order a CD online and receive it in the mail within a couple of days, or download it directly in some cases. The Internet has also greatly facilitated personalized marketing which allows a company to market a product to a specific person or a specific group of people more so than any other advertising medium.


text:
User talk:76.217.62.27

You have been blocked from editing Wikibooks for 2 hours because of disruptive edits.

--ЗAНИA talk 18:09, 12 April 2008 (UTC)

title: User talk:76.217.62.27