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General Chemistry/Balancing Equations

Balancing Equations
Chemical equations are useful because they give the relative amounts of the substances that react in a chemical equation.

$$\hbox{N}_{2} + 3{\hbox{H}_{2}} \to 2\hbox{NH}_{3}$$

In some cases, however, we may not know the relative amounts of each substance that reacts. Fortunately, we can always find the correct coefficients of an equation (the relative amounts of each reactant and product). The process of finding the coefficients is known as balancing the equation.

During a chemical reaction, atoms are neither created or destroyed. The same atoms are present before and after a reaction takes place; they are just rearranged. This is called the Law of Conservation of Matter, and we can use this law to help us find the right coefficients to balance an equation.

For example, assume in the above equation that we do not know how many moles of ammonia gas will be produced:

$$\hbox{N}_{2} + 3{\hbox{H}_{2}} \to {\color{Red}?}\hbox{NH}_{3}$$

From the left side of this equation, we see that there are 2 atoms of nitrogen gas in the molecule N2 (2 atoms per molecule x 1 molecule), and 6 atoms of hydrogen gas in the 3 H2 molecules (2 atoms per molecule x 3 molecules). Because of the Law of Conservation of Matter, there must also be 2 atoms nitrogen gas and 6 atoms of hydrogen gas on the right side. Since each molecule of the resultant ammonia gas (NH3) contains 1 atom of nitrogen and 3 atoms of hydrogen, 2 molecules are needed to obtain 2 atoms of nitrogen and 6 atoms of hydrogen.

Combustion
A combustion reaction is a reaction between a carbon chain (basically, a molecule consisting of carbons, hydrogen, and perhaps oxygen) with oxygen to form carbon dioxide and water, plus heat. Combustion reactions could get very complex: $$2\hbox{C}_6\hbox{H}_6 + 15\hbox{O}_2 \to 12 \hbox{CO}_2 + 6\hbox{H}_2\hbox{O}$$

Fortunately, there is an easy way to balance these reactions.

First, note that the carbon in C6H6 can only appear on the product side in CO2. Thus, we can write a coefficient of 6 in front of CO2.

Next, note that the hydrogen in C6H6 can only go to H2O. Thus, we put a 3 in front of H2O.

We have 15 oxygen atoms on the product side, so there are $$\frac{15}{2}$$ O2 molecules on the reactant side. To make this an integer, we multiply all coefficients by 2.

title: General Chemistry/Balancing Equations


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General Chemistry/Book Structure

Overall Structure
The general structure of information in the book was developed from pages already here. It seems to make sense logically, and works quite well as a wiki.

 * Topic 1
 * Module 1
 * Module 2
 * Topic 2
 * Module 1
 * Module 2

Each topic is one reasonably defined area of study (e.g. Acids and Bases, Basic Quantum Chemistry). Each module should contain one specific concept or idea to learn (e.g. Covalent Bonding, Lewis Acids and Bases). Each module is one page, but can contain multiple subheadings.

This layout corresponds roughly to Chapters and Subchapters in a "paper and ink" textbook.

Note: Chemistries of the Various Elements should not contain details like melting point, boiling point, mass, and a list of every possible use and source of the element. That is what Wikipedia is for. Rather, it should point out the most unique and interesting properties of the elements, and more importantly, supply some significant chemical reactions involving the elements.

Technical Points
Each page should begin with so people can find their way around.

Each page should end with to put the page in the proper category (General Chemistry).

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

title: General Chemistry/Book Structure


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General Chemistry/Orphaned Pages

 * ../Measurement/
 * ../Basic Chemical Concepts/
 * ../Properties of matter/
 * ../Atoms and molecules/
 * ../Nomenclature/
 * ../Calculations/
 * ../Molar Concepts/

''The following are a set of old modules copied from wikipedia that were previously on the front page. The current pages are much better written and form a cohesive book structure. The pages below contain a fair bit of irrelevant information (who needs 7 definitions of acid/base reactions) and are not as well suited to a textbook format (because they are from an encyclopedia).''

''Many pages were stubs. Most have been redirected to the more appropriate page, but some still remain.''

 * 1) Matter is measured
 * 2) Properties of elements are periodic
 * 3) Chemical formulas describe chemical compounds
 * 4) Gases, liquids and solids each have specific physical characteristics

title: General Chemistry/Orphaned Pages


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General Chemistry/Chemical Equilibria/Acid-Base Equilibrium

Consider an acid, HA, in water. According to the Brønsted-Lowry acid/base theory, the acid should protonate the water to form hydronium and the conjugate base, A-. There will be an equilibrium between the acid and water, and hydronium and the conjugate base.

$$\hbox{HA} + \hbox{H}_2\hbox{O} \leftrightarrow \hbox{H}_3\hbox{O}^+ + \hbox{A}^-$$

This equilibrium can be used to calculate the concentrations of species in the solution.

Acid Dissociation Constant
Like all equilibria, an acid/base dissociation will have a particular equilibrium constant which will determine the extent of the reaction (whether it lies to the left or right of the equation). As the equilibrium constant approaches zero, the reaction tends to form 100% reactants. As the equilibrium constant approaches infinity, the reaction tends to form 100% products. The equilibrium constant K=1 states that there will be 50% products and 50% reactants

Because the equilibrium is used for calculating the concentrations of weak acids, very little water actually reacts. The concentration of water during the reaction is, therefore, a constant, and can be excluded from the expression for K. This gives rise to a special equilibrium constant, Ka, known as the acid dissociation constant. It is simply K multiplied by the concentration of water.

$$K_a = \hbox{K}[\hbox{H}_2\hbox{O}] = \frac{[\hbox{H}_3\hbox{O}^+][\hbox{A}^-]}{[\hbox{HA}]}$$

The Ka of a weak acid determines how acidic it is, i.e., how far its equilibrium lies to the right. The Ka values of weak acids can and have been determined experimentally.

Base Dissociation Constant
A similar equilibrium exists when a weak base is dissolved in water. The base will accept a proton from water and form a conjugate acid, BH+.

$$\hbox{B} + \hbox{H}_2\hbox{O} \leftrightarrow \hbox{OH}^- + \hbox{BH}^+$$

This equilibrium has its own special constant, Kb, known as the base dissociation constant. Like the acid dissociation constant, it is defined as the equilibrium constant multiplied by the concentration of water.

$$K_b = K[\hbox{H}_2\hbox{O}] = \frac{[\hbox{BH}^+][\hbox{OH}^-]}{[\hbox{B}]}$$

Ion Product Constant
A special equilibrium exists between ordinary water molecules. Occasionally, one water molecule will act as an acid, and donate a proton to another water molecule (which acts as a base). This is the auto-ionization of water.

$$\hbox{H}_2\hbox{O} + \hbox{H}_2\hbox{O} \leftrightarrow \hbox{H}_3\hbox{O}^+ + \hbox{OH}^-$$

By the Le Chatelier Principle, we can see that if the hydronium ion concentration is increased (by adding an acid), the equilibrium will move to the left and there will be a lower concentration of hydroxide. Thus, hydronium and hydroxide concentration are inversely related—an increase in one will result in a decrease in the other, and vice versa.

The equilibrium expression for this reaction is given a special name, Kw. Because it simplifies to the product of hydronium and hydroxide concentration, it is sometimes called the ion product constant. The value of this constant is 1.0 × 10-14 at 25°C.

$$K_w = [\hbox{H}_3\hbox{O}^+][\hbox{OH}^-] = 1.0 \times 10^{-14}$$

This expression can be used to find the pH of pure water. Recall that pH is the negative logarithm of hydronium ion concentration. If we set the hydronium ion concentration in the above expression to be $$x$$, we can derive the pH. The hydroxide ion concentration must also be $$x$$ (since each molecule of hydroxide is a result of a molecule of hydronium forming). We have:

$$\begin{matrix}x^2 &=& 1.0 \times 10^{-14} \\ x &=& \sqrt{1.0 \times 10^{-14}} \\ x &=& 1.0 \times 10^{-7} \\ \hbox{pH} &=& -\log{1.0 \times 10^{-7}} \\ \hbox{pH} &=& 7.0\end{matrix}$$

This technique can be used to determine the pH of any solution if either one of the ion concentrations are known.

Conjugate Base Expressions
The conjugate bases of weak acids have a relationship with their parent acids. Consider the equilibrium expression of the conjugate base, A-, of the weak acid HA.

$$K_b = {\frac{[\hbox{HA}][\hbox{OH}^-]}{\hbox{A}^-}}$$

If we multiply the expression for an acid by the expression for its conjugate base, the concentrations of the acid and conjugate base cancel and we obtain the ion product constant for water! This allows us to calculate the Kb of a base if the Ka of its conjugate acid is known (and vice versa).

$$\begin{matrix} \frac{[\hbox{H}_3\hbox{O}^+][\hbox{A}^-]}{[\hbox{HA}]} &\times& \frac{[\hbox{HA}][\hbox{OH}^-]}{[\hbox {A}^-]} &=& [\hbox{H}_3\hbox{O}^+][\hbox{OH}^-] \\ K_a &\times& K_b &=& K_w \\ \end{matrix}$$

Summary
The definitions of the acid and base dissociation constants are very important. They are presented here for reference.


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Algebra/Function Graphing

Functions have an Independent Variable and a Dependent Variable
When we look at a function such as $$f(x) = \frac {1}{2}x,$$  we call the variable that we are changing—in this case  $$x\,$$ --the independent variable. We assign the value of the function to a variable we call the dependent variable. The reason that we say that $$x\,$$  is independent is because we can pick any value for which the function is defined—in this case real  $$\mathbb{R}$$  is implied—as an input into the function. Once we pick the value of the independent variable the same result will always come out of the function. We say the result is assigned to the dependent variable, since it depends on what value we placed into the function.

Equating $$y\,$$  with our function  $$y = \frac {1}{2}x,$$  then  $$ 2y = 2(\frac {1}{2}x),$$  then  $$ 2y = x,\,$$  then  $$ g(y)= 2y.\,$$

The independent variable is now $$y\,$$  and the dependent variable  $$x.\,$$
 * Note: this is a very unusual case where the ordered pair $$(g(y),y)\,$$  is reverse mapped  $$y\mapsto x\,$$  and corresponding reverses (dependent, independent), (range, domain), and now  $$x\,$$  must be singular for each and every  $$y\,$$  corresponding to an horizontal line test of function! It would be less desireable to rotate or swap the positions of axes, the order of coordinate pairs  $$(x,y)\,$$  and (abscissa, ordinate).

Have we used Algebra to change the nature of the function? Let's look at the results for three functions

If we look at the table above we can see that the independent variable for $$f(x)\,$$  gives the same results as the dependent variable of  $$g(y).\,$$  We can see what this means when we look at the values for  $$h(x).\,$$  The function  $$g(y)\,$$  is the same as the function  $$f(x),\,$$  but when we switch which variable we use as the independent variable between  $$g(y)\,$$  and  $$h(x)\,$$  we see that we have discovered that  $$g(y)\,$$  and  $$h(x)\,$$  are inverse functions.

Let's take a look at how we can draw functions in $$ x\, $$  and  $$ y\, $$  and then come back and look at this idea of independent and dependent variables again.

Explicit and Implicit Functions
Variables like $$x\,$$  and  $$y\,$$  formulate a 'relation' using simple algebra. $$f,\,$$  $$g,\,$$  and   $$h\,$$  commonly denote functions. Function notation $$f(x),\,$$  read "eff of ex", denotes a function with 'explicit' dependence on the independent variable  $$x.\,$$  By assigning variable  $$y\,$$  to  $$f(x),\,$$   $$y=f(x),\,$$   $$ y\,$$  is now an 'implicit' function of   $$x\,$$  using equation notation. If $$f(x) \,$$  is  $$\frac{x}{2},$$  then  $$y=\frac{x}{2}$$   [ $$y(x)\,$$  would denote an 'explicit' function of  $$x\,$$ ]. A relation is also a function when the dependent variable has one and only one value for each and every independent variable value.

The Cartesian Coordinate System
The Cartesian Coordinate System is a uniform rectangular grid used for plane graph plots. It's named after pioneer of analytic geometry, 17th century French mathematician René Descartes, whose Latinized name was Renatus Cartesius. Recall that each point has a unique location, different from every other point. We know that a line is a collection of points. If we pick a direction of travel for the line that starts at a point then all of the other points can be thought of as either behind our starting point or ahead of it. Finally, a plane can be thought of as a collection of lines that are parallel to each other. We can draw another line that is composed of one point from each of the lines that we chose to fill our plane. If we do this then we can locate the other lines as behind or ahead of the line with the point we chose to start on. Descartes decided to pick a line and call it the $$ x\,$$ -axis, and to then pick a line perpendicular to this line and call it the  $$ y\,$$ -axis. He then labeled this intersection point $$ (0,0)\,$$  and origin O. The points to the left (or behind) of this point each represent a negative number that we label as $$(-x,0).\,$$  The points to the right (or ahead) of this point each represent a positive number that we label as  $$ (x,0).\,$$  The points on the  $$ y\,$$ -axis that are above  $$ 0,0\,$$  are labeled as positive  $$ (0,y),\,$$  and the points on the  $$ y\,$$ -axis below  $$ 0,0\,$$  are labeled as negative  $$ (0,-y).\,$$  A point is plotted as a location on the plane using its coordinates from the grid formed by the  $$ x\,$$   and  $$ y\,$$ -axes. If you draw a line perpendicular to the $$ x\,$$ -axis from a point you pick then that point has the same  $$ x\,$$ -coordinate as the point where that line crosses the  $$ x\,$$ -axis. If you draw a line perpendicular to the $$ y\,$$ -axis from your point then it has the same  $$ y\,$$ -coordinate as the point where that line crosses the  $$ y\,$$ -axis. If you need to sharpen your knowledge in this area, this link/section should help: The Coordinate (Cartesian) Plane

An equation and its graph can be referred to as equal. This is true since a graph is a representation of a specific equation. This is because an equation is a group of one or more variables along with one or more numbers and an equal sign ( $$x=1,\,$$  $$y=x+1,\,$$  and  $$y=x^2+2x+1\,$$  are all examples of equations). Since variables were introduced as way of representing the many possible numbers that could be plugged into the equation. A graph of an equation is a way of drawing the relationship between the numbers that can be input (the independent variable) and the possible outputs that would be produced. For example, in the equation: $$y=x+2,\,$$  we could choose to make the  $$ x\,$$  the independent variable and the output number would be two more than the input number every time. The graph of this equation would be a picture showing this relationship. On the graph, each $$ y\,$$ -value (the vertical axis) would be two higher than the (horizontal)  $$ x\,$$ -value that is plugged in because of the  $$ +\, 2\,$$  in the equation.

Linear Equations and Functions
This section shows the different ways we can algebraically write a linear function. We will spend some time looking at a way called the "slope intercept form" that has the equation $$ y = f(x) = m x + b\,.\, $$
 * Linear Equations and Functions

Unless a domain for $$ x \, $$  is otherwise stated, the domain for linear functions will be assumed to be all real numbers  $$\mathbb{R}$$  and so the lines in graphs of all linear functions extend infinitely in both directions. Also in linear functions with all real number domains, the range of a linear function may cover the entire set of real numbers for $$ y\,,\, $$  one exception is when the slope  $$ m=0 \, $$  and the function equals a constant. In such cases, the range is simply the constant. Another would be a squaring function where the range would be non-negative when $$b=0\,.\,$$

The y-intercept constant b
It was shown that $$y=f(x)=mx+b\,$$  has infinite solutions (in the UK,  $$y=mx+c\,;\,$$  also common  $$y=ax+b\,,\,$$  $$y=a_1x+a_0\,$$  and  $$y=-\frac{A}{B}x-\frac{C}{B}\,$$ ). Points $$(x,y)\,$$  will be mapped with independent variable  $$x\,$$  assuming the horizontal axis and  $$y\,$$  vertical on a Cartesian grid. By assigning $$x\,$$  to a value and evaluating  $$y\,,\,$$  a (single) point coordinate solution is found. When $$x=0\,,\,$$  then by zero-product property term  $$m\times x=0\,$$,  and by additive identity terms  $$0+b=b=y\,.\,$$  The point  $$(0,b)\,$$  is the unique member of the line (linear equation's solution) where the y-axis is 'intercepted'. More about intercepts link: The  $x\,$  and  $y\,$  Intercepts

What does the m tell us when we have the equation $$ y = f(x) = m x + b \, $$ ?
$$ m \, $$ is a constant called the slope of the line. Slope indicates the steepness of the line.

Slope
Two separate points fixed anywhere defines a unique straight line containing the points. Confining this study to plane geometry ($$R^2$$) and fixing coordinates for unique points at $$(x_{1},y_{1})\,$$  and  $$(x_{2},y_{2})\,$$  a straight line is defined relating two variables in a linear-equation mappable on a graph-plot. When the two points are identical, infinite lines result, even in a single plane. When $$x_{1}=x_{2}\,$$  then a vertical-line mere relation is defined, not a function. Functions are equation-relations evaluating to singularly unique dependent values. Only when (iff) $$x_{1}\neq x_{2},\,$$  then is the line containing the points a linear 'function' of  $$x.\,$$

For a linear function, the slope can be determined from any two known points of the line. The slope corresponds to an increment or change in the vertical direction divided by a corresponding increment or change in the horizontal direction between any different points of the straight line.

Let $$\Delta y=\,$$ increment or change in the $$y\,$$ -direction (vertical)  and

Let $$ \Delta x=\,$$ increment or change in the $$x\,$$ -direction (horizontal).

For two points $$(x_{1},y_{1})\,$$  and  $$(x_{2},y_{2}),\,$$  the slope of the function line m is given by: $$m = \frac {\Delta y}{\Delta x} = \frac {y_2 - y_1}{x_2 - x_1} $$

 * This formula is called the formula for slope measure but is sometimes referred to as the slope formula.

For a linear function, fixing two unique points of the line or fixing the slope and any one point of the line is enough to determine the line and identify it by an equation. There is an equation form for a linear function called the point-slope form of a line2 which uses the slope $$m\,$$  and any one point  $$(x_{1},y_{1})\,$$  to determine a valid equation for the function's line: $$ y - y_1 = m (x - x_1) \, $$

Algebra/Slope

The Pythagorean Theorem and The Distance Formula

Other forms of linear equations?
Algebra/Standard Form and Solving Slope

Intercept Form of a Line
There is one more general form of a linear function we will cover. This is the intercept form of a line, where the constants a and b are such that (a,0) is the x-intercept point and (0,b) is the y-intercept point.

$$\frac {x}{a} + \frac {y}{b} = 1 \qquad $$   where a ≠ 0  and  b ≠ 0

Neither constant a nor b can equal 0 because division by 0 is not allowed. The intercept form of a line cannot be applied when the linear function has the simplified form y = m x because the y-intercept ordinate cannot equal 0.

Multiplying the intercept form of a line by the constants a and b will give

$$b x + a y = a b \ $$

which then becomes equivalent to the general linear equation form A x + B y + C where A = b, B = a, and C = ab. We now see that neither A nor B can be 0, therefore the intercept form cannot represent horizontal or vertical lines. Multiplying the intercept form of a line by just b gives

$$\frac {b}{a} x + y = b$$

if we subtract $$\frac {b}{a} x$$

we get:

$$y = b - \frac {b}{a} x$$

which can, in turn, be rearranged to:

$$y = - \frac {b}{a} x + b$$

which becomes equivalent to the slope-intercept form where the slope m = -b/a.

Example: A graphed line crosses the x-axis at -3 and crosses the y-axis at -6. What equation can represent this line? What is the slope?

Solution: intercept form: $$ \qquad  \frac {x}{-3} + \frac {y}{-6} = 1$$

Multiplying by -6 gives $$\frac {-6}{-3} x + y = -6$$

 * $$y = -2 x - 6 \ $$

so we see the slope m = -2.

Graph of y = - 2x – 6 showing intercepts.

The line can also be written as $$-6 x - 3 y = (-3)(-6) \ $$

$$6 x + 3 y = -18 \ $$

Example: Can the equation

$$\frac {x}{2} + \frac {y}{4} = 0$$

be transformed into an intercept form of a line, (x/a) + (y/b) =1, to find the intercepts?

Solution: No, no amount of valid mathematical manipulation can transform it into the intercept form. Instead multiplying by 4, then subtracting 2x gives

$$y = -2 x \ $$

which is of the form y = m x where m = -2. The line intersects the axes at (0,0). Since the intercepts are both 0, the general intercept form of a line cannot be used.

Example: Find the slope and function of the line connecting the points (2,1) and (4,4).

Solution: When calculating the slope of a straight line from two points with the preceding formula, it does not matter which is point 1 and which is point 2. Let's set (x1,y1) as (2,1) and (x2,y2) as (4,4). Then using the two-point formula for the slope m:

$$ m = \frac {4 - 1}{4 - 2} = \frac { 3 }{ 2 }$$

Using the point-slope form:

One substitutes the coordinates for either point into the point-slope form as x1 and y1. For simplicity, we will use x1=2 and y1=1.

$$ y - 1 = \frac {3}{2} (x - 2)$$

$$ y - 1 = \frac {3}{2} x - 3 \, $$

$$ \  y =  \frac {3}{2} x - 2  $$

Using the slope-intercept form:

Alternatively, one can solve for b, the y-intercept ordinate, in the general form of a linear function of one variable, y = m x + b.

$$ b = y - m x \ $$

Knowing the slope m, take any known point on the line and substitute the point coordinates and m into this form of a linear function and calculate b. In this example, (x1,y1) is used.

 * $$ b = y_1 - m x_1 \, $$

 * $$ b = 1 - \frac {3}{2} \cdot 2 = 1 - 3 = -2 $$

Now the constants m and b are both known and the function is written as

$$ y = \frac {3}{2} x - 2 $$      or alternatively as      $$ f(x) = \frac {3}{2} x - 2 $$

__________ end of example __________

For another explanation of slope look here:

Slope

Example: Graph the equation 5x + 2y = 10 and calculate the slope.

Solution: This fits the general form of a linear equation, so finding two different points are enough to determine the line. To find the x-intercept, set y = 0 and solve for x.

$$ 5x + 2 \cdot 0 = 10 = 5x \ $$

$$ x = 10/5 = 2 \ $$

so the x-intercept point is (2,0). To find the y-intercept, set x = 0 and solve for y.

$$ 5 \cdot 0 + 2y = 10 = 2y \ $$

$$ y = 10/2 = 5 \ $$

so the y-intercept point is (0,5). Drawing a line through (2,0) and (0,5) would produce the following graph.

Graph of 5x + 2y = 10 showing intercepts

To determine the slope m from the two points, one can set (x1,y1) as (2,0) and (x2,y2) as (0,5), or vice versa and calculate as follows:

$$m = \frac {y_2 - y_1}{x_2 - x_1} = \frac {5 - 0}{0 - 2} = - \frac {5}{2} = - 2.5 $$

__________ end of example __________

Summary of General Equation Forms of a Line
The most general form applicable to all lines on a two-dimensional Cartesian graph is

$$A x + B y = C \ $$

with three constants, A, B, and C. These constants are not unique to the line because multiplying the whole equation by a constant factor gives a new set of valid constants for the same line. When B = 0, the rest of the equation represents a vertical line, which is not a function. If B ≠ 0, then the line is a function. Such a linear function can be represented by the slope-intercept form which has two constants.

slope-intercept form: $$y = m x + b \ $$

The two constants, m and b, used together are unique to the line. In other words, a certain line can have only one pair of values for m and b in this form.

The point-slope form given here

$$ y - y_1 = m (x - x_1) \, $$

uses three constants; m is unique for a given line; x1 and y1 are not unique and can be from any point on the line. The point-slope cannot represent a vertical line.

The intercept form of a line, given here,

$$\frac {x}{a} + \frac {y}{b} = 1$$   a ≠ 0  and  b ≠ 0

uses two unique constants which are the x and y intercepts, but cannot be made to represent horizontal or vertical lines or lines crossing through (0,0). It is the least applicable of the general forms in this summary.

Of the last three general forms of a linear function, the slope-intercept form is the most useful because it uses only constants unique to a given line and can represent any linear function. All of the problems in this book and in mathematics in general can be solved without using the point-slope form or the intercept form unless they are specifically called for in a problem. Generally, problems involving linear functions can be solved using the slope-intercept form (y = m x + b) and the formula for slope.

Discontinuity in Otherwise Linear Equations
Let variable y be dependent upon a function of independent variable x
 * $$y=f(x)\, ,$$ also  $$y(x)=f \,.$$

y is also the function f, and x is also the argument. Let y be the expressed quotient function
 * $$y=\frac{2x^2-5x+3}{x-1} .$$

The graph of ys solution plots a continuous straight line set of points except' for the point where x would be 1. Evaluation of the denominator with $$x=1$$ results in division by zero, an undefined condition not a member element of R and outside algebraic closure. y has a discontinuity (break) and no solution at point 1,-1. It becomes important to treat each side of a break separately in advanced studies.

ys otherwise linear form can be expressed by an equation removed of its discontinuity'. Factor $$x-1$$ from the $$2x^2-5x+3$$ numerator (use synthetic division).
 * $$y=\frac{2x^2-5x+3}{x-1} .$$
 * $$y=\frac{(2x-3)(x-1)}{(x-1)} .$$
 * $$y=2x-3 \,,\;x\neq1$$ (for all x except 1).

Reducing its (x-1) multiplicative inverse factors (reciprocals) to multiplicative identity (unity) leaves the $$(2 x - 3)$$ factor (with implied universal-factor 1/1). Limiting this simpler function's domain; 'all $$x$$ except $$x = 1$$, where x is undefined' or simply 'and x ≠ 1' (implying 'and R2 '); equates it to the original function. This expression is a linear function of x, with slope m = 2 and a y-intercept ordinate of -3. The expression $$2x-3$$ evaluates to -1 at x = 1, but function y is undefined (division by zero) at that point. There is a discontinuity for function y at x = 1. Practically the function has a sort of one-point hole (a skip), shown on the graph as a small hollow circle around that point. Lines, rays and line segments (and arcs, chords and curves) are shown discontinuous by dashed or dotted lines.

Note: non-linear equations may also be discontinuous—see the subsequent graph plot of the reciprocal function y = 1/x, in which y is discontinuous at x = 0 not just for a point, but over a 'double' asymptotic extremum pole along the y-axis. As x is evaluated at smaller magnitudes (both – and +) closer to zero, y approaches no definition in both the – and + mappings of the function.

Example: What would the graph of the following function look like?

 * $$y = \frac {x^2 - 4}{x + 2} .$$

Solution:
 * $$y = \frac {x^2 - 4}{x + 2} = \frac {(x + 2)(x - 2)}{(x + 2)} .$$

Reduce the reciprocal (x + 2) factors to unity. This makes y = x – 2 for all x except x = -2, where there is a discontinuity. The line y = x – 2 would have a slope m = 1 and a y-intercept ordinate of -2. So for the final answer, we graph a line with a slope of 1 and a y-intercept of -2, and we show a discontinuity at x = -2, where y would otherwise have been equal to -4.

Example: Write a function which would be graphed as a line the same as y = 2 x – 3 except with two discontinuities, one at x = 0 and another at x = 1.

Solution: The function must have a denominator with the factors

denominator = (x – 0)(x – 1) = x (x – 1). to have 'zeros' at the two x values. The function's numerator also gets the factors preserving an overall factor of unity, the expressions are multiplied out:

$$y = \frac {(2 x - 3) x (x - 1)}{x (x - 1)}= \frac {2 x^3 - 5 x^2 + 3 x}{x^2 - x}. $$

__________ end of example __________

title: Algebra/Function Graphing


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User talk:Dustin

Dustin, beautiful work on the General Chemistry book! Thank you. Welcome to Wikibooks and please let me know if you have any questions. You seem like you are already a seasoned wiki person. - Karl Wick 20:53, 28 May 2004 (UTC)

Copyright problem with Image:2p orbitals cover.jpg
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title: User talk:Dustin


text:
User:Josemoya

Hello. I made a login here when Spanish and english wikibooks were put in the same server. I've done a new login at es.wikibooks (see below), but I will read english wikibooks from time to time.

My wiki experience is small: just a couple of contributions to Wikipedia and Enciclopedia Libre. I'm also "Josemoya" at these sites.

Eso tiene una desventaja: que nunca paso con la frecuencia deseable por ninguna de esas páginas.

Contributions at English wikibooks

 * I've done some spellcheck for The Wrong Way to Learn Spanish.

Thank you for your work on the Wrong Way To Learn Spanish. We need all the native spanish speakers we can get. Moonlightembrace 18:30, 25 Jan 2005 (UTC)

My page at Spanish wikibooks (Wikibooks en español)

 * Josemoya

Mis intereses
I like to know everything. But in english pages I will keep writing about Spanish or Spanish literature.

Comentarios, sugerencias, etc.
You can put messages at my postbox, but I'll probably read my es.wikibooks postbox more often.

title: User:Josemoya


text:
European History

Historia de Europa Skizo de la Historio de Eŭropo Histoire de l'Europe História da Europa Lịch sử châu Âu ইউরোপীয় ইতিহাস 欧洲历史

title: European History


text:
User:Waxmop

Modules I pay attention to:

 * Artificial Intelligence
 * Distributed Operating Systems
 * Algorithms
 * C programming
 * Data Structures
 * Swing Dancing
 * Economics
 * SQL

We don't need to economize on words to fit some book-length constraint, so I believe that textbooks here should encourage multiple explanations for topics. That extra paragraph that seems redundant to one reader because it says the same thing in just a slightly different way may be the one that finally reaches another reader.

I have an undergraduate degree in economics from the University of Houston and I'm slowly working on a master's degree in computer science at Cleveland State university.

I keep a blog at http://blog.tplus1.com.

title: User:Waxmop


text:
Cookbook:Cuisine of Colombia

| South American Cuisines

 * Ajiaco (Potato and Chicken Stew)
 * Bandeja Paisa
 * Cazuela Gallega
 * Antioquean Beans
 * Empanadas Colombianas (Colombian Dumplings)
 * Mazorca Desgranada
 * Pan de Yuca
 * Pandebono
 * Papa Turmada (Potato Casserole)
 * Pastel de Arroz (Rice Tamale)
 * Tortilla de Huevos Criolla (Latino Scramble)
 * Sancocho (Criollo Stew)

title: Cookbook:Cuisine of Colombia


text:
European History/A Background of European History

This book presents a broad overview of European history, from the end of the Roman empire to the present day, roughly from the fifth century to the twentieth century. This is, of course, a somewhat arbitrary period, but not a wholly unuseful one. Though archaeological evidence shows a Europe of continuous settlement over tens of thousands of years, the evidence for the political and sociological order of these communities is scarce if not completely absent. The history of early Europeans is, therefore, a history of their tools and settlements.

Neolithic Era (7,000 BCE to 3,000 BCE)
The Neolithic period was a period of human development characterized by the spread of agriculture and pottery. Tools such as the stone ax and microflint arose during this era, and are common throughout despite their origins predating the Neolithic era. The stone ax, though rudimentary in its design, is perfectly capable of forest clearance. It is only natural to presume that tools made of wood were also available to Neolithic man, and with the rise of agriculture, the need for wood yokes and plows. However, wood is not a durable material, and therefore archaeological finds are rare. Evidence for agrarian societies dates back to 9,000 BCE on the Asian continent and it is believed that agricultural practices spread throughout two millennia into Europe. There is an argument for independent development of technologies not influenced by any other source except the internal ingenuity of the community itself, though further investigation will be needed to give full credence to this idea. For our purposes, we will view the spread of Neolithic technology as reliant on societies being in contact with one another and will adhere to the east to west advancement theories.

The Aegean Sea and Southern Europe
The earliest evidence of European agriculture can be found at Knossos, Crete, and dates from the sixth millennium BCE. The Aegean Sea, with its numerous islands within a short distance and clement weather, provided ideal conditions for settlement. Successive waves of migration by different tribal peoples from the east led to the establishment of communities throughout the Aegean and also along with the coastal areas of modern day Turkey and Greece. It is no coincidence that the Romans believed themselves, descendants of Aeneas, the legendary hero of Troy, for in the legends and myths of Classical Europe lies the eastern pantheon of gods and heroes. Crete, one of the largest islands in the Aegean and situated between the coasts of North Africa, the Southern European peninsula, and the coast of Asia Minor, had more natural resources than smaller Aegean islands as well as enough landmass to support a sizeable population. These factors made it ideal for colonization. The settlers of Crete arriving in 6,500 BCE brought with them both the agricultural practices of the East and seafaring skills. These played a large part in their eventual rise as a major force in the Aegean.

On the mainland of central Greece in the region of Thessaly, hundreds of Neolithic dwellings have been discovered through field archaeology, aerial mapping, and spectrum analysis by satellites. The alluvial plains of the region were attractive to communities based on agriculture and animal husbandry. The analysis of sherds and dwellings show a common culture amongst these sites with the pottery showing a distinctive style and the "wattle and daub" technique of architecture.

Northern Europe
The prehistoric Pile Dwellings of Vingelz in Switzerland (3,000 BCE) with its stone ax "factory" and dwellings near a water source are typical of other agrarian sites found in France, Germany, Austria and many other parts of Europe. These neolithic "villages" built near the lake shores and wetlands of the Alps show a continuous settlement of 4,500 years. The Alps were no impediment to trade with archaeological finds of sherds, stone tools and dwelling structures showing similarities that can only be explained by contact and mutual exchange. These neolithic "villages" were probably trading and intermarrying across the Alps though this was in some respects limited by small population numbers and structured by family and tribal identification rather than nationhood.

It must be borne in mind by the reader that the regions of Europe developed at different times though extensive trade networks across Europe existed at this period. This fact does allow for a certain degree of uniformity within periods so that we can group early European periods according to archaeological finds, normally to do with tools, utensils, and weapons.

The Bronze Age (3,000 BCE to 700 BCE)
The rise of the Minoan (3000 BCE to 1100 BCE) and Mycenaean civilizations and their influence on the Aegean islands and mainland Greece shapes the earliest history of Southern Europe. The discovery of these early civilizations was a result of investigations by a new breed of 19th century archaeologists who studied the classical literature of the Ancient world for its historical value rather than its mythical content. The descriptions of kings and cities mentioned in the Iliad were investigated for ethnographical and geographical clues as to their location. Heinrich Schliemann and Arthur Evans were two pioneers in this type of investigation. Schliemann went in search of the city of Troy on the coast of modern day Turkey and Arthur Evans searched for the lost cities of the Aegean islands. Schliemann who had made his fortune from Crimean War contracts turned to an area of Turkey now called Hisarlik. Earlier investigations of the Hisarlik site had led some to propose that this was the fabled lost city of Troy mentioned in the Iliad which thousands of years later is still known to modern audiences by the tale of the wooden horse. Schliemann was the first major excavator of this site and his discovery of Troy is considered to be the first modern investigation of classical history in relation to its myths and literature. It was not long after Schliemann's discovery that Arthur Evans discovered the Minoan civilization on Crete. Schliemann was to make one more great discovery; the Mycenean civilization that dominated the Aegean islands. Of special importance to Europeans is the discovery of the Linear B tablets at Knossos (Crete) by Arthur Evans. These clay tablets contain a form of archaic Greek from the 2nd millennium BCE. This is the oldest Greek (European) written language known to us and therefore its importance in relation to Homer's Iliad is fundamental to our understanding and investigation of the Aegean world during the Bronze Age.

The Classical Period (700 BCE to 500 CE)

The rise of the Greek city states from 1000 BCE onwards constitute in part a continuation of the hegemony that the Minoan and Mycenaean civilizations imposed upon the region. However the Greeks took a distinctly different view of the city state and developed forms of government that relied on citizen participation of which Athens in the fifth century BCE is considered to be the high point. One of the unusual aspects of Greek society was the existence of the city state of Sparta situated on the Southern Peloponnese. Sparta was a formidable regional military power and a rival to Athens throughout the classical period. It was of interest to such Athenians as the philosopher Plato who examines the various forms of government in his book The Republic including the warrior caste system as found in Sparta. The action of the 300 Spartans who defended the pass at Thermopylae against the invading army of Persia is still considered as encapsulating the essence of the Spartan city state. In the other city states of Greece, whether ruled by a dictator, king or democracy, the Spartan austerity and discipline directed towards the formation of a military state ruled by an elite oligarchy was both admired and feared. The history of mainland Greece is tied to the conflict between Athens and Sparta; both forming alliances with other Greek city states to create leagues. The Delian League representing the interests of Athens and her allies with its counterpart The Peloponnesian League led by Sparta. The idea of the city state as an independent power was so strong that Greek city states only overcame their enmity when faced with a foreign invasion that equally threatened them all. Once a foreign threat was removed the city states reverted back to open hostility towards one another. Eventually this internal friction between the Greek city states allowed the Macedonians to gain control of all Greece. The Romans viewed this Macedonian expansion as a threat and the conflict that would later arise would lead to Roman domination of all Greece.

The height of Athens in the 5th century BCE is also the starting point for the expansion of a Rome which was then a city surrounded by other tribes and city states. Rome's expansion started initially as a form of alliances or aggressions against other Latium kingdoms with a view to securing a central role in the region. Eventually as Rome's strength grew so did its expansion and soon the city states of Magna Graecia (Greek cities of southern Italy) felt obliged to form an alliance with Rome. Internal tribal divisions in Italy led to Rome's war with the Samnites (central Italian tribe) who were threatening the Greek city states of Italy. Rome's wars with Carthage further entrenched Rome's position in the region. Rome had subdued the Etruscans north of its city and the surrounding tribes. It had fought and been victorious against tribes in the central region. Southern Italy now came under Rome's hegemony as a guard against further Carthaginian expansion through their political alliances with Aegean kingdoms opposed to Roman power. Rome had become master of all Italy by 272 BCE. This domination of Italy led Rome to feel confident enough to launch military operations against Carthage. Italy which had for so long defended itself against Carthage's military was now on the offensive. Eventually Rome would attack Carthage itself in 148 BCE and divest them of their Aegean interests and Spain after they sued for peace. Rome had now extended its control to the whole of the Aegean and coastal areas of Spain. Carthage was destroyed and habitation of the city was disallowed by the Romans.

Rome's empire reached its zenith during the Augustan period (50 BCE to 20 BCE). Though Rome as a republic already had extensive territorial conquests, it was during the reign of Augustus that the empire achieved its greatest territorial expansion. From that period on Rome the Republic would never exist in anything except the imaginations of its historians and people. The reign of Augustus heralded the end of the republic and the beginning of the empire. Though the banner "SPQR" (Senatus Populusque Romanus - The Senate and People Of Rome) flew, its stated republican values now bowed to the will of the emperor. The Augustan period is seen as a golden age for Rome - the relative peace and stability during the reign of Augustus allowed Roman writers and arts to flourish. For the next 400 years Rome would be the greatest power in the world, though eventually, during the 3rd and 4th centuries CE, the pressures of empire led to Rome relinquishing its hold over its provinces.

Whilst the collapse of the Roman empire from the 4th century CE onward saw a contraction of the connections that tied the urbanized centers of Europe together (as evidenced in the decline of road building and maintenance), it did lay the foundation for the inward-looking reorganization of Roman provinces, which by the eleventh century had created an abundance of small regional states. From the eleventh century onward these small regional states started to change into the precursors of today's modern European nations. This was in part driven by the invention of banking, international commerce and the rise of mercantile states like Venice which controlled the trade from the East. The fifth century CE is, then, a useful starting point, for it forms a historical boundary between the ancient and modern.

Geography
Europe is not a discrete geographical unit, and it is too easy to see it as such, when in reality the cultures of Europe flow across its borders. The medieval peasants of Italy or Spain, for instance, shared far more in common with their close neighbours in North Africa than they did with their counterparts in Germany or England. Similarly, large parts of Eastern Europe, most particularly Russia, show significant cultural influence from Asian cultures, and were historically more closely connected with the east than the west.

A significant problem, therefore, is the porous nature of Europe's geographical borders. In the south, Europe's Mediterranean countries are only a short sail from the ports of North Africa. In the south-east, Europe is separated from Asia by nothing more than the short channel of the Bosphorus, and the most significant cultures in this region, such as the Byzantine Empire and the Ottoman Turks, were spread between modern Greece and Turkey. In Europe's far eastern reaches the continent is separated from Asia by the Ural mountains, and weather, rather than geography, is the most significant bar for travel between east and west. It is only to Europe's west, with the Atlantic Ocean, that we see a clear and significant geographical barrier.

The response to this problem is to accept that treating Europe as a discrete unit is somewhat arbitrary. It is essential, in any history, to define one's field of study, and to treat Europe as a unit is one way of achieving this aim. Providing one bears this in mind, the problem of European geography is not a problem at all. The somewhat arbitrary geographical borders of Europe need not detract from attempts to investigate the history of individual regions within the continent.

How To Approach Historical Investigation
History is frequently seen in narrative terms, as a story concerning the activities of our collective ancestors. It is true, to a great extent, that history is a form of storytelling, after all. However, unlike other forms of storytelling, history is closely based on real events and, as such, is shaped by certain rules and guidelines.

The most important of these govern how the historian reads source material. Before the historian can confidently say that a particular event occurred, he or she requires evidence. Much of the evidence is documentary in form, such as written records left by past generations in the course of everyday life. It is rare for these documents to have been written for the consumption of future generations, so it is important for the historian to understand the forces that shaped the production of these documents.

There are a number of criteria for this:

 * What motivated this person to act?
 * What were the prevailing attitudes at the time regarding this issue?
 * How did previous events cause this one?
 * What are similar events and how did they turn out?
 * How will this event matter in the future?
 * Why does this event occur now rather than earlier or later?
 * How was this event or person affected by forces in society? These forces could include the church, economic conditions, the government, geography, the education of the person as well as the general education of society, technology, nationalism, culture and traditions, and the class of the person.

This can be summed up in the 'ADAPTIL' method of evaluating historical sources:

 * 'A' - consideration of the author of the source
 * 'D' - consideration of the date of the source
 * 'A' - consideration of the source's intended audience
 * 'P' - consideration of the purpose of the source
 * 'T' - consideration of the tone of the source
 * 'I' - consideration of what the source implies (inference)
 * 'L' - consideration of the limitations of the source

Using these techniques will allow you to better grasp the events documented throughout this wiki book, and they will allow you to apply your knowledge outside of simple memorization.

All of our contributors hope you find this wiki book a useful method to learn more about European history - whether you are preparing for the Advanced Placement European History Test, a college final, or even if you simply desire to learn more about European history.

Fall of the Roman Empire

It is normal to speak of the fall of the Roman empire, but in many ways this description is too simple, and can be misleading. Certainly, the centralized state ruled by Augustus Caesar and his successors disappeared from history. However, the laws and language that Rome had given to a wide area of Europe persisted in its influence long after the empire had collapsed.

The Roman empire's later period was riddled with political and social turmoil. Much of the turmoil involved the failing Western empire. From the 3rd century, the Roman empire was under constant attack by internal and external forces. Tribes that Rome had never fully brought under Pax Romana saw in a weakened Rome the chance to either expand or raid. The cost of maintaining the furthest flung outposts and borders of the empire had meant that a professional army had to be maintained abroad permanently; a costly endeavour for any state in any period. Internally the militarization of the Roman empire meant that the army had become a powerful political force and in the Empire's later stages they played a major role in the choice of Emperor.

Though Rome had conquered east and west, this small city on the Alban Hills in the region of Latium was now finding itself stretched. With a large empire came the need to extend citizenship and this in turn led to emperors being drawn from all over the empire during its last years. These strains upon the very fibre of empire were drawing a line between the eastern and western provinces. The fact that the City of Rome had entered a period of decline and the perennial importance of grain imports from Egypt and other eastern provinces meant that Byzantium (now Istanbul, Turkey, and previously called Constantinople - the city of Constantine) would be used to bind the eastern Roman empire to the west. Constantine I, in attempting to use Constantinople as representative of full intergration of the eastern Roman empire into the Latin western empire, had inadvertently set up the ideal conditions for the breaking of the empire. After Constantine's death the empire in the west saw a period of turmoil as his three sons fought each other for control of Rome. Constantius II eventually became sole emperor after his two brothers were assassinated, and raised his cousin Constantius Gallus to the rank of Ceasar of the eastern empire. The empire was now two distinct powers; a fact that would shape the history of the Medieval period and the course of Christianity for the next thousand years.

The diminishing influence and power of Rome led to a retraction of the roman culture that had once dominated the provinces. Throughout Roman territories, non-Italian citizens whose forbears had adopted Rome as the axis of their world began to give more emphasis to their local identities—Gauls, Spaniards, Britons, North Africans filled the void left by the departure of Rome's armies and administration by reasserting their own culture and laws. Pax Romana as decreed by Augustus had already allowed local laws and customs to take precedence over Rome's laws as long as the provinces accepted military and taxation control. In this respect the end of the Roman empire meant that the provinces were now free to organise their own military and economic affairs. However they did not discard all that Rome had offered. Latin had by then become the lingua franca of politics and trade, and the better traits of Roman law were keenly adopted, especially in regards to codification of laws that were then made accessible to the public and applicable to all citizens. Hostile tribes who had once been repelled by Rome's military might now invaded across the Roman frontier, settling in imperial lands. These tribes were not alien to Roman culture with many of their leaders and people having constant contact with Rome through the political and economic machine of empire. After setting up parallel societies in the areas they invaded, it was only natural that they slowly mixed with the existing Roman population. This process went on for two to three centuries and resulted in a sweeping change in the makeup of European society within Rome's old boundaries. The Greek-speaking eastern portion looked now to Byzantium as its centre, and this division of the empire would become permanent in the Europe of later centuries.

Most significant of Constantine's acts as Emperor was to make a death bed conversion to Christianity. Although the empire moved to Constantinople, the new found papacy remained in Rome, as envisaged by Saint Peter - the symbolic first Pope. To some extent papal power became synonymous with that of the Western Emperor. The secular capital of the Western empire though was at Ravenna. Poor leadership and stress from invasion led to the fall of Rome in 410 to the Visigoths. The Western empire itself fell in 476 at Ravenna. The remaining Eastern empire was now referred to as the Byzantine empire, after its capital. Italy would not be a unified state again until the 19th century.

Barbarians Unite
As Roman power moved away, tribes from outside the old borders moved in to fill the vacuum. Visigoths set up a new kingdom in Iberia, while Vandals settled eventually in north Africa. In Britain, Germanic tribes arrived and settled on the east coast in 5CE. These settlers eventually formed small Anglo-Saxon kingdoms which filled the vacuum left by the departure of the Romans. Post-Roman Italy itself came under the sway of the Ostrogoths, whose most influential king, Theoderic, was hailed as a new emperor by the Roman Senate, and had good relations with the Christian pope, but kept his seat of power at Ravenna in northern Italy. Southern Italy and Sicily were under the sway of the Byzantine Emperor for several centuries during this period.

The province of Gaul, which had been the most prosperous of the Western Roman provinces, came into the possession of the Franks. The Roman-Gaulish society and its leaders eventually assimilated with the Franks, relying on their warriors for security and cementing connections of marriage with Frankish clans. The new Frankish kingdom came to include much of modern western Germany and northern France, its power within this region made clear by victories over the Visigoths and other barbarian rivals. Today the German word for France, Frankreich, pays homage to the Frankish kingdom of times past. The Merovingian dynasty, named after the legendary tribal king Merovech, was first to rule over the Frankish realm. Their most skilled and powerful ruler, Clovis, converted to Catholic Christianity in 496 as a promise for victory in battle. At Clovis's death, he divided his kingdom up among his many sons, whose rivalries touched off a century of intermittent and bloody civil war. Some, like Chilperic, were insane, and none were willing to give up lands or power to reunify the kingdom. The fortunes of the Merovingians ebbed and they faded into irrelevance.

The rise of a new ruling power, the Carolingian dynasty, was the result of the expanded power of the Majordomo, or "head of the house". Merovingian kings gave their majordomo extensive power to command and control their estates, and some of them used this power to command and control entire territories. Pepin II was one of the first to expand his power so much that he held power over almost all of Gaul. His son, also a majordomo, Charles Martel, won the Battle of Tours against invading Islamic armies, keeping Muslim influence out of most of Europe. "Martel", meaning "The Hammer", was a reference to his weapon of choice.

Martel's son, Pepin III the Short, after requesting support of the papacy, disposed of the Merovingian "puppets". The papacy gave Pepin permission to overthrow the Merovingians in order to secure Frankish support of the papal states, and protection against Lombard incursion. Pepin was declared rex Dei gratia, "King by the grace of God", thereby setting a powerful precedent for European absolutism, by arguing that it was the Christian God's will to declare someone a king.

(Both Pepin II and Pepin III were known as Pepin the Younger, as Pepin I was Pepin the Elder. However, Pepin III is also known as Pepin the Short, so that is the name that will be used here.)

Pepin the Short was the founder of the Carolingian dynasty, which culminated in his son Charlemagne, "Charles the Great". Charlemagne was also the first king crowned Holy Roman Emperor—the supposed successor to the Caesars and the protector of the Catholic Church. Charlemagne, whose empire expanded by conquest to encompass most of present-day Germany and France, created something of a renaissance for the intellectual world in the Frankish kingdom. Charlemagne set up monasteries and had monks copy out the Bible, in illuminated manuscripts, in rooms called scriptoria. For women, the cloth was one of the few ways they were allowed to expand their intellectual horizons and do something aside from birth children and work in the fields.

Charlemagne's sons followed Germanic tradition after his death and divided his kingdom between them during the 9th century. "East and West Francia" emerged, which in the following centuries would be called France and Germany (the latter known as the Holy Roman Empire). During this time, western Europe's settled regions came under increasing attack by the Vikings or Northmen, independent bands of seagoing warriors from Denmark and Scandinavia whose gods included Thor and Odin, and whose raids on wealthy Christian cities and churches were but one feature of their trade networks, which spanned the Atlantic and Europe from Newfoundland to Byzantium. Viking activity continued until Norway and Sweden reluctantly accepted Christianity in the 11th and 12th centuries. England, Ireland and French Normandy all saw substantial Viking settlement in this period, with important historical consequences.

In Spain, a Visigothic elective kingdom flourished, with continuous conflicts, until 711, when much of Spain fell quickly to the Muslim invaders. A great Muslim empire, called the Califato de Córdoba, flourished culturally and by arms. In the north mountains, small Christian kingdoms, Galiza, Asturias, Navarra and Aragón, persisted. The border with the Frankish empire, called Marca Hispanica, included Barcelona since 801—the origin of the Principality of Catalonia. The Muslims lost their last Spanish kingdom, Granada, in 1492, when the king of Aragón, Fernando, married the Queen of Castille, Isabel, in 1469. The dynastic union, producing grandson Carlos V, maintained the internal borders and different nationalities, laws and institutions of every one of the ancient kingdoms until the 18th century. Navarra was annexed in 1515 and still maintains its own laws and fiscality, as the Basque Country. Italy would begin to split into smaller kingdoms ruled by various different forms of government. However, the papacy would still be able to exert great force over most European people.

Eastern Europe, more thinly populated and more remote from the Roman borders, experienced numerous invasions and migrations during the 6th to 10th centuries. From their homeland in southern Russia, Slavic peoples expanded westward in the wake of the Germanic migrations, settling in the Balkans, Bohemia, Poland and eastern Germany and dividing their allegiances among Rome's successors. The Poles, Czechs, Croats adopted Latin Christianity, while the Bulgarian, Serbian and Russian kings accepted the Greek Byzantine rite. In the middle Bosnians, a small tribe of South Slavs, in highlands of Dinaric Alps, were neither, instead they adopted liturgy of their own and during the 10th century established their Bosnian Church, often confused with Bogomilism since Bosnian rulers showed great deal of tolerance toward them and even protected these people, sometimes with tragic consequences to their own way of life and certainly with historical consequences for entire Balkans. The Magyars, a tribe of mounted warriors ethnically akin to the Huns, entered Europe from the Russian steppes in the 9th century, fought a series of wars with the German emperors who succeeded Charlemagne, and eventually made their home in Hungary as a Latin-Christian kingdom (their king, Stephen, achieving sainthood). The peoples of the Baltic region, such as the Letts and Prussians, remained largely untouched by the Christian expansion during the early Middle Ages. In Russia, a kingdom at Kiev had been formed, incorporating both Slavic peoples and Scandinavian elements.

Great Britain was divided into various kingdoms after the Anglo-Saxon-Jute invasion, until the period of the Danish invasions. King Alfred unified much of England into one kingdom in the late 9th century. On October 10, 1066 William the Bastard, later William the Conqueror, initiated a Norman invasion of England with the Battle of Hastings. French control of England during this period is shown by much of the formal vocabulary in English, which stems from French.

In 1215 the barons of England forced King John to sign the Magna Carta. King John was particularly despotic. He abused his vassals, killed his child nephew Arthur, and brought the wrath of the Church upon the country in the form of a Papal Interdict (no church services were performed: no marriages, funerals, or masses). His barons ultimately united against him. While he was out hunting, they surrounded him giving him the option of signing the Magna Carta or death. The Magna Carta enshrined in law certain rights that the barons were entitled to. Absolute royal power had given way to rule by consensus and the cornerstone for Parliament had been placed.

Romanesque and Gothic Art

Romanesque is the name given to the architectural style found across Europe from the 10th to 13th century. Borrowing elements from Ancient Roman and Oriental Byzantine architecture in a fusion of opulence and utility. These are the churches of a rich and expanding Christian religion. The rise of religous orders and monasticism; the need to cater for larger congregations as populations expanded; the desire for patronage and remembrance as a christian benefactor and the all powerful need to build churches that reflect the beauty and power of religious sentiment are all key factors in Romanesque architecture. You can be in no doubt when entering the Cathedral at Notre Dame that your corporeal flesh is firmly of terra firma but your soul belongs to the realms of heaven. Many churches remain from this period which is a testament to the development of new methods of vaulting in masonry rather than wood. The style of the basilica gives way to loftier expressions of religious architecture. Romanesque architecture is designed to convey the spiritual power and authority of the church. Many castles were also begun at this time and extended later. The chief characteristics of Romanesque architecture are round-headed arches in windows, doors and arcades, rectangular piers, cylindrical columns, barrel vaults, groin vaults and paired windows. Late Romanesque churches may have ribbed vaults. Walls are thick and buttresses are flat. Doorways are often surmounted by sculptured tympanum, particularly with sculpture of Christ in Majesty. Interior wall were often frescoed with scenes from the Bible.

Gothic architecture developed out of Romanesque architecture in France in the early 12th century and spread across Europe, being modified in different regions. It lasted in some parts of Europe into the 1500s. Gothic architecture, as seen in many of the great churches and cathedrals of Europe, is distinguished by the use of the pointed, ribbed vault. Later ribbed vaulting became very complex and decorative, particularly in England, Spain and Eastern Europe. The development of projecting buttresses and flying buttresses allowed openings in the walls to be larger and vaults to be higher. Columns and piers often had a cluster of shafts supporting the ribs and mouldings above. Doors and windows have pointed arches. Doors were often set into richly carved portals and surrounded by figurative sculpture. Windows became increasingly large and were filled with stone tracery supporting the glazing. Stained glass became a more important art form than fresco in most of Europe except Italy.

Manuscript illumination was a major artistic form during the Late Middle Ages. Manuscripts were copied by hand, often by monks working at the scriptoria in a monastery. The books that were thus produced were often illuminated (decorated with illustrations and ornate borders). Often the initial letter on a page was illuminated with colour and small scenes. The style of decoration developed local characteristics, ranging from abstract and floral borders, to detailed landscapes and interior scenes of the Duc du Berry's Tres Riches Heures by the Limbourg brothers.

Poetry and music were carried between cities by travelling musicians called troubadours. These troubadours would give news from across Europe as they travelled and told songs and stories. One of the longest Germanic epic poems, Beowulf tells the tale of the killing of a giant beast that plagued a town. Another of these epics was the Song of Roland (based on the attack on the rear guard of Charlemagne's army by the Basques as Charlemagne led his men out of Spain).
 * Merry and bold is now that Emperour,
 * Cordres he holds, the walls are tumbled down,
 * His catapults have battered town and tow'r.
 * Great good treasure his knights have placed in pound,
 * Silver and gold and many a jewelled gown.
 * In that city there is no pagan now
 * But he been slain, or takes the Christian vow.

Papal Power in the Middle Ages
The papacy was a powerful institution in medieval Europe. Religion permeated every aspect of society and equally applied to king or farm worker. In medieval Europe the Pope was power-broker and peace-maker with immense spiritual authority in an age where the existence of heaven and hell and one's path to either was examined constantly. The shortness of life, diseases, infant mortality and plague all added to a preoccupation with death and the eternal soul; faith was the medieval mind's medicine. The Popes appointed Bishops, received tithes, issued canonical laws throughout Christendom. They could call for Crusades against any enemy of Christendom; raising armies led by the kings and knights of Europe.

This must be seen in context by considering the nature of papal power. Royal power derived from ownership of land, which was essential for raising armies of knights. The papacy held fewer direct assets, and to complicate matters, these could not be passed on by inheritance. This meant that a pope nearing death could only enrich his family by donating Church property, thus siphoning off the assets of the papal court instead of enlarging them. Therefore, the Pope had to rely on other forms of power, and his spiritual authority was always a significant alternative to direct territorial control.

The Pope also attempted through political means to control Europe. He pushed many kings into placing his bishops into powerful secular positions. In response to this European kings chose to put their own bishops into religious positions in a process called lay investiture. Emperor Henry IV sold many bishoprics until he attempted to depose Pope Gregory VII. In response Gregory VII excommunicated him and absolved his vassals of responsibility to him. Henry attempted to hold his kingdom together, but in the face of an angry population terrified for their souls, he went to the Alpine monastery of Canossa to beg the Pope to absolve him, standing outside the walls barefoot in the snow for three days before Gregory VII relented and removed his excommunication. For years the Pope and Holy Roman Empire battled over investiture until Pope Calixtus II and Holy Roman Emperor Henry V sat down and executed the Concordat of Worms. The Concordat gave the Pope the authority to put bishops into religious positions, and the Emperor the authority to put bishops into secular positions. This struggle that ended in the Concordat of Worms was known as The Investiture Controversy.

The ascetic aspects of Christ had always been prevalent in the lives of christians; from St Augustine to St Francis of Assisi the intellectual introspection and rejection of luxuries had always appealed to those seeking to draw closer to Jesus. This withdrawal to common community had existed in the Middle East at the birth of Christianity. The Jewish sect who hid the Dead Sea Scrolls in caves were just one of many early ascetic communities devoting themselves to a religious life. The Medieval period saw the development of ascetic orders based around monasteries. New orders were formed to follow certain religious aims. St Benedict was the first to mandate a way in which all orders should function. The Benedictine order followed very specific rituals. Among these rituals were precise times for each prayer; it is suggested that this led to the development of the clock, which was developed around the time of the Fourth Lateran Council. The Fourth Lateran Council set rules for worship for all monastic orders. These rules were heavily based on those of the Cistercians. Citaeux, the father of the Cistercian order, put all of the monasteries into an "Order", but also allowed them their own regional features. Monks not only developed the intellectual policies of Medieval Europe, but also helped among cities by offering food and services to those in need.

Society in the Middle Ages
Guilds, which controlled who could or could not work in a profession through education and vertical integration, were the first to form universities. Among them were the universities of Salamanca, Paris, and Bologna. Scholasticism, an education philosophy that emphasised teachings of the Bible and Aristotle, was the common form in education. Renaissance brought forth Humanism and the Liberal Arts.

Cities began to form as a place to house shops and guilds. Cities were required as centres for trade, and many of the famous medieval cities of Europe were ports. Within these cities money capitalism began to thrive. In the north, the Gilder became a powerful, almost universal currency. Europeans also reopened long-dormant trade routes with China and the Middle East. The Silk Road was traversed once again, and much of Europe's period of exploration and discovery was meant to find easier and faster routes to the East.

Advances in agriculture were also coming about. Fields became ploughed by large metal grates pulled by draft animals. Europeans also began to leave ground fallow to allow nutrients to come back into the soil, thus starting the concept of crop rotation. The systems of feudalism and manorialism grew with the new farming methods. Feudalism, the political arm of the Manorial system, organized society into a pyramid structure, with Lords ruling, and below them vassals. A vassal would receive land and in return, they would protect, and answer to those above them. Manorialism, the economic end of this system, was the way in which these large properties were managed. Vassals would farm the land and then trade grain to their Lord for protection, access to storage, a mill to grind the grain, and access to ovens to bake bread produced from this grain. Food was thus provided. However, the flour milled from medieval grain was often very coarse and harsh on the teeth. Water supplies were often polluted, so grain was also used to make ale which was the main beverage of northern Europeans, while wine remained the potable of choice for southern Europeans.

História da Europa/Antecedentes da História da Europa vi:Lịch sử châu Âu/Mở đầu:Hoàn cảnh lịch sử châu Âu


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User:Alexbl

Alexbl is interested in working on history texts, and is trying to get a good European_History text started. he also has interests in World History and Art History texts.

Contact: drax@stampede.org AIM: DraXVAZ

title: User:Alexbl


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Cookbook:Sauces

Sauces are thickened liquid additions to a recipe, used to enhance the flavor or appearance, or to add moisture. In the 1800s, the French chefs Antonin Carême and Auguste Escoffier developed a systematic categorization of sauces, based on five "mother sauces". All sauces were considered to be a variation of one of these sauces. As other cuisines and ingredients have become more common to world cuisine, the number of mother sauces and sauce families has expanded slightly, but most sauces can be seen as a variation or adaptation of a few basics.

The sauces below are grouped by their family. The first five groups are the traditional sauce families, with the modern additions following. Except for the Dessert family, the top sauce in each family is that family's "mother sauce". Many of the mother sauces use a roux as the thickening agent. Roux itself is not really a sauce.

White milk-based sauces

 * Béchamel Sauce
 * Alfredo Sauce
 * Cheese Sauce
 * Sauce Mornay
 * Vegetarian "Bacon" Gravy
 * White Sauce

White stock-based sauces

 * Velouté sauce

Brown sauces

 * Espagnole Sauce
 * Ham sauce
 * Mirepoix sauce
 * Mushroom sauce
 * Neapolitan Sauce
 * Roman Sauce

Egg-based emulsions

 * Hollandaise
 * Béarnaise
 * Mock Béarnaise
 * Mayonnaise
 * Louis Dressing
 * Tartar Sauce
 * Thousand Island Dressing

Tomato-based sauces

 * Tomato Sauce
 * Barbecue Sauce
 * Barbecue Sauce - Kansas City style
 * Barbecue Sauce - Sweet
 * Chicken Barbecue Sauce
 * Basic Indian Tomato Gravy
 * Ketchup
 * Marinara Sauce
 * Pico de Gallo
 * Pizza Sauce
 * Puttanesca Sauce
 * Spaghetti Sauce
 * Tomato Pasta
 * Tomato Sauce Piquante
 * Salsa

Purees

 * Pesto
 * Coconut Chutney
 * English Field Bean Pate
 * Guacamole
 * Onion Chutney
 * Peanut Sauce
 * Red Chile Sauce
 * Saus Kacang - Indonesian Peanut Sauce
 * Sriracha

Fermented sauces

 * Soy Sauce
 * Fish Sauce (Vegan)
 * Nước maøu
 * Worcestershire Sauce

Vinaigrettes

 * Vinaigrette
 * Beurre Blanc
 * Caramelized Sweet and Sour Onion Marmalade
 * Champagne Vinaigrette
 * French Dressing
 * Honey Mustard Salad Dressing
 * Mango Atchar
 * Neapolitan Anchovy Sauce
 * Red Pepper and Goat Cheese sauce
 * Senf - German Style Sweet Mustard
 * Sweet and Sour

Dessert sauces

 * Black Pepper Syrup
 * Caramel Sauce
 * Creme Anglaise
 * Hot Fudge
 * Lemon Curd

title: Cookbook:Sauces


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Cookbook talk:Household Cyclopedia Sauces

I moved this page from Sauces to Household Cyclopedia Sauces to enable the creation of a Sauces recipe index page. I've linked to this page from Sauces. loum 06:38, 29 May 2004 (UTC)


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Computer Programming/Physics

About This Document
I have programmed a (working) 2D physics system before and I am currently in college as a Physics Engineering major, so I have experience, but I am by no means an industry expert, so I'm going to start this document and hope that either someone who knows this stuff steps in and takes over or guides me so that I can do this correctly.

Also, if my formatting looks bad, any one can feel free to change it.

`External Party - Added more to the portion of physics. Hopefully not too lengthy or off-topic.

= About Physics =

Computer programs today simulate physical systems for various purposes, such as realism in games and testing mechanical prototypes before construction. Added with the current trend of increasing processing power, these physical simulations are continually becoming more complicated and elaborate. As such where physics is applicable, programs without these capabilities appear more simple and primitive and thus less desirable compared to programs with these abilities.

Not all programs require the full range of computational capabilities. Most games only require the ability to accurately calculate mechanical processes. Programs like the ones used to test objects need more complex abilities, like calculating the heat given off by the friction between two surfaces. Some programs may even find it desirable to perform pseudo-calculations, calculations based upon physics yet tweaked to achieve a more desirable behavior, or they may just represent a very narrow range of processes that can occur.

When programming physics, various different mechanisms of physics, including branching decisions such as collisions between objects, gravity and any other exertional forces taking effect upon objects present, along with various other elements depending upon the given situation, need to be taken into account and compensated for accordingly.

Each different object has its own mass/weight, which is usually dependent upon the size of the object itself. Effectively programming physics generally entails keeping track of the weight of each object, factoring in other forces such as gravity and collisions while calculating the force of each collision, depending on the object weights and perform an action accordingly.

One example of the weight of objects taking an effect on the world around them would be where, when placed upon another object held up by a weak force would then cause that object to lower towards the ground. The types of physics examples that could be programmed to demonstrate this are virtually endless and, for beginners, it is recommended that they attempt basic examples of physics like gravity.

= Concepts and Programming in Physics =

Prerequisites
To program for physics or using physics one should be familiar with some widely-applicable concepts and techniques:

 * Algebra
 * Coordinate Systems
 * 2D
 * Cartesian (x, y)
 * polar - distance and direction (r, θ)
 * parametric
 * 3D
 * Cartesian (x, y, z)
 * polar - distance and two angles (ρ, θ, φ)
 * parametric
 * (Higher dimensions to be addressed under "Relativity")
 * Trigonometry
 * Series Approximations (Pre-Calculus)
 * Numerical Methods for Calculus and Differential Equations
 * Iteration
 * Matrices and Matrix Math
 * Determinant
 * Matrix Multiplication
 * Boundary Conditions
 * Complex Math (involving $$\sqrt{-1}$$)
 * Addition of Complex Numbers
 * Multiplication of Complex Numbers
 * Complex Exponentiation and Logarithms
 * Hyperbolic Trigonometry
 * Complex Exponentiation and Logarithms
 * Hyperbolic Trigonometry

Mechanics

 * Frames of Reference
 * Vectors
 * Vector Addition
 * Dot Product
 * Cross Product
 * Matrix Product
 * Forces
 * Momentum
 * Energy
 * Collisions and Geometric Forces
 * Simple Harmonic Motion, Springs
 * Friction
 * Static Friction
 * Kinetic Friction
 * Rolling, Wheels
 * Friction in Air and Other Fluids
 * Hydrodynamics
 * Computer programming/Physics/Position of an accelerating body function
 * Computer programming/Physics/Position of an accelerating body function (constant acceleration)
 * Computer programming/Physics/Motion of an object through space (piecemeal approximation)

Thermodynamics

 * Brownian Motion
 * Heat and Temperature
 * "Zeroth Law"
 * First Law
 * Second Law
 * Third Law

title: Computer Programming/Physics


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Using GNOME/Font

< ../Preferences

GNOME makes changing fonts easy. Here's how.

First, Select Menu, Preferences, Font.

Font types
There are Four types of font : Application Font, Desktop Font, Window Title Font and Terminal Font. To change them, just click them.

Rendering
Rendering allows you to control the look of the fonts

Details
Details allow you to control advanced properties of the fonts.

title: Using GNOME/Font


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Civ/Civilization

This is a guide to the original computer game Civilization, designed by Sid Meier.

Table of Contents

 * The Basics

Appendices

 * /Forms of Government/
 * /Unit Index/
 * /Patch History/

title: Civ/Civilization