Elementary algebra

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Introduction

For this introduction, knowledge of arithmetic (including the use of parentheses) is assumed. We will use * to denote multiplication and / to denote division.

Algebra is a topic in mathematics covering the process of manipulation mathematical statements, expressions and equations. Specifically, while the process of arithmetics computes a well formed numeric expression to a simplified result, algebra adds the notion of variables and statements of equality. For example, if one is asked to compute:

2 + 7 * 3

one will arrive at

23

as the result of the computation. The phrases "2 + 7 * 3" and "23" are examples of mathematical expressions. If one expression is the result of a computation of another expression, the two can be concatenated with an "=" in between then to form a true mathematical statement. In this case:

2 + 7 * 3 = 23

Now, we also know that:

5 * 5 - 2

gives the result of

23

So we also have the following true mathematical statement:

5 * 5 - 2 = 23

Using a technique called "transitivity" which can be applied to true mathematical statements of these forms we can combine the two statements to produce the following true statement:

2 + 7 * 3 = 5 * 5 - 2

The notion of variables allows us to make a group of expressions or statements. For example a statement like:

2 + y * 3

Simultaneously represents the following expressions: "2 + 1 * 3", "2 + 2 * 3", "2 + 3 * 3", ... and so on (where the symbol y is representing the values 1, 2, 3, ... in each respective instance.) In the notion of algebra, although this is a multitude of expressions, it is ordinarily thought of as a single expression on its own level. Now if we combine two expressions thusly:

2 + y * 3 = 5 * 5 - 2

then we have an algebraic statement. For the moment, we can think of this as the following collection of statements:

2 + 1 * 3 = 5 * 5 - 2

2 + 2 * 3 = 5 * 5 - 2

2 + 3 * 3 = 5 * 5 - 2

...

etc. Now these statements were not formed as a result of any computation, but rather simply by considering all the expressions that "2 + y * 3" represents. Now we know that the statement

2 + 7 * 3 = 5 * 5 - 2

is a true mathematical statement. Thus the value of 7 for y, can generate this true statement. The other statements listed above are not formed from equivalent calculations, and in fact cannot be. Those statements are false mathematical statement. So 7 is the only value of y that leads to a true mathematical statement.

Ordinarily, if one asserts a mathematical statement such as:

2 + y * 3 = 5 * 5 - 2

it is understood that we are only interested in the true instances of this statement. True statements are also called equations. As we noted above, the value 7 being what y is, is the only way to make this a true statement, so one can immediately follow this up with the following true statement:

y = 7

The process of deducing values for variables from an equation, is called solving an equation.

The mechanics of algebra typically involves the manipulation of equations or expressions to form other equations or expressions, or to solve the equations. However the mechanics for all these processes is the same. Remembering that we have the equations:

5 * 5 - 2 = 23

and

2 + y * 3 = 5 * 5 - 2

from above, we can again use the transitivity property to arrive at the following equation:

2 + y * 3 = 23

Now, for this to be true, we know that the value for y will have to be such that the computation of the expression "2 + y * 3" will yield the result of 23. So if we consider the expression "23 - 2" we can also write it as "(2 + y * 3) - 2". So we have

(2 + y * 3) - 2 = 23 - 2

Which (using arithmetic: ((a + b) - c is the same as (a + (b - c)) and transitivity) is then same as

y * 3 + 2 - 2 = 23 - 2

Which (using arithmetic: 2 - 2 is the same as 0, and 23 - 2 is 21 and transitivity) is the same as

y * 3 + 0 = 21

or just

y * 3 = 21

Now if we consider the expression 21/3 we have:

(y * 3)/3 = 21/3

or

y * (3/3) = 7

or

y * 1 = 7

or just

y = 7

which matches our original deduction for y above.

Expressions or statement may contain many variables, from which you may or may not be able to deduce the values for some of the variables. For example:

(x - 1) * (x - 1) = y * 0

After some algebraic steps (not covered here), we can deduce that x = 1, however we cannot deduce what the value of y is. Try some values of x and y (which may lead to either true or false statements) to get a feel for this.


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