# Algebra

Algebra is an ancient form of mathematical analytical methodology and is one of the most fundamental in our modern practice of analysis.

## Digits

Numbers are made of digits. Here are their names:

0 - zero
1 - one
2 - two
3 - three
4 - four
5 - five
6 - six
7 - seven
8 - eight
9 - nine


## Rules of arithmetic and algebra

The following laws are true for all $a,b,c$ whether these are numbers, variables, functions, or more complex expressions involving numbers, variable and/or functions.

• Commutative Law: $a+b=b+a$ .
• Associative Law: $(a+b)+c=a+(b+c)$ .
• Additive Identity: $a+0=a$ .
• Additive Inverse: $a+(-a)=0$ .

### Subtraction

• Definition: $a-b=a+(-b)$ .

### Multiplication

• Commutative law: $a\times b=b\times a$ .
• Associative law: $(a\times b)\times c=a\times (b\times c)$ .
• Multiplicative identity: $a\times 1=a$ .
• Multiplicative inverse: $a\times {\frac {1}{a}}=1$ , whenever $a\neq 0$ • Distributive law: $a\times (b+c)=(a\times b)+(a\times c)$ .

### Division

• Definition: ${\frac {a}{b}}=a\times {\frac {1}{b}}$ , whenever $b\neq 0$ .

Let's look at an example to see how these rules are used in practice.

 ${\frac {(x+2)(x+3)}{x+3}}$ $=\left[(x+2)\times (x+3)\right]\times \left({\frac {1}{x+3}}\right)$ (from the definition of division) $=(x+2)\times \left[(x+3)\times \left({\frac {1}{x+3}}\right)\right]$ (from the associative law of multiplication) $=((x+2)\times (1)),\qquad x\neq -3$ (from multiplicative inverse) $=x+2,\qquad x\neq -3$ (from multiplicative identity)

Of course, the above is much longer than simply cancelling $x+3$ out in both the numerator and denominator. But, when you are cancelling, you are really just doing the above steps, so it is important to know what the rules are so as to know when you are allowed to cancel. Occasionally people do the following, for instance, which is incorrect:

${\frac {2\times (x+2)}{2}}={\frac {2}{2}}\times {\frac {x+2}{2}}=1\times {\frac {x+2}{2}}={\frac {x+2}{2}}$ .

The correct simplification is

${\frac {2\times (x+2)}{2}}=\left(2\times {\frac {1}{2}}\right)\times (x+2)=1\times (x+2)=x+2$ ,

where the number $2$ cancels out in both the numerator and the denominator.