Algebra

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

Learning projects

Pre-University Level Courses

• Pre-Algebra
• Speak Math Now!
• College Algebra

Undergraduate Level Courses

• Introduction to Linear Algebra
• Introduction to Logic and Proofs
• Abstract Linear Algebra
• Introduction to Group Theory

Graduate Level Courses

• Higher Algebra
  • Group Theory
  • Ring Theory
  • Virasoro Algebra

Algebra Resources

Wikiversity

• Continuum mechanics/Tensor algebra identities
• Portal:Mathematics
• Powers

Wikipedia

• Algebra
• Geometry
• Pre-algebra

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.

Addition

• 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.