Maths - Rules when working of equations

There are a set of rules which can be applied to parts of an equation which will not affect the truth of the equation.

rule name description
x + y = y + x commutative - additive An operation is commutative if the order of its operands can be reversed without effecting the result
x * y = y * x commutative - multiplicative
x + (y + z) = (x + y) + z associative - additive An operation is associative if the order of doing multiple operations is not important
x * (y * z) = (x * y) * z associative - multiplicative
x * (y + z) = (x * y) + (x * z) distributive When there are two operations in this algebra say, + and *, then * is said to be distributive over +
x + 0 = x identity operator - additive (right) adding 0 does not change result
0 + x = x identity operator - additive (left)
x * 1 = x identity operator - multiplicative (right) multiplying by 1 does not change result
1 * x = x identity operator - multiplicative (left)

if z = x + y

then x = z - y

subtracting is the inverse of adding subtract y from both sides of the equation (this is like adding -y)

if z = x * y

then x = z / y

dividing is the inverse of multiplying divide both both sides of the equation by y (this is like multiplying by 1/y)

where: x,y and z can be numbers, variables, any expression enclosed in brackets, or any expression that could be enclosed in brackets without changing the value of the equation.

These rules all apply to real number algebra, some of these rules don't apply in some other algebras. For example in matrix and quaternion algera the commutative (multiplicative) rule does not apply.

Algebras

  * distributive over + Multiplication commutative Multiplication associative
Real Numbers yes yes yes
Complex Numbers yes yes yes
Quaternions yes no yes
Octonions yes no no
Vectors yes no no
Matricies yes no yes
Multi Vectors yes no no


For further information see group theory.


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