# Mathematical Programming (Manipulation) Tips

Commutative means that the order of elements contained within an expression do not alter a  certain mathematical result.

For example:

$2\cdot 3=3\cdot 2=6\\ \\ 1\cdot 2\cdot 3=2\cdot 1\cdot 3=3\cdot 1\cdot 2=6\\ \\ Or:\\ \\ 1+2=2+1=3\\ \\ 1+2+3=3+2+1=2+1+3=6$

Knowing this, you can say that:

$a\cdot b\cdot c=c\cdot a\cdot b=b\cdot a\cdot c=c\cdot b\cdot a\\ \\ Or:\\ \\ a+b+c=c+a+b=b+a+c=c+b+a$

—————————————————————–

Associative means that you can alter the grouping of certain elements in an expression without changing its result.

$\left( 2\cdot 3 \right) \cdot 4=2\cdot \left( 3\cdot 4 \right) =24\\ \\ Or:\\ \\ \left( 2+3 \right) +4=2+\left( 3+4 \right) =9$

Knowing this you could say that:

$\left( a\cdot b \right) \cdot c=a\cdot \left( b\cdot c \right) \\ \\ Or:\\ \\ \left( a+b \right) +c=a+\left( b+c \right)$

——————————————————————

Warning:

Commutative and Associative manipulation should not be used when subtracting  and dividing.