- Use the commutative and associative properties of numbers to solve math problems.
For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.
Commutative Properties
[latex]a+b=b+a[/latex]
The commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.
[latex]a\cdot b=b\cdot a[/latex]
- [latex]\left(-2\right)+7=5\text{ and }7+\left(-2\right)=5[/latex]
- [latex]\left(-11\right)\cdot\left(-4\right)=44\text{ and }\left(-4\right)\cdot\left(-11\right)=44[/latex]
It is important to note that neither subtraction nor division is commutative.
Non-examples:
- [latex]17 - 5[/latex] is not the same as [latex]5 - 17[/latex].
- [latex]20\div 5\ne 5\div 20[/latex].
Associative Properties
[latex]a\left(bc\right)=\left(ab\right)c[/latex]
The associative property of addition tells us that numbers may be grouped differently without affecting the sum.
[latex]a+\left(b+c\right)=\left(a+b\right)+c[/latex]
- [latex]\left(3\cdot4\right)\cdot5=60\text{ and }3\cdot\left(4\cdot5\right)=60[/latex]
- [latex][15+\left(-9\right)]+23=29\text{ and }15+[\left(-9\right)+23]=29[/latex]
Non-examples:
-
[latex]\begin{align}8-\left(3-15\right) & \stackrel{?}{=}\left(8-3\right)-15 \\ 8-\left(-12\right) & \stackrel{?}=5-15 \\ 20 & \neq 20-10 \\ \text{ }\end{align}[/latex]
-
[latex]\begin{align}64\div\left(8\div4\right)&\stackrel{?}{=}\left(64\div8\right)\div4 \\ 64\div2 & \stackrel{?}{=}8\div4 \\ 32 & \neq 2 \\ \text{ }\end{align}[/latex]
Note: neither subtraction nor division is associative.