Introduction to Real Numbers: Learn It 3

Properties of Real Numbers

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

The commutative property of addition states that numbers may be added in any order without affecting the sum.

[latex]a+b=b+a[/latex]
For example: [latex]\left(-2\right)+7=5\text{ and }7+\left(-2\right)=5[/latex]

 

Similarly, 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]
For example: [latex]\left(-11\right)\cdot\left(-4\right)=44\text{ and }\left(-4\right)\cdot\left(-11\right)=44[/latex]

commutative properties

Property Example In Words
Commutative Property of Addition
[latex]a + b = b + a[/latex]		 [latex]3 + 7 = 7 + 3[/latex] Numbers can be added in any order
Commutative Property of Multiplication
[latex]a \times b = b \times a[/latex]		 [latex]10 \times 4 = 4 \times 10[/latex] Numbers can be multiplied in any order
It is important to note that neither subtraction nor division is commutative. For example, [latex]17 - 5[/latex] is not the same as [latex]5 - 17[/latex]. Similarly, [latex]20\div 5\ne 5\div 20[/latex].

Associative Properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

[latex]a\left(bc\right)=\left(ab\right)c[/latex]

Consider this example: [latex]\left(3\cdot4\right)\cdot5=60\text{ and }3\cdot\left(4\cdot5\right)=60[/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]

This property can be especially helpful when dealing with negative integers.

Consider this example: [latex][15+\left(-9\right)]+23=29\text{ and }15+[\left(-9\right)+23]=29[/latex]

associative properties

Property Example In Words
Associative Property of Addition
[latex]a + (b + c) = (a + b) + c[/latex]		 [latex]4 + (3 + 8) = (4 + 3) + 8[/latex] Doesn’t matter which pair of numbers is added first
Associative Property of Multiplication
[latex]a \times (b \times c) = (a \times b) \times c[/latex]		 [latex]2 \times (5 \times 7) = (2 \times 5) \times 7[/latex] Doesn’t matter which pair of numbers is multiplied first
Are subtraction and division associative?To answer this, review these 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]
As we can see, neither subtraction nor division is associative.