Properties of Real Numbers

Learning Objectives

  • Use properties of real numbers to simplify expressions

A General Note: Properties of Real Numbers

The following properties hold for real numbers a, b, and c.

Addition Multiplication
Commutative Property [latex]a+b=b+a[/latex] [latex]a\cdot b=b\cdot a[/latex]
Associative Property [latex]a+\left(b+c\right)=\left(a+b\right)+c[/latex] [latex]a\left(bc\right)=\left(ab\right)c[/latex]
Distributive Property [latex]a\cdot \left(b+c\right)=a\cdot b+a\cdot c[/latex]
Identity Property There exists a unique real number called the additive identity, 0, such that, for any real number a

[latex]a+0=a[/latex]
 There exists a unique real number called the multiplicative identity, 1, such that, for any real number a

[latex]a\cdot 1=a[/latex]
Inverse Property Every real number a has an additive inverse, or opposite, denoted [latex]–a[/latex], such that

[latex]a+\left(-a\right)=0[/latex]
 Every nonzero real number a has a multiplicative inverse, or reciprocal, denoted [latex]\Large\frac{1}{a}[/latex], such that

[latex]a\cdot \left(\Large\frac{1}{a}\normalsize\right)=1[/latex]

These properties allow us to simplify expressions. We simplify an expression by removing grouping symbols and combining like terms. Like terms have exactly the same variable factors. For example, [latex]3x[/latex] and [latex]5x[/latex] are like terms, because they each have exactly one factor of [latex]x[/latex]. On the other hand, [latex]3x^2[/latex] and [latex]5x[/latex] are not like terms because [latex]3x^2[/latex] has two factors of [latex]x[/latex] while [latex]5x[/latex] has just one factor of [latex]x[/latex]. The constant factor in a term is called its coefficient. The coefficient of [latex]3x[/latex] is [latex]3[/latex]. The coefficient of [latex]5x[/latex] is [latex]5[/latex]. The distributive property lets us combine like terms by adding their coefficients.

[latex]3x+5x=(3+5)x=8x[/latex]

Example

Simplify each expression.

  1. [latex]3x+5+4x-1[/latex]
  2. [latex]2x+3(5x-2)[/latex]

In the following video you will be shown how to combine like terms using the idea of the distributive property.

Example

Combine like terns:

[latex]3x^2-5x-2+x^2+7x-3[/latex]

In the video that follows, you will be shown another example of combining like terms.