# 12.2: Properties of Real Numbers

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## Commutative Property

$$a + b = b + a$$

$$3 + 4 = 4 + 3$$

Multiplication

$$ab = ba$$

$$4 \cdot 3 = 3 \cdot 4$$

## Associative Property

$$a+(b+c)=(a+b)+c$$

$$4+(3+5)=(4+3)+5$$

Multiplication

$$a(bc)=(ab)c$$

$$4(3⋅5)=(4⋅3)5$$

## Distributive Property

$$a(b+c)=ab+ac$$

$$4(x+3)=4x+12$$

$$(b+c)a=ab+bc$$

$$(x+3)4=4x+12$$

## Properties of Zero

$$a \cdot 0 = 0$$

$$0 \cdot a = 0$$

If $$a =\not = 0$$, then $$\dfrac{0}{a} = 0$$ and $$\dfrac{a}{0} = 0$$ is undefined.

## Double Negative Property

$$-(-a) = a$$

This page titled 12.2: Properties of Real Numbers is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .