$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 2.4: The Distributive property of Multiplication over Addition and/or Subtraction

• • Contributed by David Guichard
• Professor (Mathematics) at Whitman College
$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## The Distributive property of Multiplication over Addition and/or Subtraction

(0,8)(5,-5) (10,0)$$\overbrace{5\ \cdot\ (4}\ +\ 3)\ =(5\ \cdot \ 4\ )\ +\ (5\ \cdot \ 3 )$$ (11.05,1.5)(0,1)1.5 (11.05,3.0)(1,0)5.5 (16.5,3.0)(0,-1)2.0 (10.1,-1.5)(1,0)3.2 (10.1,-0.5)(0,-1)1.0 (13.3,-0.5)(0,-1)1.0 (11.7,-1.5)(0,-1)1.0 (11.7,-2.5)(1,0)8.8 (20.4,-2.5)(0,1)2.2

$$5(4+3)=5(7)=35$$
and
$$(5)(4)+(5)(3)=20+15=35$$
also.

The picture illustrates the Distributive property of multiplication over addition.

In general

$\fbox{\Large \boldmath a(b+c)=ab+ac}$
where $$a$$, $$b$$, and $$c$$ are any real numbers.

A common mistake:

(0,4)(0,-5) (0,-2)$$x(y\pm z)=xy\pm z$$ (0,-4)$$5(7+2)=5(7)+2$$ (0,-6)$$5(9)=35+2$$ (0,-8)$$45=37$$ (0,-8)(1,1)7 (0,-1)(1,-1)7

***************************************************
The picture illustrates the Distributive property of multiplication over Subtraction.

(0,8)(10,-4) (10,0)$$\overbrace{5\ \cdot\ (4\ }-\ 3)\ =(5\ \cdot \ 4\ )\ -\ (5\ \cdot \ 3 )$$ (10.4,-1.4)(0,1)1 (10.4,-1.4)(1,0)4.7 (15.0,-1.4)(0,1)1 (25.6,-2.5)(0,1)2 (13.4,-2.4)(0,1)1 (13.4,-2.6)(1,0)12.3

(11.6,1.5)(0,1)1.1 (11.7,2.5)(1,0)7.6 (19.4,2.5)(0,-1)1.0

$$5(4-3)=5(1)=5$$ and $$(5)(4)-(5)(3)=20-15=5$$ also.

In general

$\fbox{\large \boldmath a(b+c)=ab+ac}$
[-15pt]

where $$a$$, $$b$$, and $$c$$ are any real numbers.

***************************************************
Waning:

You cannot distribute multiplication over multiplication.

(0,8)(10,-4) (10,0)$$\overbrace{5\ \cdot\ (4\ }\cdot \ 3)\ =(5\ \cdot \ 4\ )\ \cdot\ (5\ \cdot \ 3 )$$ (10.4,-1.4)(0,1)1 (10.4,-1.4)(1,0)4.3 (14.7,-1.4)(0,1)1 (24.4,-2.5)(0,1)2 (13.4,-2.4)(0,1)1 (13.4,-2.6)(1,0)11.1 (11.6,1.5)(0,1)1.1 (11.7,2.5)(1,0)7.6 (19.4,2.5)(0,-1)1.0 (10,-3)(5,2)15 (10,3)(5,-2)15

$$5(4\cdot3)=5(12)=60$$ (Left side only)
and
$$(5)(4)\cdot(5)(3)=20\cdot 15=300$$ (Right side only)
The example shows that the distributive property does not apply to multiplication over multiplication.

***************************************************

(0,8)(-10,-4) (9.8,0)$$\overbrace{16\ \div}\ (4\ \div\ 2)\ =(16\ \div \ 4\ )\ \div\ (16\ \div \ 2 )$$ (14.2,-0.5)(0,-1)1 (14.2,-1.5)(1,0)8.7 (23.0,-1.5)(0,1)1 (10.9,1.5)(0,1)2 (10.9,2.5)(1,0)2.0 (12.9,2.5)(0,-1)1 (10.9,3.5)(1,0)4.5 (15.3,3.5)(0,-1)2.0

$$16\div (4\div 2)=16\div 2=8$$
and
$$(16\div 4)\div(16\div 2)=4\div 8=\displaystyle \frac{1}{2}$$
which is different.

The example shows that the distributive property does not apply to division over division.

***************************************************