# Multiplication and division of real numbers

 Page 1 / 1
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to multiply and divide signed numbers. By the end of the module students should be able to multiply and divide signed numbers and be able to multiply and divide signed numbers using a calculator.

## Section overview

• Multiplication of Signed Numbers
• Division of Signed Numbers
• Calculators

## Multiplication of signed numbers

Let us consider first, the product of two positive numbers. Multiply: $\text{3}\cdot \text{5}$ .

$\text{3}\cdot \text{5}$ means $5+5+5=\text{15}$

This suggests In later mathematics courses, the word "suggests" turns into the word "proof." One example does not prove a claim. Mathematical proofs are constructed to validate a claim for all possible cases. that

$\left(\text{positive number}\right)\cdot \left(\text{positive number}\right)=\left(\text{positive number}\right)$

More briefly,

$\left(+\right)\left(+\right)=\left(+\right)$

Now consider the product of a positive number and a negative number. Multiply: $\left(3\right)\left(-5\right)$ .

$\left(3\right)\left(-5\right)$ means $\left(-5\right)+\left(-5\right)+\left(-5\right)=-\text{15}$

This suggests that

$\left(\text{positive number}\right)\cdot \left(\text{negative number}\right)=\left(\text{negative number}\right)$

More briefly,

$\left(+\right)\left(-\right)=\left(-\right)$

By the commutative property of multiplication, we get

$\left(\text{negative number}\right)\cdot \left(\text{positive number}\right)=\left(\text{negative number}\right)$

More briefly,

$\left(-\right)\left(+\right)=\left(-\right)$

The sign of the product of two negative numbers can be suggested after observing the following illustration.

Multiply -2 by, respectively, 4, 3, 2, 1, 0, -1, -2, -3, -4.

We have the following rules for multiplying signed numbers.

## Rules for multiplying signed numbers

Multiplying signed numbers:
1. To multiply two real numbers that have the same sign , multiply their absolute values. The product is positive.
$\left(+\right)\left(+\right)=\left(+\right)$
$\left(-\right)\left(-\right)=\left(+\right)$
2. To multiply two real numbers that have opposite signs , multiply their abso­lute values. The product is negative.
$\left(+\right)\left(-\right)=\left(-\right)$
$\left(-\right)\left(+\right)=\left(-\right)$

## Sample set a

Find the following products.

$\text{8}\cdot \text{6}$

$\left(\begin{array}{ccc}|8|& =& 8\\ |6|& =& 6\end{array}}$ Multiply these absolute values.

$8\cdot 6=\text{48}$

Since the numbers have the same sign, the product is positive.

Thus, $8\cdot 6\text{=+}\text{48}$ , or $8\cdot 6=\text{48}$ .

$\left(-8\right)\left(-6\right)$

$\left(\begin{array}{ccc}|-8|& =& 8\\ |-6|& =& 6\end{array}}$ Multiply these absolute values.

$8\cdot 6=\text{48}$

Since the numbers have the same sign, the product is positive.

Thus, $\left(-8\right)\left(-6\right)\text{=+}\text{48}$ , or $\left(-8\right)\left(-6\right)=\text{48}$ .

$\left(-4\right)\left(7\right)$

$\left(\begin{array}{ccc}|-4|& =& 4\\ |7|& =& 7\end{array}}$ Multiply these absolute values.

$4\cdot 7=\text{28}$

Since the numbers have opposite signs, the product is negative.

Thus, $\left(-4\right)\left(7\right)=-\text{28}$ .

$6\left(-3\right)$

$\left(\begin{array}{ccc}|6|& =& 6\\ |-3|& =& 3\end{array}}$ Multiply these absolute values.

$6\cdot 3=\text{18}$

Since the numbers have opposite signs, the product is negative.

Thus, $6\left(-3\right)=-\text{18}$ .

## Practice set a

Find the following products.

$3\left(-8\right)$

-24

$4\left(\text{16}\right)$

64

$\left(-6\right)\left(-5\right)$

30

$\left(-7\right)\left(-2\right)$

14

$\left(-1\right)\left(4\right)$

-4

$\left(-7\right)7$

-49

## Division of signed numbers

To determine the signs in a division problem, recall that

$\frac{\text{12}}{3}=4$ since $\text{12}=3\cdot 4$

This suggests that

## $\frac{\left(+\right)}{\left(+\right)}=\left(+\right)$

$\frac{\left(+\right)}{\left(+\right)}=\left(+\right)$ since $\left(+\right)=\left(+\right)\left(+\right)$

What is $\frac{\text{12}}{-3}$ ?

$-\text{12}=\left(-3\right)\left(-4\right)$ suggests that $\frac{\text{12}}{-3}=-4$ . That is,

## $\frac{\left(+\right)}{\left(-\right)}=\left(-\right)$

$\left(+\right)=\left(-\right)\left(-\right)$ suggests that $\frac{\left(+\right)}{\left(-\right)}=\left(-\right)$

What is $\frac{-\text{12}}{3}$ ?

$-\text{12}=\left(3\right)\left(-4\right)$ suggests that $\frac{-\text{12}}{3}=-4$ . That is,

## $\frac{\left(-\right)}{\left(+\right)}=\left(-\right)$

$\left(-\right)=\left(+\right)\left(-\right)$ suggests that $\frac{\left(-\right)}{\left(+\right)}=\left(-\right)$

What is $\frac{-\text{12}}{-3}$ ?

$-\text{12}=\left(-3\right)\left(4\right)$ suggests that $\frac{-\text{12}}{-3}=4$ . That is,

## $\frac{\left(-\right)}{\left(-\right)}=\left(+\right)$

$\left(-\right)=\left(-\right)\left(+\right)$ suggests that $\frac{\left(-\right)}{\left(-\right)}=\left(+\right)$

We have the following rules for dividing signed numbers.

## Rules for dividing signed numbers

Dividing signed numbers:
1. To divide two real numbers that have the same sign , divide their absolute values. The quotient is positive.
$\frac{\left(+\right)}{\left(+\right)}=\left(+\right)$ $\frac{\left(-\right)}{\left(-\right)}=\left(+\right)$
2. To divide two real numbers that have opposite signs , divide their absolute values. The quotient is negative.
$\frac{\left(-\right)}{\left(+\right)}=\left(-\right)$ $\frac{\left(+\right)}{\left(-\right)}=\left(-\right)$

## Sample set b

Find the following quotients.

$\frac{-\text{10}}{2}$

$\left(\begin{array}{ccc}|-10|& =& 10\\ |2|& =& 2\end{array}}$ Divide these absolute values.

$\frac{\text{10}}{2}=5$

Since the numbers have opposite signs, the quotient is negative.

Thus $\frac{-\text{10}}{2}=-5$ .

$\frac{-\text{35}}{-7}$

$\left(\begin{array}{ccc}|-35|& =& 35\\ |-7|& =& 7\end{array}}$ Divide these absolute values.

$\frac{\text{35}}{7}=5$

Since the numbers have the same signs, the quotient is positive.

Thus, $\frac{-\text{35}}{-7}=5$ .

$\frac{\text{18}}{-9}$

$\left(\begin{array}{ccc}|18|& =& 18\\ |-9|& =& 9\end{array}}$ Divide these absolute values.

$\frac{\text{18}}{9}=2$

Since the numbers have opposite signs, the quotient is negative.

Thus, $\frac{\text{18}}{-9}=2$ .

## Practice set b

Find the following quotients.

$\frac{-\text{24}}{-6}$

4

$\frac{\text{30}}{-5}$

-6

$\frac{-\text{54}}{\text{27}}$

-2

$\frac{\text{51}}{\text{17}}$

3

## Sample set c

Find the value of $\frac{-6\left(4-7\right)-2\left(8-9\right)}{-\left(4+1\right)+1}$ .

Using the order of operations and what we know about signed numbers, we get,

$\begin{array}{ccc}\hfill \frac{-6\left(4-7\right)-2\left(8-9\right)}{-\left(4+1\right)+1}& =& \frac{-6\left(-3\right)-2\left(-1\right)}{-\left(5\right)+1}\hfill \\ & =& \frac{18+2}{-5+1}\hfill \\ & =& \frac{20}{-}\hfill \\ & =& -5\hfill \end{array}$

## Practice set c

Find the value of $\frac{-5\left(2-6\right)-4\left(-8-1\right)}{2\left(3-\text{10}\right)-9\left(-2\right)}$ .

14

## Calculators

Calculators with the key can be used for multiplying and dividing signed numbers.

## Sample set d

Use a calculator to find each quotient or product.

$\left(-\text{186}\right)\cdot \left(-\text{43}\right)$

Since this product involves a $\left(\text{negative}\right)\cdot \left(\text{negative}\right)$ , we know the result should be a positive number. We'll illustrate this on the calculator.

 Display Reads Type 186 186 Press -186 Press × -186 Type 43 43 Press -43 Press = 7998

Thus, $\left(-\text{186}\right)\cdot \left(-\text{43}\right)=7,\text{998}$ .

$\frac{\text{158}\text{.}\text{64}}{-\text{54}\text{.}3}$ . Round to one decimal place.

 Display Reads Type 158.64 158.64 Press ÷ 158.64 Type 54.3 54.3 Press -54.3 Press = -2.921546961

Rounding to one decimal place we get -2.9.

## Practice set d

Use a calculator to find each value.

$\left(-51\text{.}3\right)\cdot \left(-21\text{.}6\right)$

1,108.08

$-2\text{.}\text{5746}÷-2\text{.}1$

1.226

$\left(0\text{.}\text{006}\right)\cdot \left(-0\text{.}\text{241}\right)$ . Round to three decimal places.

-0.001

## Exercises

Find the value of each of the following. Use a calculator to check each result.

$\left(-2\right)\left(-8\right)$

16

$\left(-3\right)\left(-9\right)$

$\left(-4\right)\left(-8\right)$

32

$\left(-5\right)\left(-2\right)$

$\left(3\right)\left(-\text{12}\right)$

-36

$\left(4\right)\left(-\text{18}\right)$

$\left(\text{10}\right)\left(-6\right)$

-60

$\left(-6\right)\left(4\right)$

$\left(-2\right)\left(6\right)$

-12

$\left(-8\right)\left(7\right)$

$\frac{\text{21}}{7}$

3

$\frac{\text{42}}{6}$

$\frac{-\text{39}}{3}$

-13

$\frac{-\text{20}}{\text{10}}$

$\frac{-\text{45}}{-5}$

9

$\frac{-\text{16}}{-8}$

$\frac{\text{25}}{-5}$

-5

$\frac{\text{36}}{-4}$

$8-\left(-3\right)$

11

$\text{14}-\left(-\text{20}\right)$

$\text{20}-\left(-8\right)$

28

$-4-\left(-1\right)$

$0-4$

-4

$0-\left(-1\right)$

$-6+1-7$

-12

$\text{15}-\text{12}-\text{20}$

$1-6-7+8$

-4

$2+7-\text{10}+2$

$3\left(4-6\right)$

-6

$8\left(5-\text{12}\right)$

$-3\left(1-6\right)$

15

$-8\left(4-\text{12}\right)+2$

$-4\left(1-8\right)+3\left(\text{10}-3\right)$

49

$-9\left(0-2\right)+4\left(8-9\right)+0\left(-3\right)$

$6\left(-2-9\right)-6\left(2+9\right)+4\left(-1-1\right)$

-140

$\frac{3\left(4+1\right)-2\left(5\right)}{-2}$

$\frac{4\left(8+1\right)-3\left(-2\right)}{-4-2}$

-7

$\frac{-1\left(3+2\right)+5}{-1}$

$\frac{-3\left(4-2\right)+\left(-3\right)\left(-6\right)}{-4}$

-3

$-1\left(4+2\right)$

$-1\left(6-1\right)$

-5

$-\left(8+\text{21}\right)$

$-\left(8-\text{21}\right)$

13

## Exercises for review

( [link] ) Use the order of operations to simplify $\left({5}^{2}+{3}^{2}+2\right)÷{2}^{2}$ .

( [link] ) Find $\frac{3}{8}\text{of}\frac{\text{32}}{9}$ .

$\frac{4}{3}=1\frac{1}{3}$

( [link] ) Write this number in decimal form using digits: “fifty-two three-thousandths”

( [link] ) The ratio of chlorine to water in a solution is 2 to 7. How many mL of water are in a solution that contains 15 mL of chlorine?

$\text{52}\frac{1}{2}$

( [link] ) Perform the subtraction $-8-\left(-\text{20}\right)$

Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!