1.9 Properties of real numbers (Page 3/10)

Elementary algebra Page 3 / 10

We have the statement that 2 times a blank space equals 1. Then it is stated that “We know 2 times 1/2 equals 1.”

Notice that in each case, the missing number was the reciprocal of the number!

We call $\frac{1}{a}$ the multiplicative inverse of a . The reciprocal of $a$ number is its multiplicative inverse. A number and its reciprocal multiply to one, which is the multiplicative identity. This leads to the Inverse Property of Multiplication that states that for any real number $a, a \neq 0, a \cdot \frac{1}{a} = 1 .$

We’ll formally state the inverse properties here:

Inverse property

\begin{matrix} of addition & For any real number a, & a + (− a) = 0 \\ − a . is the additive inverse of a . \\ A number and its opposite add to zero. \\ of multiplication & For any real number a, a \neq 0 & a \cdot \frac{1}{a} = 1 \\ \frac{1}{a} is the multiplicative inverse of a . \\ A number and its reciprocal multiply to one. \end{matrix}

Find the additive inverse of ⓐ $\frac{5}{8}$ ⓑ $0.6$ ⓒ $−8$ ⓓ $- \frac{4}{3} .$

Solution

To find the additive inverse, we find the opposite.

ⓐ The additive inverse of $\frac{5}{8}$ is the opposite of $\frac{5}{8} .$ The additive inverse of $\frac{5}{8}$ is $- \frac{5}{8} .$
ⓑ The additive inverse of 0.6 is the opposite of 0.6. The additive inverse of 0.6 is $−0.6 .$
ⓒ The additive inverse of $−8$ is the opposite of $−8 .$ We write the opposite of $−8$ as $− (−8),$ and then simplify it to 8. Therefore, the additive inverse of $−8$ is 8.
ⓓ The additive inverse of $- \frac{4}{3}$ is the opposite of $- \frac{4}{3} .$ We write this as $− (- \frac{4}{3}),$ and then simplify to $\frac{4}{3} .$ Thus, the additive inverse of $- \frac{4}{3}$ is $\frac{4}{3} .$

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Find the additive inverse of: ⓐ $\frac{7}{9}$ ⓑ $1.2$ ⓒ $−14$ ⓓ $- \frac{9}{4} .$

ⓐ $- \frac{7}{9}$ ⓑ $−1.2$ ⓒ $14$ ⓓ $\frac{9}{4}$

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Find the additive inverse of: ⓐ $\frac{7}{13}$ ⓑ $8.4$ ⓒ $−46$ ⓓ $- \frac{5}{2} .$

ⓐ $- \frac{7}{13}$ ⓑ $−8.4$ ⓒ $46$ ⓓ $\frac{5}{2}$

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Find the multiplicative inverse of ⓐ $9$ ⓑ $- \frac{1}{9}$ ⓒ $0.9 .$

Solution

To find the multiplicative inverse, we find the reciprocal.

ⓐ The multiplicative inverse of 9 is the reciprocal of 9, which is $\frac{1}{9} .$ Therefore, the multiplicative inverse of 9 is $\frac{1}{9} .$
ⓑ The multiplicative inverse of $- \frac{1}{9}$ is the reciprocal of $- \frac{1}{9},$ which is $−9 .$ Thus, the multiplicative inverse of $- \frac{1}{9}$ is $−9 .$
ⓒ To find the multiplicative inverse of 0.9, we first convert 0.9 to a fraction, $\frac{9}{10} .$ Then we find the reciprocal of the fraction. The reciprocal of $\frac{9}{10}$ is $\frac{10}{9} .$ So the multiplicative inverse of 0.9 is $\frac{10}{9} .$

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Find the multiplicative inverse of ⓐ $4$ ⓑ $- \frac{1}{7}$ ⓒ $0.3$

ⓐ $\frac{1}{4}$ ⓑ $−7$ ⓒ $\frac{10}{3}$

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Find the multiplicative inverse of ⓐ $18$ ⓑ $- \frac{4}{5}$ ⓒ $0.6 .$

ⓐ $\frac{1}{18}$ ⓑ $- \frac{5}{4}$ ⓒ $\frac{5}{3}$

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Use the properties of zero

The identity property of addition says that when we add 0 to any number, the result is that same number. What happens when we multiply a number by 0? Multiplying by 0 makes the product equal zero.

Multiplication by zero

For any real number a .

a \cdot 0 = 0 0 \cdot a = 0

The product of any real number and 0 is 0.

What about division involving zero? What is $0 \div 3 ?$ Think about a real example: If there are no cookies in the cookie jar and 3 people are to share them, how many cookies does each person get? There are no cookies to share, so each person gets 0 cookies. So,

0 \div 3 = 0

We can check division with the related multiplication fact.

12 \div 6 = 2 because 2 \cdot 6 = 12 .

So we know $0 \div 3 = 0$ because $0 \cdot 3 = 0 .$

Division of zero

For any real number a , except $0,$ $\frac{0}{a} = 0$ and $0 \div a = 0 .$

Zero divided by any real number except zero is zero.

Now think about dividing by zero. What is the result of dividing 4 by 0? Think about the related multiplication fact: $4 \div 0 = ?$ means $? \cdot 0 = 4 .$ Is there a number that multiplied by 0 gives 4? Since any real number multiplied by 0 gives 0, there is no real number that can be multiplied by 0 to obtain 4.

Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

how can I help

Sir

hmm can we speak here?

Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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