Multiplying and Dividing Real Numbers
After addition and subtraction, the next operations you learned how to do were multiplication and division. You may recall that multiplication is a way of computing “repeated addition,” and this is true for negative numbers as well.
Multiplication and division are inverse operationsA mathematical operation that can reverse or “undo” another operation. Addition and subtraction are inverse operations. Multiplication and division are inverse operations. , just as addition and subtraction are. You may recall that when you divide fractions, you multiply by the reciprocal.
Multiplying real numbers is not that different from multiplying whole numbers and positive fractions. However, you haven't learned what effect a negative sign has on the product.
With whole numbers, you can think of multiplication as repeated addition. Using the number line, you can make multiple jumps of a given size. For example, the following picture shows the product `3 * 4` as `3` jumps of `4` units each.
So to multiply `3(-4)`, you can face left (toward the negative side) and make three “jumps” forward (in a negative direction).
Use the interactive number line to see how to multiply integers.
The product of a positive number and a negative number (or a negative and a positive) is negative. You can also see this by using patterns. In the following list of products, the first number is always `3`. The second number decreases by `1` with each row (`3`, `2`, `1`, `0`, `-1`, `-2`). Look for a pattern in the products of these numbers. What numbers would fit the pattern for the last two products?
`3(3) = 9`
`3(2) = 6`
`3(1) = 3`
`3(0) = 0`
`3(-1)` = ?
`3(-2)` = ?
Notice that the pattern is the same if the order of the factorsA number or mathematical symbol that is multiplied by another number or mathematical symbol to form a product. For example, in the equation `4 * 5 = 20`, `4` and `5` are factors. is switched:
`3(3) = 9`
`2(3) = 6`
`1(3) = 3`
`0(3) = 0`
`-1(3)=?`
`-2(3)=?`
Take a moment to think about that pattern before you read on.
As the factor decreases by `1`, the product decreases by `3`. So `3(-1) = -3` and `3(-2) = -6`.
If you continue the pattern further, you see that multiplying `3` by a negative integer gives a negative number. This is true in general.
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The Product of a Positive Number and a Negative Number.
To multiply a positive number and a negative number, multiply their absolute values. The product is negative. |
You can use the pattern idea to see how to multiply two negative numbers. Think about how you would complete this list of products.
`-3(3) = -9`
`-3(2) = -6`
`-3(1) = -3`
`-3(0) = 0`
`-3(-1)=?`
`-3(-2)=?`
As the factor decreases by `1`, the product increases by `3`. So `-3(-1) = 3`, `-3(-2) = 6`.
Multiplying `-3` by a negative integer gives a positive number. This is true in general.
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The Product of Two Numbers with the Same Sign (both positive or both negative).
To multiply two positive numbers, multiply their absolute values. The product is positive.
To multiply two negative numbers, multiply their absolute values. The product is positive. |
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Example |
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Problem |
Find `-3.8(0.6)`. |
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Multiply the absolute values as you normally would. `3.8` times `0.6`.
Place the decimal point by counting place values.
`3.8` has `1` place after the decimal point, and `0.6` has `1` place after the decimal point, so the product has `1 + 1` or `2` places after the decimal point. |
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Answer |
`-3.8(0.6) = -2.28` |
The product of a negative and a positive is negative. |
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Example |
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Problem |
Find `(-3/4)(-2/5).` |
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Multiply the absolute values of the numbers. |
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`(3/4)(2/5)=6/20=3/10`
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First, multiply the numerators together to get the product's numerator. Then, multiply the denominators together to get the product's denominator. Rewrite in lowest terms, if needed. |
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Answer |
`(-3/4)(-2/5)=3/10` |
The product of two negative numbers is positive. |
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Example |
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Problem |
Find `43y` when `y = -3`. |
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`43(-3)` |
Substitute `-3` for `y` in the expression. |
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`43(3) = 129` |
Multiply `43` and `3`. |
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Answer |
`43(-3) = -129` |
The product of a positive number and a negative number is negative. |
To summarize:
positive `*` positive: The product is positive.
negative `*` negative: The product is positive.
negative `*` positive: The product is negative.
positive `*` negative: The product is negative.
You can see that the product of two negative numbers is a positive number. So, if you are multiplying more than two numbers, you can count the number of negative factors.
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Multiplying More Than Two Negative Numbers
If there are an even number (`0`, `2`, `4`, ...) of negative factors to multiply, the product is positive. If there are an odd number (`1`, `3`, `5`, ...) of negative factors, the product is negative. |
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Example |
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Problem |
Find `3(-6)(2)(-3)(-1).` |
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`3(6)``(2)(3)(1)` `18``(2)``(3)(1)` `36``(3)``(1)` `108``(1)` `108` |
Multiply the absolute values of the numbers. |
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`3(-6)(2)(-3)(-1)` |
Count the number of negative factors. There are three (`-6`, `-3`, `-1`). |
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Answer |
`3(-6)(2)(-3)(-1) = -108` |
Since there are an odd number of negative factors, the product is negative. |
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Find `(-30)(-0.5)`.
A) `-150`
B) `-15`
C) `15`
D) `150`
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There is a number that can be added, again and again, without ever changing the sum. That number, `0`, is called the additive identity.
There is also a number that can be included as a factor as many times as you want, and it will never change the value of the product. That number, `1`, is called the multiplicative identity.
| `7(1)=7` | `-7(1)=-7` | |
| `1(3.6)=3.6` | `-2/23(1)=-2/23` | |
| `x(1)=x` | `(1)x=x` |
The identity property of 1 When you multiply any number by `1`, the product is the same as the original number. For example, `9(1) = 9`.states that `x*1 = x` and `1*x = x`.
You can think of it in this way: Multiplying by `1` lets the other number keep its identity.
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What is `1(y)`, when `y = -3`?
A) `-3`
B) `1`
C) `3`
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You might recall that two numbers are additive inverses if their sum is `0`, the additive identity.
`3` and `-3` are additive inverses because `3 + (-3) = 0`.
Two numbers are multiplicative inversesTwo numbers are multiplicative inverses if their product is `1`. For example, `3/1*1/3=3/3=1`. if their product is `1`, the multiplicative identity.
`2/3` and `3/2` are multiplicative inverses because `2/3(3/2)=6/6=1`.
You may remember that when you divided fractions, you multiplied by the reciprocalA number that when multiplied by a given number gives a product of 1. For example, `2/7` and `7/2` are reciprocals of each other.. Reciprocal is another name for the multiplicative inverse (just as opposite is another name for additive inverse).
An easy way to find the multiplicative inverse is to just “flip” the numerator and denominator as you did to find the reciprocal. Here are some examples:
The reciprocal of `4/9` is `9/4` because `4/9(9/4)=36/36=1`.
The reciprocal of `3` is `1/3` because `3/1(1/3)=3/3=1`.
The reciprocal of `-5/6` is `-6/5` because `-5/6(-6/5)=30/30=1`.
The reciprocal of `1` is `1` as `1(1) = 1`.
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What is the reciprocal, or multiplicative inverse, of `-12`?
A) `-12`
B) `1`
C) `1/12`
D) `12`
E) `-1/12`
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When you divided by positive fractions, you learned to multiply by the reciprocal. You also do this to divide real numbersAll rational or irrational numbers..
Think about dividing a bag of `26` marbles into two smaller bags with the same number of marbles in each. You can also say each smaller bag has one half of the marbles.
`26-:2=26(1/2)=13`
Notice that `2` and `1/2 ` are reciprocals.
Try again, dividing a bag of `36` marbles into smaller bags.
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Number of bags |
Dividing by number of bags |
Multiplying by reciprocal |
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`3` |
`36/3=12` |
`36(1/3)=36/3=(12(3))/3=12` |
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`4` |
`36/4=9` |
`36(1/4)=36/4=(9(4))/4=9` |
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`6` |
`36/6=6` |
`36(1/6)=36/6=(6(6))/6=6` |
Dividing by a number is the same as multiplying by its reciprocal. (That is, you use the reciprocal of the divisorThe number that you are dividing by in a division problem. In the problem `8-:2=4`, `2` is the divisor. , the second number in the division problem.)
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Example |
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Problem |
Find `28-:4/3.` |
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`28-:4/3=28(3/4)` |
Rewrite the division as multiplication by the reciprocal. The reciprocal of `4/3` is `3/4.` |
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`28/1(3/4)=(28(3))/4=(4(7)(3))/4=7(3)=21` |
Multiply. |
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Answer |
`28-:4/3=21` |
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Find `6/7-:3/10`. Write the answer in lowest terms.
A) `9/35`
B) `7/20`
C) `27/10`
D) `20/7`
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Now let's see what this means when one or more of the numbers is negative. A number and its reciprocal have the same sign. Since division is rewritten as multiplication using the reciprocal of the divisor, and taking the reciprocal doesn’t change any of the signs, division follows the same rules as multiplication.
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Rules of Division
When dividing, rewrite the problem as multiplication using the reciprocal of the divisor as the second factor.
When one number is positive and the other is negative, the quotientThe result of a division problem. In the problem `8/2=4`, 4 is the quotient. is negative.
When both numbers are negative, the quotient is positive.
When both numbers are positive, the quotient is positive. |
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Example |
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Problem |
Find `24-:(-5/6).` |
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`24-:(-5/6)=24(-6/5)` |
Rewrite the division as multiplication by the reciprocal. |
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`24/1(-6/5)=-144/5` |
Multiply. Since one number is positive and one is negative, the product is negative. |
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Answer |
`24-:(-5/6)=-144/5` |
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Example |
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Problem |
Find `4x-:(-6)` when `x=-2/3.` |
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`4(-2/3)-:(-6)` |
Substitute `-2/3` for `x` in the expression. |
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`4/1(-2/3)(-1/6)` |
Rewrite the division as multiplication by the reciprocal. |
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`(4(2)(1))/(3(6))=8/18` |
Multiply. There is an even number of negative numbers, so the product is positive. |
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Answer |
`4x-:(-6)=4/9text( when )x=-2/3` |
Write the fraction in lowest terms. |
Remember that a fraction bar also indicates division. So, a negative sign in front of a fraction goes with the numerator, the denominator, or the whole fraction: `-3/4=(-3)/4=3/(-4).` In each case, the overall fraction is negative because there's only one negative in the division.
Situations that require multiplication or division may use negative numbers and rational numbers.
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Example |
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Problem
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Carl didn't know his bank account was at exactly `0` when he wrote a series of `$100` checks. As each check went through, `$125` was charged against his account. (In addition to the `$100` for the check, there was a `$25` overdraft charge.) After `1` check, his account was `-$125` dollars. After `6` of these checks, what was his account balance? |
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`-$125(6)`
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Each check reduces the account by `$125`; this is represented by `-$125`. To find the amount it reduces for multiple checks, multiply the number of checks by the amount charged. |
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`-$125(6) = -$750`
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Multiply. Since there is one negative number, the product is negative.
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Answer |
Carl’s account balance is now `-$750`. |
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Example |
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Problem
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Brenda thought she was taking `3` chocolate bars to a picnic with `5` friends. When she brought out the chocolate, she discovered her brother had eaten half of one bar, so she only had `2 1/2` bars to divide among the `6` people (herself and her `5` friends). If each person gets the same amount, how much of a full bar does each person get? |
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`2 1/2-:6` |
Since the candy is being shared among `6` people, divide the amount of chocolate by `6.` |
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`2 1/2 (1/6)` |
Rewrite the problem as multiplication, using the reciprocal of the divisor. |
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`5/2(1/6)` |
Change the mixed number to an improper fraction. Multiply. |
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Answer |
Each person gets `5/12` of a full bar. |
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Example |
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Problem
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During a storm, the temperature dropped by `1/2` degree every minute. At the beginning of the storm, the temperature was `83^@text(F)`. An expression giving the temperature `t` minutes after the storm began is `-1/2t+83`. What was the temperature after `8` minutes? |
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`-1/2(8)+83` |
Substitute `8` for `t` in the expression. |
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`-1/2(8/1)+83=` `-8/2+83=-4+83`
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First, multiply. You may find it helpful to rewrite `8` as `8/1` first. Notice that since you are multiplying one negative and one positive number, the product is negative. Finally, add. Use the rules for adding numbers with different signs. |
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Answer |
The temperature was `79^@text(F)` after `8` minutes. |
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Over the course of an `18`-year research project, the height of an oceanside cliff actually dropped due to soil erosion. At the end of this period, its height was measured as `-3` inches compared to what it had been at the beginning of the research project. What was the average amount that the cliff’s height changed each year?
A) `-6` inches
B) `-1/6` inches
C) `1/6` inches
D) `6` inches
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With multiplication and division, you can find the sign of the final answer by counting how many negative numbers are used in the product or quotient. If there are an even number of negatives, the result is positive. If there are an odd number of negatives, the result is negative. Division can be rewritten as multiplication by using the reciprocal or multiplicative inverse of the divisor.