Introduction

In this chapter, you will learn how to multiply and divide with negative numbers.

This video illustrates the lesson material below. Watching the video is optional.

• Multiplication and Division with Negative Numbers (02:53 mins) | Transcript

Multiplying by Negatives

Multiplication with negative numbers is the same as normal multiplication, except for a few additional rules:
\begin{align*} Positive \times Positive &= Positive\\\\ Negative \times Negative &= Positive\\\\ Positive \times Negative &= Negative\\\\ Negative \times Positive &= Negative \end{align*}

Multiplication Examples
\begin{align*}-3\times -2 &= 6 &\color{red}\small\text{Negative \(\times\) Negative = Positive}\\\\3\times -2 &=- 6 &\color{red}\small\text{Positive \(\times\) Negative = Negative}\\\\-3\times 2 &=- 6 &\color{red}\small\text{Negative \(\times\) Positive = Negative }\\\end{align*}

Dividing by Negatives

The same rules apply for division with negative numbers:
\begin{align*} Positive \div Positive &= Positive\\\\ Negative \div Negative &= Positive\\\\ Positive \div Negative &= Negative\\\\ Negative \div Positive &= Negative \end{align*}

Division Examples
\begin{align*}-10\div -5 &= 2 &\color{red}\small\text{Negative \(\div\) Negative = Positive}\\\\10\div -5 &=-2 &\color{red}\small\text{Positive \(\div\) Negative = Negative}\\\\-10\div 5 &=- 2 &\color{red}\small\text{Negative \(\div\) Positive = Negative }\\\end{align*}

Things to Remember

• A negative number multiplied or divided by a negative number will equal a positive number.
• A negative number multiplied or divided by a positive number will equal a negative number.

Practice Problems

1. If a positive number is multiplied by a negative number, what do you know about the result? (
   Solution

   x

   Solution:
   b
   )
   
   1. The result will always be positive.
   2. The result will always be negative.
   3. The result will be negative if the negative value is bigger than the positive value.
   4. There is not enough information to draw a conclusion about the result.
2. If a negative number is divided by a negative number, what do you know about the result? (
   Solution

   x

   Solution:
   a
   )
   
   1. The result will always be positive.
   2. The result will always be negative.
   3. The result will be greater than -1 and less than 1.
   4. There is not enough information to draw a conclusion about the result.
   Evaluate the following expressions:
   
3. \(8 × (-9) = ?\) (
   Solution

   x

   Solution: -72
   Details:
   Notice the pattern as you move down the number line, the answers get more negative. A positive number times a negative will always give a negative answer.
   
   \(8 × 1 = 8\)
   \(8 × 0 = 0\)
   \(8 × -1 = -8\)
   \(8 × -2 = -16\)
   \(8 × -3 = -24\)
   \(8 × -4 = -32\)
   \(8 × -5 = -40\)
   \(8 × -6 = -48\)
   \(8 × -7 = -56\)
   \(8 × -8 = -64\)
   \(\color{red}8 × -9 = -72\)
   )
4. \((-15) × 7 = ?\) (
   Video Solution

   x

   Solution: -105
   Details:
   

   
   (Multiplication and Division with Negative Numbers #4 (00:54 mins) | Transcript)
   | Transcript)
5. \((-45) × (-7) = ?\) (
   Solution

   x

   Solution:
   315
   )
6. \(15 ÷ (-3) = ?\) (
   Solution

   x

   Solution: -5
   Details:
   Notice the pattern, once you divide by a negative, the answer is the opposite. When 15 is divided by 3 the answer is 5, but when 15 is divided by -3 the answer is the opposite: -5.
   
   \(15 ÷ 5 = 3\)
   \(15 ÷ 3 = 5\)
   \(15 ÷ 1 = 15\)
   \(\color{red}15 ÷ -1 = -15\)
   \(\color{red}15 ÷ -3 = -5\)
   \(\color{red}15 ÷ -5 = -3\)
   )
7. \((-63) ÷ (7) = ?\) (
   Solution

   x

   Solution:
   -9
   )
8. \((-81) ÷ (-9) = ?\) (
   Video Solution

   x

   Solution: 9
   Details:
   

   
   (Multiplication and Division with Negative Numbers #8 (00:29 mins) | Transcript)
   | Transcript)

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