Chapter Notes: Negative Numbers: Multiplication And Division
You already know how to multiply and divide positive numbers. But what happens when you bring negative numbers into the picture? At first, multiplying or dividing with negatives might seem confusing, but there are simple rules that make it easy. Once you understand the patterns and practice a few examples, you'll be able to work with negative numbers just as confidently as you do with positive ones. These skills are important for understanding temperatures below zero, financial losses and gains, and even motion in opposite directions.
Understanding the Sign Rules
When you multiply or divide two numbers, the signs of those numbers determine whether your answer is positive or negative. There are four basic situations you need to know, and each follows a clear rule.
Multiplying Two Positive Numbers
This is what you already know well. When you multiply two positive numbers, the result is always positive.
Example: Multiply 4 × 5.
What is the product?
Solution:
Both numbers are positive.
4 × 5 = 20
The product is 20, which is positive.
Multiplying Two Negative Numbers
This is where things get interesting. When you multiply two negative numbers, the result is always positive. This might seem strange at first, but there's a logical pattern behind it.
Think of it this way: if owing money is negative, then removing debts (a negative action on a negative thing) results in a gain, which is positive.
Example: Multiply (-3) × (-4).
What is the product?
Solution:
Both numbers are negative.
When we multiply two negative numbers, the negatives cancel out and the result is positive.
(-3) × (-4) = 12
The product is 12, which is positive.
Multiplying One Positive and One Negative Number
When you multiply a positive number by a negative number (or a negative by a positive), the result is always negative. It doesn't matter which number comes first.
Example: Multiply 6 × (-2).
What is the product?
Solution:
One number is positive (6) and one number is negative (-2).
When we multiply a positive and a negative, the result is negative.
6 × (-2) = -12
The product is -12, which is negative.
Example: Multiply (-5) × 7.
What is the product?
Solution:
One number is negative (-5) and one number is positive (7).
When we multiply a negative and a positive, the result is negative.
(-5) × 7 = -35
The product is -35, which is negative.
The Sign Rules for Multiplication
Here's a summary of the multiplication sign rules you can memorize:
A simple way to remember: If the signs are the same, the answer is positive. If the signs are different, the answer is negative.
Multiplying Negative Numbers Step by Step
When you multiply numbers with negative signs, follow these steps:
- Look at the signs of both numbers.
- Determine what the sign of your answer will be using the rules above.
- Multiply the absolute values (the numbers without their signs).
- Put the correct sign on your answer.
Example: Multiply (-8) × (-3).
What is the product?
Solution:
Step 1: Both numbers are negative.
Step 2: Negative × Negative = Positive, so the answer will be positive.
Step 3: Multiply the absolute values: 8 × 3 = 24
Step 4: Put a positive sign on the answer: +24 or just 24
The product is 24.
Example: Multiply (-9) × 4.
What is the product?
Solution:
Step 1: One number is negative (-9) and one is positive (4).
Step 2: Negative × Positive = Negative, so the answer will be negative.
Step 3: Multiply the absolute values: 9 × 4 = 36
Step 4: Put a negative sign on the answer: -36
The product is -36.
Multiplying More Than Two Numbers
Sometimes you need to multiply three or more numbers together, and some of them might be negative. The key is to count how many negative signs you have:
- If you have an even number of negative signs, the answer is positive.
- If you have an odd number of negative signs, the answer is negative.
Example: Multiply (-2) × (-3) × (-4).
What is the product?
Solution:
Step 1: Count the negative signs: there are 3 negative signs.
Step 2: Since 3 is odd, the answer will be negative.
Step 3: Multiply the absolute values: 2 × 3 × 4 = 24
Step 4: Put a negative sign on the answer: -24
The product is -24.
Example: Multiply (-1) × 5 × (-2) × (-3).
What is the product?
Solution:
Step 1: Count the negative signs: there are 3 negative signs (on -1, -2, and -3).
Step 2: Since 3 is odd, the answer will be negative.
Step 3: Multiply the absolute values: 1 × 5 × 2 × 3 = 30
Step 4: Put a negative sign on the answer: -30
The product is -30.
Division with Negative Numbers
The sign rules for division are exactly the same as the sign rules for multiplication. This makes sense because division is the opposite operation of multiplication.
The Sign Rules for Division
Just like with multiplication: If the signs are the same, the answer is positive. If the signs are different, the answer is negative.
Dividing Positive by Positive
Example: Divide 20 ÷ 4.
What is the quotient?
Solution:
Both numbers are positive.
20 ÷ 4 = 5
The quotient is 5, which is positive.
Dividing Negative by Negative
Example: Divide (-18) ÷ (-3).
What is the quotient?
Solution:
Both numbers are negative.
When we divide two negative numbers, the negatives cancel out and the result is positive.
(-18) ÷ (-3) = 6
The quotient is 6, which is positive.
Dividing Positive by Negative
Example: Divide 15 ÷ (-5).
What is the quotient?
Solution:
One number is positive (15) and one is negative (-5).
When we divide a positive by a negative, the result is negative.
15 ÷ (-5) = -3
The quotient is -3, which is negative.
Dividing Negative by Positive
Example: Divide (-24) ÷ 6.
What is the quotient?
Solution:
One number is negative (-24) and one is positive (6).
When we divide a negative by a positive, the result is negative.
(-24) ÷ 6 = -4
The quotient is -4, which is negative.
Dividing Negative Numbers Step by Step
When you divide numbers with negative signs, follow these steps:
- Look at the signs of both the dividend and divisor.
- Determine what the sign of your answer will be using the rules above.
- Divide the absolute values (the numbers without their signs).
- Put the correct sign on your answer.
Example: Divide (-42) ÷ (-7).
What is the quotient?
Solution:
Step 1: Both numbers are negative.
Step 2: Negative ÷ Negative = Positive, so the answer will be positive.
Step 3: Divide the absolute values: 42 ÷ 7 = 6
Step 4: Put a positive sign on the answer: 6
The quotient is 6.
Example: Divide 56 ÷ (-8).
What is the quotient?
Solution:
Step 1: One number is positive (56) and one is negative (-8).
Step 2: Positive ÷ Negative = Negative, so the answer will be negative.
Step 3: Divide the absolute values: 56 ÷ 8 = 7
Step 4: Put a negative sign on the answer: -7
The quotient is -7.
Real-World Applications
Understanding multiplication and division with negative numbers helps you solve many practical problems.
Temperature Changes
When the temperature drops (a negative change) multiple times, you can use multiplication with negatives to find the total change.
Example: The temperature drops 3 degrees each hour for 4 hours.
The temperature change each hour is -3 degrees.What is the total temperature change?
Solution:
We need to multiply the hourly change by the number of hours.
Total change = (-3) × 4
One number is negative and one is positive, so the answer is negative.
(-3) × 4 = -12
The total temperature change is -12 degrees (a drop of 12 degrees).
Money and Debt
Financial situations often involve negative numbers for expenses or debts.
Example: Marcus owes his friend $15, which we can represent as -15.
He pays back his debt in 3 equal payments.How much is each payment?
Solution:
We need to divide the total debt by the number of payments.
Each payment = (-15) ÷ 3
One number is negative and one is positive, so the answer is negative.
(-15) ÷ 3 = -5
Each payment is -5 dollars, meaning Marcus pays $5 each time.
Elevation Changes
When moving below sea level or descending repeatedly, negative numbers represent downward motion.
Example: A submarine descends 50 feet per minute.
We can represent this as -50 feet per minute.What is the submarine's position after 6 minutes if it started at sea level?
Solution:
Total change = (-50) × 6
One number is negative and one is positive, so the answer is negative.
(-50) × 6 = -300
The submarine is at -300 feet, or 300 feet below sea level.
Common Mistakes to Avoid
Here are some errors students often make when working with negative numbers:
- Forgetting that negative × negative = positive: Many students think two negatives make a bigger negative, but they actually create a positive.
- Mixing up the signs: Always check whether the signs are the same (positive answer) or different (negative answer).
- Dropping the negative sign: When writing your answer, don't forget to include the negative sign if your answer should be negative.
- Confusing subtraction with negative numbers: Remember that -3 × 4 is different from 3 - 4. The multiplication sign and the negative sign are different operations.
Connecting Multiplication and Division
Multiplication and division are inverse operations, which means they undo each other. This relationship holds true with negative numbers too.
If you know that \( (-6) \times 3 = -18 \), then you also know:
- \( (-18) \div 3 = -6 \)
- \( (-18) \div (-6) = 3 \)
Example: Use the fact that (-4) × (-5) = 20 to write two division facts.
What are the division facts?
Solution:
If (-4) × (-5) = 20, then we can reverse the operation with division.
First division fact: 20 ÷ (-4) = -5
Second division fact: 20 ÷ (-5) = -4
The two division facts are 20 ÷ (-4) = -5 and 20 ÷ (-5) = -4.
Working with Variables and Negative Numbers
When you work with algebraic expressions, you'll often multiply or divide variables by negative numbers. The same sign rules apply.
Example: Simplify (-3) × \( x \).
What is the simplified expression?
Solution:
When we multiply a negative number by a variable, we write the negative sign in front.
(-3) × \( x \) = -3\( x \)
The simplified expression is -3\( x \).
Example: If \( y = -4 \), find the value of (-5) × \( y \).
What is the value?
Solution:
Substitute \( y = -4 \) into the expression.
(-5) × (-4)
Both numbers are negative, so the answer is positive.
(-5) × (-4) = 20
The value is 20.
Summary of Key Points
Let's review the most important ideas about multiplying and dividing negative numbers:
- When multiplying or dividing two numbers with the same sign, the answer is positive.
- When multiplying or dividing two numbers with different signs, the answer is negative.
- When multiplying more than two numbers, count the negative signs: an even number of negatives gives a positive answer, and an odd number of negatives gives a negative answer.
- The sign rules for division are exactly the same as the sign rules for multiplication.
- Always determine the sign of your answer first, then perform the operation on the absolute values, and finally attach the correct sign.
With practice, working with negative numbers in multiplication and division will become second nature. Remember the simple pattern: same signs give positive, different signs give negative. This rule will serve you well throughout all your future math courses.
