Multiplication of Integers: Definition & Examples

Key Concepts

• Multiply a negative integer by a positive integer
• Multiply a positive integer by a negative integer
• Multiply a negative integer by a negative integer

1.5 Multiplication of Integers 

Definition: 

Multiplication of any two integers is the repeated addition.  

Multiplication can be classified as follows:  

• Multiplication between any two positive numbers 
• Multiplication between any two negative numbers 
• Multiplication between a positive number and a negative number. 

The following table shows the different rules of multiplication: 

Rules of Multiplication
Type of Numbers	Operation	Result	Example
Positive x positive	Multiply	Positive +	1 × 2 = 2
Negative x negative	Multiply	Positive +	−1 × −5 = 5
Positive x negative	Multiply	Negative −	1 × − 4 = − 4
Negative x positive	Multiply	Negative −	−1 × 7 = − 7

Example: 

• 4 × (−2) = four times (− 2) = (− 2) + (− 2) + (− 2) + (− 2) = – 8 
• 4 × 2 = four times 2 = 2 + 2 + 2 + 2 = 8 

1.5.1 Multiplication of Two Positive Integers 

Rule: 

Product of any two positive integers is always a positive. 

Example: 

Multiply: 2 × 3 

To represent this on the number line, we start at 0 and put 2 groups of 3 on the number line.  

We end up at 6. So, the product is 6. 

∴∴

2 x 3 = 6 (Positive × Positive = Positive) 

1.5.2 Multiplication of a Positive Integer and a Negative Integer 

To multiply a positive integer with a negative integer, we can multiply them as a whole number and then put the negative sign before their product. 

So, the product of a negative and a positive integer will always be a negative integer. 

For two integers, p and q,  

p × (− q) = (− p) × q = − (p × q) = − pq 

Example: 

4 × (− 10) = (− 4) × 10 = − (4 × 10) = − 40 

1.5.3 Multiplication of Two Negative Integers 

To multiply two negative integers, we can multiply them as a whole number and then put the positive sign before their product. 

Hence, if we multiply two negative integers, then the result will always be a positive integer. 

For two integers, p and q, 

(− p) × ( − q) = ( − p) ( − q) = p × q 

Example: 

(− 10) × (− 3) = 30 

Exercise:

1. From sea level, a submarine descends 40 feet per minute. Where is the submarine in relation to sea level 5 minutes after it starts descending?
2. The price of one share of a stock fell 4 dollars each day for 8 days. How much value did one share of the stock lose after 8 days?
3. As a cold front passed through Temple, the temperature changed steadily over 6 hours.  Altogether it changed 18 degrees.  What was the change in temperature each hour for the 6 hours?
4. Bob bought a Ford Expedition for his family. After five years, the SUV was worth $10,000 less than what he paid for it. What integer represents how much value the SUV lost each year?
5. The football team lost 8 yards on every play for 5 plays. Which integer shows the total number of yards the team has lost? 
6. A mountain climber climbed up a cliff 50 feet at a time. He did this 5 times in one day. What was the overall change in his elevation?
7. Find the product of – 3 and – 4.
8. The price of one share of Acme Company declined $5 per day for 4 days in a row. How much did the price of one share change in total after the 4 days?
9. Explain the process for finding the product of two integers.
10. The product of three integers is –3. Determine all of the possible values for the three factors.

What we have learned:

• Multiply two negative integers
• Multiply a positive integer by a negative integer
• Understand the multiplication properties
• Understand how to find the product of integers
• Model multiply and divide integers using integer chips and horizontal and vertical number lines.

Concept Map