### 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:

- 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?
- 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?
- 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?
- 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?
- The football team lost 8 yards on every play for 5 plays. Which integer shows the total number of yards the team has lost?
- 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?
- Find the product of – 3 and – 4.
- 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?
- Explain the process for finding the product of two integers.
- 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

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