Reference

Multiplying and Dividing Integers

Method

Multiplying Integers With the Same Sign

The product of two integers is always positive if and only if the factors have the same sign. This means that when the factors are both positive or both negative, ignore the signs and multiply them as if they were whole numbers. To illustrate this process, consider the following multiplication of integers.

- 3 \times (- 4)

There are two steps to follow when the factors have the same sign.

Verify the Signs of the Factors

If both factors are positive, keep the numbers as they are. On the other hand, if the factors are both negative, ignore the signs of the factors to reduce the multiplication to a multiplication of two positive integers. In this example, the signs of the integers will be removed because both are negative.

- 3 \times (- 4) ⇕ 3 \times 4

The expression is now a multiplication of two whole numbers.

Find the Product Using a Number Line

The first factor indicates how many intervals will be needed to find the product. The second factor indicates the size of each interval. The intervals are drawn on a number line starting from zero. In this case,

3 \times 4

indicates that three equal intervals of length

4

are needed.

The endpoint represents the product of the given multiplication of integers with the same sign.

3 \times 4 = 12 \Leftrightarrow - 3 \times (- 4) = 12

The result of multiplication of integers with the same sign is always positive.

Method

Multiplying Integers With Different Signs

The product of two integers is always negative if and only if one factor is negative and the other is positive. Change the negative factor to its opposite and perform the multiplication as if both were whole numbers. Next, change the result to its opposite to get the final product. Consider the following multiplication of integers.

- 7 \times 2

Follow these three steps to find the product of two integers with different signs.

Check the Signs of the Factors

Verify that one factor is positive and the other is negative. Change the negative factor to its opposite. The result is a multiplication of two positive integers. For the given example, the first factor is negative

7,

so change it to its opposite

7 .

- 7 \times 2 \Rightarrow 7 \times 2

Find the Product of the Resulting Multiplication

Next, find the product of the multiplication of two whole numbers. In this case,

7 \times 2

means that seven equal intervals of length

2

are needed. Use a number line to find this product.

The endpoint is the product of the resulting multiplication. This means that

7 \times 2 = 14 .

Change the Product to Its Opposite

Change the product from the previous step to its opposite to get the product of the initial multiplication of integers with different signs. For the given example, the product of whole numbers is

14 .

Its opposite is

- 14 .

7 \times 2 = 14 \Rightarrow - 7 \times 2 = - 14

In summary, multiply two integers with different signs as if they were whole numbers and change the result to its opposite to get the final product.

Method

Dividing Integers With the Same Sign

The quotient of dividing an integer

a

by an integer

b

is always positive if those integers have the same sign. This means the dividend and the divisor are both negative or both positive. If this condition is met, ignore their signs and perform the division as if they were whole numbers. This will be illustrated using the following division of integers.

- 15 \div (- 3)

Follow these two steps when dividing two integers with the same sign.

Verify the Signs of the Dividend and the Divisor

If the dividend and the divisor are positive, keep the numbers as they are. On the other hand, if the dividend and the divisor are negative, ignore their sings to reduce the division to a division of two positive integers. In the given example, the signs of the integers will be removed because both are negative.

- 15 \div (- 3) \Leftrightarrow 15 \div 3

The expression is now a division of two whole numbers.

Find the Quotient Using a Number Line

The quotient can be found by moving to the right starting from zero on a number line. Use intervals the size of the divisor until the dividend is reached. The number of jumps needed to reach the dividend equals the quotient. In this case, move to the right of zero in intervals of

3

since the divisor is

3 .

It took

5

jumps of

3

units to reach

15 .

This means that the quotient of the initial division is

5 .

15 \div 3 = 5 \Leftrightarrow - 15 \div (- 3) = 5

The quotient of two numbers with the same sign is always positive.

Method

Dividing Integers With Different Signs

The quotient of dividing two integers is always negative if and only if one is negative and the other is positive. Change the negative number to its opposite and perform the division as if they were whole numbers. Next, change the result to its opposite to get the quotient of the initial division. This process will be illustrated with the following division.

- 12 \div 4

There are three steps to follow to find the quotient when dividing two numbers with different signs.

Check the Signs of the Integers

Identify which of the numbers involved in the division is negative. Next, change it to its opposite to get a division of two positive integers. In the given example, the dividend

- 12

is negative. Its opposite is

12 .

- 12 \div 4 \Rightarrow 12 \div 4

Find the Quotient of the Resulting Division

Find the quotient of the division of two whole numbers. For this example,

12 \div 4

means how many jumps of

4

are needed to reach

12

on a number line, starting from zero.

It took three jumps of

4

units to reach

12 .

This means that

12 \div 4 = 3 .

Change the Quotient to Its Opposite

Change the quotient found in the previous step to its opposite to get the quotient of the initial division of integers with different sings. For the given example, the quotient is

3 .

Its opposite is

- 3 .

Therefore, the quotient of

- 12 \div 4

- 3 .

12 \div 4 = 3 \Rightarrow - 12 \div 4 = - 3

In summary, divide two integers with different signs as if they were whole numbers and change the result to its opposite to get the final quotient.

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