Integer - Multiplying and dividing

I think you will agree with me when I say some students find it challenging when multiplying and dividing an integer by another integer.

But it needn’t be if you know what they are and how to carry out their multiplication and division efficiently.

So….

What is an integer and how do we multiply and divide them?

Integers are negative and positive numbers, including zero.

….., -5,  -4,  -3,  -2,  -1,  0,  1,  2,  3,  4,  5,  …..

Here are some key points to remember when multiplying integers.

  • Multiplying two integers of the same signs gives a positive answer.
  • Multiplying two integers of different signs gives a negative answer.

Example (1)      4 x 3 = 12                  (4 and 3 are both positive integers)

Example (2)      -4 x -3 = 12                (-4 and -3 are both negative integers)

Example (3)     -4 x 3 = -12                 (-4 and 3 are integers with different signs)

Example (4)       4 x -3 = -12               (4 and -3 are integers with different signs)

We can use pattern spotting to explain the multiplication of integers.

Consider the calculations below for multiplying integers when the signs are the same and when the signs are different.

2 x 3 = 6

2 x 2 = 4

2 x 1 = 2

2 x 0 = 0

2 x -1 = -2

2 x -2 = -4

2 x -3 = -6

You may spot the pattern of the numbers being multiplied by 2 decreasing by 1 each time and the results decreasing by 2 each time.

You can see that a positive integer multiplied by a positive integer gives a positive answer

+   x   +  =  +

You can also see that a positive integer multiplied by a negative integer gives a negative answer

+   x   –  =  –

Now consider the calculations below.

-2 x 3 = -6

-2 x 2 = -4

-2 x 1 = -2

-2 x 0 = 0

-2 x -1 = 2

-2 x -2 = 4

-2 x -3 = 6

You may spot the pattern of the numbers being multiplied by -2 decreasing by 1 each time and the results increasing by 2 each time.

You can see that a negative integer multiplied by a positive integer gives a negative answer.

–   x   +  =  –

You can also see that a negative integer multiplied by a negative integer gives a positive answer.

–   x   –  =  +

So far we have looked at the multiplication of two integers, but what if you were multiplying three or more integers. Will the same rule apply?

Let us consider -2 x -3 x -4. They all have the same signs. But will it give a positive answer?

Multiplying the first two integers (-2 x -3) gives 6, then 6 x -4 gives -24 because a positive integer (6) multiplied by a negative integer (-4) gives a negative answer.

Further investigation will reveal that you will get a positive answer if the amount of negative integers being multiplied is an even number, and you’ll get a negative answer if the amount of negative integers being multiplied is an odd number.

Example          Two negative integers (even)  – x – = +

Three negative integers (odd) – x – x – = –

Four negative integers (even) – x – x – x – = +

Five negative integers (odd)   – x – x – x – x – x = –

Six negative integers (even)    – x – x – x – x – x – = +

Here are some key points to remember when dividing integers.

  • The same rule that applies to multiplication applies to the division.
  • Dividing two integers of the same signs gives a positive answer.
  • Dividing two integers of different signs gives a negative answer.
  • The integer being divided appears before the division symbol. It is called the dividend.
  • The integer we are dividing by appears after the division symbol. It is called the divisor.
  • The result of the calculation is called the quotient.

Example (1)      4 ÷ 2 = 2                    (4 and 2 are both positive integers)

Example (2)      -4 ÷ -2 = 2                  (-4 and -2 are both negative integers)

Example (3)     -4 ÷ 2 = -2                   (-4 and 2 are integers with different signs)

Example (4)       4 ÷ -2 = -2                 (4 and -2 are integers with different signs)

We can use pattern spotting to explain the division of integers.

Consider the calculations below for dividing integers when the signs are the same and when the signs are different. In these calculations, we have used inverse operations for the calculations used previously for multiplication.

6 ÷ 3 = 2

4 ÷ 2 = 2

2 ÷ 1 = 2

0 ÷ 0 is undefined.

-2 ÷ -1 = 2

-4 ÷ -2 = 2

-6 ÷ -3 = 2

You may spot the pattern of the numbers being divided (dividend) decreasing by 2 each time and the number we are dividing by (divisor) decreasing by 1 each time, giving an answer (quotient) of 2 each time, with the exception of 0 divided by 0 which is undefined.

You can see that a positive integer divided by a positive integer gives a positive answer

+   ÷   +  =  +

You can also see that a negative integer divided by a negative integer gives a positive answer

–   ÷   –  =  +

Now consider the calculations below.

-6 ÷ 3 = -2

-4 ÷ 2 = -2

-2 ÷ 1 = -2

0 ÷ 0 is undefined

2 ÷ -1 = -2

4 ÷ -2 = -2

6 ÷ -3 = -2

You may spot the pattern of the numbers being divided (dividend) increasing by 2 each time and the number we are dividing by (divisor) decreasing by 1 each time, giving an answer (quotient) of -2 each time, with the exception of 0 divided by 0 which is undefined.

You can see that a negative integer divided by a positive integer gives a negative answer

–   ÷   +  =  –

You can also see that a positive integer divided by a negative integer gives a negative answer

+   ÷   –  =  –

In summary, there are several methods used to explain the multiplication and division of integers. We have covered some of the methods here, and hope you have found this blog useful. It was written by Key Stage Tutors a one-to-one tutoring website providing online tutoring in several academic subjects, covering the UK. You can click here to see a blog on adding and subtracting integers.

Comments

comments