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Integers

    Multiplication Of Integers

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7th class > Integers > Multiplication Of Integers

Multiplication Of Integers

We can add and subtract integers. Let us now learn how to multiply integers.

Multiplication of a Positive and a Negative Integer

We know that multiplication of whole numbers is repeated addition.For example,

5 + 5 + 5 = 3 × 5 = 15

Can you represent addition of integers in the same way?

We have from the following number line, (–5) + (–5) + (–5) = –15

Now, try this out yourself.

Instruction

1.Find the given Values in Number Line.

(i) 4 x (-8) =

(ii) 8 x (-2) =

(iii) 3 x (-7) =

(iv) 10 x (-1) =

Instruction

But we can also write:

(-5) + (-5) + (-5) = 3 × (-5)

Therefore,

3 × (-5) =

Similarly

(- 4) + (- 4) + (- 4) + (- 4) + (- 4) = 5 × (- 4) =

Now, try it again by yourself.

Instruction

We see that:

When a number is added to itself, a certain number of times - we get the resulting number to be equal to:

Resulting Number = Number of repetitions × Number Value

So:

−3+−3+−3+−3 = × =

(Enter the number of repetitions first)

Again,

(–7) + (–7) + (–7) = × =

(Enter the number of repetitions first)

Now, let us see how to find the product of a positive integer and a negative integer without using number line.

Let us find 3 × (–5) in a different way. First, find 3 × 5 and then put minus sign (–) before the product obtained. We get –.

Similarly, 5 × (– 4) = –(5 × 4) =

Find the products of:

4 × (– 8) = -(4 × ) =

3 × (– 7) = -( × ) =

6 × (– 5) = -( × ) =

2 × (– 9) = -( × ) =

10 × (– 43) = -( × ) =

Till now we multiplied integers as (positive integer) × (negative integer). Now, let's try multiplying them as (negative integer) × (positive integer).

1. Find:

(i) 6 × (–19) =

(ii) 12 × (–32) =

(iii) 7 × (–22) =

We first find: –3 × 5.

Observe the pattern as we move along.

Instruction

3×5=15

  • Now, 2 × 5 =
  • 1 x 5 =
  • 0 x 5 =
  • We notice that as we decrease the given integer, the multiple of the integer with the constant i.e. 5 can also be written as the subtraction of 5 from the preceeding multiple.
  • Likewise, -1 x 5 =
  • -2 x 5 =
  • -3 x 5 =
  • This pattern is useful in understanding how addition, subtraction and multiplication are co-related.

So, we get −3×5 = –15 =3×−5

Using such patterns, we also find that:

−5×4= =5×−4

−4×8= = 8×−4

−3×7 = = 7×−3

−6×5 = = 5×−6

−2×9 = = 9×−2

We thus find that while multiplying a positive integer and a negative integer,

multiply them as whole numbers and put a minus sign (–) before the product. We thus, get a integer.

1. Find:

(a) 15 × (–16) = .

(b) 21 × (–32) = .

(c) (– 42) × 12 = .

(d) –55 × 15 = .

2. Check if

(a) 25 × (–21) = and (–25) × 21 =

Therefore, both are

(b) (–23) × 20 = and 23 × (–20)=

Therefore, both are

3. Write five more such examples for commutative property (as seen above).

1)17×(−15) = ()x()

2)35×(−42) = ()x()

3)(−56)×17 = ()x()

4)64×(−29) = ()x()

5)49×(−81) = ()x()

In general, for any two positive integers a and b we can say:

a × (– b) = (– a) × b = –(a × b)

Multiplication of two Negative Integer

Similarly, let's try to understand the pattern when dealing with the multiplication of two negative integers.

What is the product of (–3) × (–2) ?

Let's find out.

Observe the below pattern:

Instruction

−3×4=

  • Now, -3 × 4 =
  • -3 x 3 =
  • -3 x 2 =
  • -3 x 1 =
  • -3 x 0 =
  • -3 x (-1) =
  • -3 x (-2) =
  • This pattern is useful in understanding how addition, subtraction and multiplication are co-related.

Do you see any pattern? Observe how the products change. Based on this observation, complete the following:

Finding the product of -4 x -3:

Instruction

−4x4=

  • Now, -4 x 4 =
  • -4 x 3 =
  • -4 x 2 =
  • -4 x 1 =
  • -4 x 0 =
  • -4 x (-1) =
  • -4 x (-2) =
  • -4 x (-3) =
  • And the pattern continues

Instruction

(i) Starting from (–5) × 4, find (–5) × (– 6) =

−5x4=

  • Now, - 5 × 4 =
  • -5 × 3 =
  • -5 × 2 =
  • -5 × 1 =
  • -5 × 0 =
  • -5 × (-1) =
  • -5 × (-2) =
  • -5 × (-3) =
  • And the pattern continues
  • -5 × (-6) =

Instruction

(ii) Starting from (– 6) × 3, find (– 6) × (–7) =

−6x−7=

  • Now, - 6 × 3 =
  • -6 × 2 =
  • -6 × 1 =
  • -6 × 0 =
  • -6 × (-1) =
  • -6 × (-2) =
  • -6 × (-3) =
  • And the pattern continues
  • -6 × (-7) =

From these patterns we can find out that,

(–3) × (–1) = = 3 × 1

(–3) × (–2) = = 3 × 2

(–3) × (–3) = = 3 × 3

Similarly,

(– 4) × (–1) = = 4 × 1

So,

(– 4) × (–2) = 4 × 2 =

(– 4) × (–3) = 4 × 3 =

So, observing these products we can say that:

The product of two negative integers is a positive integer. We multiply the two negative integers as whole numbers and put the positive sign before the product.

Thus, we have

(–10) × (–12) =

(–15) × (– 6) =

In general, for any two positive integers a and b,

(-a) × (– b) = a × b

1. Find:

a)(–31) × (–100) = .

b)(–25) × (–72) =

c)(–83) × (–28) =