Classwork Series and Exercises {Mathematics- JSS2}: Inverse and Identity

Additive Inverse

If we add 0 (zero) to any number, the result is the same as the given number.

 For example,

3 + 0 = 3,       0 + 8 = 8

We say that 0 is the identity for addition. If the sum of two numbers is 0, we say that each number is the additive inverse of the other.

For example, (+3) + (-3) = 0.

(-3) is the additive inverse of (+3).

(+3) is the additive inverse of (-3).

(-8) + (+8) = 0.

(+8) is the additive inverse of (-8).

(-8) is the additive inverse of (8+).

Example 1

State the additive inverse of:

1. -19
2. 0.32
3. -7/8
4. 6 x 10⁷
5. – 3.1 x 10^-5

Given number                    Additive inverse

1. -19                                       + 19
2. 0.32                                    -0.32
3. -7/8                                    +7/8
4. 6 x 10⁷                                -6 x 10⁷
5. – 3.1 x 10^-5                       +3.1 x 10^-5           

In parts d and e of example 1, remember that the power of 10 in a number in standard form places the decimal the decimal point. It is not significant in deciding whether the number is positive or negative.

Solve the following equations:

1. x + 7 = 2

Solution

x + 7 = 2 is the same as x + (+7) = 2

Add (-7) to both sides.

X + (+7) + (-7) = 2 + (-7)

               X + 0 = 2 – 7

                     X = – 5

Multiplicative Inverse

If we multiply any number by 1 the result is the same as the given number. For example,

1 x 9 = 9, -5 x 1 = 15, 1 x /34 = ¾

We say that 1 is the identity for multiplication.  If the product of two numbers is the  multiplicative inverse  of the other. For example,

9 x 1/9 = 1

1/9 is the multiplicative inverse of 9.

9 is the multiplicative inverse of 1/9.

(-5) x(-1/5) = 1

-1.5 is the multiplicative inverse of 3/4.

-5 is the multiplicative inverse of -1/5.

¾ x 4/3 = 1

4/3 is the multiplicative inverse of ¾.

¾ is the multiplicative inverse of 4/3.

You have already used multiplicative inverses. In Book 1 you used reciprocals. The reciprocal of a fraction is that fraction turned upside down. The reciprocal of 2/3 is 3/2. Thus the multiplicative inverse of a number is the same as its reciprocal. 1/8 is the reciprocal of 8/1 or 8. 1/8 is the multiplicative inverse of 8.

Example

Find the multiplicative inverses of the following:

1. -32/ b. 0.3 c. 2 ½  d. n

Reciprocal of -3/2 = -2/3

-2/3 is the multiplicative inverse of -3/2.

Check: (-3/2) x (-2/3) = + (3/2 x 2/3) = 1.

2. 0.3 = 3/10

Reciprocal of 3/10 = 10/3.

10/3 (or 3 1/3) is the multiplicative inverse of 0.3.

Check: 0.3 x 3 1/3 = 3/10 x 10/3 = 1.

3. 2 ½ = 5/2

2/5 is the multiplicative inverse of 2 ½.

The check is left as an exercise.

4. i/n is the multiplicative inverse of n.

n x 1/n = 1.

Example

Solve -5x = 20.

Method I:

Notice that -5 is the multiplying x. Multiply both sides by the multiplicative inverse of -5.

Multiply both sides by -1/5.

(-1/5) x (-5) X x = (-1/5) x (+20)

                  1 X x = -(1/5 x 20)

                        x = -4

Method II:

Notice that multiplying by -1/5 is equivalent to dividing by -5. The example can be solved as follows.

-5x = 20

Divide both sides by -5.

(-5) X x/(-5) = +20/-5

             1 X x = -(20/5)

                   x = -4

The second method is usually quicker.

Inverse Operation

Do the following:

1. Stand up. Sit down.
2. Add 3 to 15. Subtract 3 from the result.
3. Multiply 7 by 2. Divide the result by 2.

In each case you should end where you start. When this happens, we say that the two actions are inverse operations.

Sitting down is the inverse operation of standing up. Adding a number is the inverse operation of subtracting the same number. Multiplying a number and dividing by the same number are inverse operations.

operation	Inverse operation
Shut the door	Open the door
Add 20	Subtract 20
Subtract -3	Add -3
Multiply by 4	Divide by 4
Divide by 0.3	Multiply by 0.3

EXERCISES

Lets see how much you’ve learnt, attach the following answers to the comment below

1. 6 times a number is 48. What is the number?
2. Find the number which, when multiplied by 10, gives 70.
3. A number divide by 5 gives 9, what is the n umber?
4. Solve this equation: 4x = 28
5. Solve this equation: 7x = 4 2/3

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