**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:

- -19
- 0.32
- -7/8
- 6 x 10
^{7 } - – 3.1 x 10
^{-5 }

**Given number** **Additive inverse**

- -19 + 19
- 0.32 -0.32
- -7/8 +7/8
- 6 x 10
^{7 }-6 x 10^{7} - – 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:

- 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:

- -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:

- Stand up. Sit down.
- Add 3 to 15. Subtract 3 from the result.
- 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

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