Numbers such as -7 and -500, the additive inverses of whole numbers, are included with all the whole numbers and called integers.

Fractions can be negative too, e.g.- $\frac{3}{4}$ and 3,46.

negative 7.

Equation

Solution

Required property of negative numbers

$17 + x = 10$

$x = -7$ because $17 + (-7) = 17 - 7$

$= 10$

1. Adding a negative number is just like subtracting the corresponding positive number

$5 -x = 9$

$x=-4$ because $5 -(-4) = 5 + 4 = 9$

2. Subtracting a negative number is just like adding the corresponding positive number

$20 + 3x = 5$

$x =-5$ because $3 \times (-5) = -15$

3. The product of a positive number and a negative number is a negative number

• In each case, state what number will make the equation true. Also state which of the properties of integers in the table above, is demonstrated by the equation.

1. $20 - x = 50$

2. $50 + x = 20$

3. $20 - 3x = 50$

4. $50 + 3x = 20$

• Examples: $(-5) + (-3)$ and $(-20) - (-7)$

$(-5) + (-3)$ can also be written as $-5 + (-3)$ or as $-5 + -3$

Examples: $5 - 9$ and $29 - 51$

We know that $-9 = (-4) + (-5)$

$-51 = (-29) + (-22)$

How much will be left of the 51, after you have subtracted 29 from 29 to get 0?How can we find out? Is it $51 - 29$?Examples: $7 + (-5); 37 + (-45)$ and $(-13) + 45$$20 + (a~ certain ~number) = 15$ true must have the following strange property:add this number, it should have the same effect as subtracting 5.So mathematicians agreed that the number called negative 5 will have the property that if you add it to another number, the effect will be the same as subtracting the natural number 5.negative 5 to a number, you may subtract 5.

Adding a negative number has the same effect as subtracting a corresponding natural number.

For example: $20 + (-15) = 20 - 15 = 5$.

We may say that for each "positive" number there is a corresponding or opposite negative number. Two positive and negative numbers that correspond, for example 3 and (-3), are called additive inverses.

• $-7 + 18$
• $24 - 30 - 7$
• $-15 + (-14) - 9$
• $35 - (-20)$
• $30 - 47$
• $(-12) - (-17)$
• Calculate.

1. $-7 + -7 + -7 + -7 + -7 + -7 + -7 + -7 + -7 + -7$

2. $-10 + -10 + -10 + -10 + -10 + -10 + -10$

3. $10 \times (-7)$
4. $7 \times (-10)$
• Say whether you agree (âœ“) or (âœ—) disagree with each statement.

1. $10 \times (-7) = 70$

2. $9 \times (-5) = (-9) \times 5$

3. $(-7) \times 10 = 7 \times (-10)$

4. $9 \times (-5) = -45$

5. $(-7) \times 10 = 10 \times (-7)$
6. $5 \times (-9) = 45$

• Multiplication of integers is commutative:

$(-20) \times 5 = 5 \times (-20)$

• Calculate each of the following. Note that brackets are used for two purposes in these expressions: to indicate that certain operations are to be done first, and to show the integers.

1. $20 + (-5)$
2. $4 \times (20 + (-5))$
3. $4 \times 20 + 4 \times (-5)$
4. $(-5) + (-20)$
5. $4 \times ((-5) + (-20))$
6. $4 \times (-5) + 4 \times (-20)$
• If you worked correctly, your answers for question 1 should be 15; 60; 60; -25; -100 and -100. If your answers are different, check to see where you went wrong and correct your work.

• Calculate each of the following where you can.

1. $20 + (-15)$

2. $4 \times ((20 + (-15))$

3. $4 \times 20 + 4 \times (-15)$

4. $(-15) + (-20)$

5. $4 \times ((-15) + (-20))$

6. $4 \times (-15) + 4 \times (-20)$

7. $10 + (-5)$

8. $(-4) \times (10 + (-5))$

9. $(-4) \times 10 + ((-4) \times (-5))$
• What property of integers is demonstrated in your answers for questions 3(a) and (g)?

In question 3 (i) you had to multiply two negative numbers. What was your guess?

We can calculate (-4) $\times$ (10 + (-5)) as in (h). It is (-4) $\times$ 5 = -20

If we want the distributive property to be true for integers, then (-4) $\times$ 10 + (-4) $\times$ (-5) must be equal to -20.

(-4) $\times$ 10 + (-4) $\times$ (-5) = -40 + (-4) $\times$ (-5)

Then (-4) $\times$ (-5) must be equal to 20.

• Calculate:

1. $10 \times 50 + 10 \times (-30)$

2. $50 + (-30)$

3. $10 \times (50 + (-30))$

4. $(-50) + (-30)$

5. $10 \times (-50) + 10 \times (-30)$

6. $10 \times ((-50) + (-30))$

• The product of two positive numbers is a positive number, for example $5 \times 6 = 30$.
• The product of a positive number and a negative number is a negative number, for example $5 \times (-6) = -30$.
• The product of a negative number and a positive number is a negative number, for example $(-5) \times 6 = -30$.
1. Underline the numerical expression below which you would expect to have the same answers. Do not do the calculations.

$16 \times (53 + 68)$ $53 \times (16 + 68)$ $16 \times 53 + 16 \times 68$ $16 \times 53 + 68$

2. What property of operations is demonstrated by the fact that two of the above expressions have the same value?

1. Does multiplication distribute over addition in the case of integers?

• Underline the numerical expression below which you would expect to have the same answers. Do not do the calculations now.

$10 \times ((50) -(-30))$ $10 \times (50) (30)$ $10 \times (-50) - 10 \times (-30)$

• Do the three sets of calculations given in question 8.

• Calculate $(-10) \times (5 + (-3))$.

• Now consider the question whether multiplication by a negative number distributes over addition and subtraction of integers. For example, would $(-10) \times 5 + (-10) \times (-3)$ also have the answer $-20$, like $(-10) \times (5 + (-3))$?

To make sure that multiplication distributes over addition and subtraction in the system of integers, we have to agree that

(a negative number) $\times$ (a negative number) is a positive number,

for example $(-10) \times (-3) = 30$.

• Calculate each of the following.

1. $(-20) \times (-6)$

2. $(-20) \times 7$

3. $(-30) \times (-10) + (-30) \times (-8)$

4. $(-30) \times ((-10) +(-8))$

5. $(-30) \times (-10) - (-30) \times (-8)$

6. $(-30) \times ((-10) - (-8))$

• When a number is added to its additive inverse, the result is 0. For example, (+12) + (-12) = 0.
• Adding an integer has the same effect as subtracting its additive inverse. For example, 3 + (-10) can be calculated by doing 3 - 10, and the answer is -7.
• Subtracting an integer has the same effect as adding its additive inverse. For example, 3 - (-10) can be calculated by calculating 3 + 10 is 13.
• The product of a positive and a negative integer is negative. For example, $(-15) \times 6 = -90$.
• The product of a negative and a negative integer is positive. For example $(-15) \times (-6) = 90$.
• Calculate

1. $5 \times (-7)$

2. $(-3) \times 20$
3. $(-5) \times (-10)$

4. $(-3) \times (-20)$

1. $(-35) \div 5$

2. $(-35) \div (-7)$

3. $(-60) \div 20$

4. $(-60) \div (-3)$

5. $50 \div (-5)$

6. $50 \div$ (-10)

7. $60 \div (-20)$)

8. $60 \div (-3)$

• The quotient of a positive number and a negative number is a negative number.
• The quotient of two negative numbers is a positive number.
• Calculate.

1. $20(-50 + 7)$
2. $20 \times (-50) + 20 \times 7$
3. $20(-50 + -7)$
4. $20 \times (-50) + 20 \times -7$
5. $-20(-50 + -7)$
6. $-20 \times -50 + -20 \times -7$
• Calculate.

1. $40 \times (-12 + 8) -10 \times (2 + -8) - 3 \times (-3 - 8)$

2. $(9 + 10 - 9) \times 40 + (25 - 30 - 5) \times 7$

3. $-50(40 - 25 + 20) + 30(-10 + 7 + 13)- 40(-16 + 15 - 2)$

4. $-4 \times (30 - 50) + 7 \times (40 - 70) - 10 \times (60 - 100)$

5. $-3 \times (-14 - 6 + 5) \times (-13 - 7 + 10) \times (20 - 10 - 15)$

• without using a calculator.
• Complete the tables.

1.  x 1 2 3 4 5 6 7 8 9 10 11 12 $x^{2}$ $x^{3}$
2.  x -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 $x^{2}$ $x^{3}$

The symbol $\sqrt{~}$ means that you must take the positive square root of the number.

$3^{2}$ is 9 and $(-3)^{2}$ is also 9.

$3^{3}$ is 27 and $(-5)^{3}$ is âˆ’125.

Both (âˆ’3) and 3 are square roots of 9.

3 may be called the positive square root of 9, and (âˆ’3) may be called the negative square root of 9.

3 is called the cube root of 27, because $3^{3}= 27$.

âˆ’5 is called the cube root of âˆ’125 because $(-5)^{3} = âˆ’125$.

$10^{2}$ is 100 and $(âˆ’10)^{2}$ is also 100. Both 10 and (-10) are called square roots of 100.

• Calculate the following:

1. $\sqrt{4} - \sqrt{9}$

2. $\sqrt{27} +(- \sqrt{64})$

3. $-(3^{2}$)

4. $(-3) ^{2}$

5. $4^{2} - 6^{2} + 1^{2}$

6. $3^{3}- 4^{3}- 2^{3} -1^{3}$
7. $\sqrt{81} - \sqrt{4} \times \sqrt {125}$

8. $-(4^{2})(-1) ^{2}$
9. $\frac{(-5) ^2}{\sqrt{37 - 12}}$
10. $\frac{-\sqrt{36}}{-1^{3} - 2^{3}}$
• Determine the answer to each of the following:

1. The overnight temperature in Polokwane drops from 11 $^\circ$C to -2 $^\circ$C. By how many degrees has the temperature dropped?

2. The temperature in Estcourt drops from 2 $^\circ$C to -1 $^\circ$C in one hour, and then another two degrees in the next hour. How many degrees in total did the temperature drop over the two hours?

3. A submarine is 75 m below the surface of the sea. It then rises by 21 m. How far below the surface is it now?

4. A submarine is 37 m below the surface of the sea. It then sinks a further 15 m. How far below the surface is it now?