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?

• Consider your answers for question 5.

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

2. Illustrate your answer with two examples.

• 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. $$(-3) \times (-20)$$

• Use your answers in question 1 to determine the following:

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. $$\sqrt[3]{27} +(- \sqrt[3]{64})$$

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

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

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

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

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?