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 (✓) or (✗) 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)?

    Explain your answer.

    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 −125.

    Both (−3) and 3 are square roots of 9.

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

    3 is called the cube root of 27, because \(3^{3}= 27\).

    −5 is called the cube root of −125 because \((-5)^{3} = −125\).

    \(10^{2}\) is 100 and \((−10)^{2}\) 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?