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Operations on Integers

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Addition

A sum operation is an operation that involves the $+$ sign. On the number line, integers added with positive integers will shift to the right (the value gets bigger). Meanwhile, if it is added to a negative number, then the position of the number will shift to the left (the value is getting smaller).

Addition with a Number Line

Using a number line, the numbers that we will add are represented as arrows originating from the zero point ($0$) with the number value indicated by the length of the arrow. Positive numbers to the right, while negative numbers to the left. Here are some examples of solving integer addition operations with the help of a number line.

  1. $5 + 3$

    2. The number $5$ is represented by an arrow to the right from the point $0$ along 5 units. The number is added $3 $ by making arrows from the end of the first arrow to the right along 3 units. The result of the sum can be obtained by drawing an arrow from the point $0$ to the end of the second arrow, or by looking at the scale on the tip of the second arrow. The result is $8$. For more details, see the following image:

  2. $6 + \left( { - 8} \right)$

    3. Look at the following image:

    By looking at the number line, the sum is $-2$.

  3. $\left( { - 3} \right) + \left( { - 4} \right)$

    4. Look at the following image:

    By looking at the number line, the sum is $-7$.

Properties of Addition

  1. Closed

    2. An integer if added to another integer, the result will be an integer as well.

  2. Commutative

    3. If $a$ and $b$ are integers then the following applies:

    $ \boxed{a + b = b + a}$

    Example:

    $\begin{array}{l} 2 + 3 = 3 + 2\\ 4 + \left( { - 5} \right) = \left( { - 5} \right) + 4 \end{array}$
  3. Associative

    4. If $a$, $b$, and $c$ are integers then the following applies:

    $\boxed{\left( {a + b} \right) + c = a + \left( {b + c} \right)}$

    Example:

    $\begin{array}{l} \left( {2 + 3} \right) + 4 = 2 + \left( {3 + 4} \right)\\ \left( {6 + 7} \right) + \left( { - 4} \right) = 6 + \left( {7 + \left( { - 4} \right)} \right) \end{array}$
  4. Identity

    5. The identity element of addition is zero ($0$) and for $a$ integers are as follows:

    $\boxed{\begin{array}{l} a + 0 = a\\ 0 + a = a \end{array}}$

    Example:

    $\begin{array}{l} 4 + 0 = 4\\ - 5 + 0 = - 5\\ 0 + 7 = 7 \end{array}$
  5. Inverse

    6. If $a$ is an integer, then the inverse of $a$ is $-a$ and applies:

    $\boxed{\begin{array}{l} a + \left( { - a} \right) = 0\\ \left( { - a} \right) + a = 0 \end{array}}$

    Example:

    The inverse of $6$ is $-6$ and applies $6 + \left( { - 6} \right) = 0$.

    The inverse of $-4$ is $4$ and applies $ - 4 + 4 = 0$.

Practical Formulas for Adding Integers

  • $\left( { - a} \right) + \left( { - b} \right) = - \left( {a + b} \right)$
  • $\left( { - a} \right) + b = b - a\left( {\text{if a < b}} \right)$
  • $\left( { - a} \right) + b = - \left( {a - b} \right)\left( {\text{if a > b}} \right)$

Subtraction

In integers, subtracting a number is the same as adding the number to the opposite of the subtractor. For every $a$ and $b$ an integer applies:

$\boxed{a - b = a + \left( { - b} \right)}$

Example:

  • $8 - 3 = 8 + \left( { - 3} \right) = 5$
  • $7 - 9 = 7 + \left( { - 9} \right) = - 2$
  • $ - 5 - 7 = - 5 + \left( { - 7} \right) = - 12$
  • $ - 9 - 2 = - 9 + \left( { - 2} \right) = - 11$
  • $8 - \left( { - 2} \right) = 8 + 2 = 10$
  • $ - 9 - \left( { - 6} \right) = - 9 + 6 = - 3$

Multiplication

The multiplication operation is usually symbolized by a cross $\left( \times \right)$ or a period $\left( . \right)$. The concept of multiplication actually comes from repeated addition operations.

$\boxed{a \times b = \underbrace {b + b + \ldots + b}_{\text{a times}}}$

Example:

$\begin{array}{l} 3 \times 2 = 2 + 2 + 2 = 6\\ 2 \times \left( { - 3} \right) = \left( { - 3} \right) + \left( { - 3} \right) = - 6 \end{array}$

Properties of Multiplication

  1. Closed

    2. An integer is multiplied by another integer and the result is an integer.

  2. Commutative

    3. If $a$ and $b$ are integers then the following applies:

    $ \boxed{a \times b = b \times a}$

    Example:

    $\begin{array}{l} 2 \times 3 = 3 \times 2\\ 5 \times \left( { - 6} \right) = \left( { - 6} \right) \times 5 \end{array}$
  3. Associative

    4. If $a$, $b$, and $c$ are integers then the following applies:

    $\boxed{\left( {a \times b} \right) \times c = a \times \left( {b \times c} \right)}$

    Example:

    $\begin{array}{l} \left( {2 \times 3} \right) \times 4 = 2 \times \left( {3 \times 4} \right)\\ \left( {6 \times 7} \right) \times \left( { - 4} \right) = 6 \times \left( {7 \times \left( { - 4} \right)} \right) \end{array}$
  4. Identity

    5. The identity element of multiplication is one ($1$) and for $a$ integers are as follows:

    $\boxed{\begin{array}{l} a \times 1 = a\\ 1 \times a = a \end{array}}$

    Contoh:

    $\begin{array}{l} 4 \times 1 = 4\\ - 5 \times 1 = - 5\\ 1 \times 7 = 7 \end{array}$
  5. Distributive

    6. If $a$, $b$, and $c$ are integers, then:

    $\begin{array}{l} \boxed{a \times \left( {b + c} \right) = \left( {a \times b} \right) + \left( {a \times c} \right)} \Rightarrow \left( { \text{distributive to addition}} \right)\\ \boxed{a \times \left( {b - c} \right) = \left( {a \times b} \right) - \left( {a \times c} \right)} \Rightarrow \left( { \text{distributive to subtraction}} \right) \end{array}$

    Example:

    $\begin{array}{l} 2 \times \left( {4 + 5} \right) = \left( {2 \times 4} \right) + \left( {2 \times 5} \right)\\ 3 \times \left( {6 - 2} \right) = \left( {3 \times 6} \right) - \left( {3 \times 2} \right)\\ 5 \times 999 &= 5 \times \left( {1.000 - 1} \right)\\ &= \left( {5 \times 1.000} \right) - \left( {5 \times 1} \right)\\ &= 5.000 - 5\\ &= 4.995 \end{array}$

Integer Multiplication Result

  • Multiplication of integers with an equal sign is always positive.
    • $\begin{array}{l} \boxed{+} \times \boxed{+} = \boxed{+} \\ \boxed{-} \times \boxed{-} = \boxed{+} \end{array}$
  • Multiplication of integers with different signs is always negative.
    • $\begin{array}{l} \boxed{+} \times \boxed{-} = \boxed{-} \\ \boxed{-} \times \boxed{+} = \boxed{-} \end{array}$

Division

The division operation is the opposite of multiplication. If $2 \times 4 = 8$ then $8 \div 4 = 2$ or $8 \div 2 = 4$. So it can be concluded as follows:

If $a$, $b$, and $c$ are integers and $b \ne 0$ then $\boxed{a \div b = c}$ if and only if $\boxed{a = b \times c}$

Example:

$\begin{array}{l} 3 \times 4 = 12 \Rightarrow 12 \div 4 = 3\\ \left( { - 3} \right) \times 5 = - 15 \Rightarrow - 15 \div 5 = - 3\\ 2 \times \left( { - 4} \right) = - 8 \Rightarrow - 8 \div \left( { - 4} \right) = 2\\ \left( { - 5} \right) \times \left( { - 4} \right) = 20 \Rightarrow 20 \div \left( { - 4} \right) = - 5 \end{array}$

Integer Division Result

  • Dividing integers with an equal sign is always positive.
    • $\begin{array}{l} \boxed{+} \times \boxed{+} = \boxed{+} \\ \boxed{-} \times \boxed{-} = \boxed{+} \end{array}$
  • Dividing integers with different signs is always negative.
    • $\begin{array}{l} \boxed{+} \times \boxed{-} = \boxed{-} \\ \boxed{-} \times \boxed{+} = \boxed{-} \end{array}$

Dividing Integers by Zero

For every integer $a$, $a \div 0$ undefined.

Example:

$\begin{array}{l} 4 \div 0 = \text{undefined}\\ \frac{{ - 2}}{0} = \text{undefined} \end{array}$

Division of Zero

For every integer $a$, apply $\boxed{0 \div a = 0}$.

Example:

$\begin{array}{l} 0 \div 5 = 0\\ 0 \div \left( { - 7} \right) = 0 \end{array}$
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nurhamim86
nurhamim86 A Mathematics Teacher who also likes the IT world.

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