\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)

Integers

Definition


Imagine a world with two types of particles: positives (+) and negatives (-). They interact in specific ways.
  • When particles of the same type meet, they join forces.
  • When a positive and a negative particle meet, they cancel each other out, leaving nothing. This is called a zero pair.
  • Let’s see what happens if 2 positives meet 1 negative. One zero pair is formed, leaving 1 positive.
  • To show which type of particle we have, we put a sign in front of the number:
    • The + sign for a group of positives.
    • The - sign for a group of negatives.
  • Now, let’s see what happens when 3 positives meet 1 negative.
    There are 2 positives left.
  • Finally, let’s see what happens when 2 positives meet 2 negatives.
    There are \(0\) particles left.

Definition Integers
The integers are the set that contains the natural numbers (\(1, 2, 3, \dots\)), their opposites (\(-1, -2, -3, \dots\)), and \(0\).
  • Positive numbers (\(\textcolor{colordef}{+1}, \textcolor{colordef}{+2},\dots\)) are written with a positive sign \((+)\). This sign is often omitted (\(\textcolor{colordef}{+2}=\textcolor{colordef}{2}\)).
    \(\textcolor{colordef}{+2}=\)
  • Negative numbers (\(\textcolor{colorprop}{-1}, \textcolor{colorprop}{-2},\dots\)) are written with a negative sign \((-)\).
    \(\textcolor{colorprop}{-3}=\)
  • Zero (\(0\)) is neither positive nor negative.
  • Two numbers are opposites if their sum is \(0\).

    \(\textcolor{colorprop}{-2}\) is the opposite of \(\textcolor{colordef}{+2}\).
  • To avoid confusion between a number's sign and an operation sign, we often use parentheses. For example, \(\textcolor{colordef}{+1}+\textcolor{colorprop}{-2}\) can be written as \(\textcolor{colordef}{(+1)}+\textcolor{colorprop}{(-2)}\).
Example
Calculate \(\textcolor{colordef}{(+1)}+\textcolor{colorprop}{(-2)}\).

  • So, \(\textcolor{colordef}{(+1)}+\textcolor{colorprop}{(-2)}=\textcolor{colorprop}{-1}\).

Definition Absolute Value
The absolute value of a number is the number without its sign.
  • The absolute value of \(\textcolor{colordef}{+2}=\) is \(2\).
  • The absolute value of \(\textcolor{colorprop}{-3}=\) is \(3\).

Rules of Addition

Method Rules of Addition
  • When you add two positive numbers, add their absolute values. The sum is also a positive number:$$\textcolor{colordef}{(+2)}+\textcolor{colordef}{(+4)}=\textcolor{colordef}{+6} \quad \text{as }2+4=6.$$
  • When you add two negative numbers, add their absolute values. The sum is also a negative number:$$\textcolor{colorprop}{(-5)}+\textcolor{colorprop}{(-3)}=\textcolor{colorprop}{-8} \quad \text{as }5+3=8.$$
  • When you add a positive number and a negative number, subtract the smaller absolute value from the larger one and use the sign of the number with the larger absolute value.
    \(\textcolor{colorprop}{(-2)}+\textcolor{colordef}{(+5)}=\textcolor{colordef}{+3} \quad \text{as }5-2=3\)
    \(\textcolor{colordef}{(+2)}+\textcolor{colorprop}{(-6)}=\textcolor{colorprop}{-4} \quad \text{as }6-2=4\)
Example
Calculate \(\textcolor{colorprop}{(-10)}+\textcolor{colordef}{(+3)}\).

  • \(\textcolor{colorprop}{(-10)}+\textcolor{colordef}{(+3)}=\textcolor{colorprop}{-7} \quad \text{as }10-3=7.\)

Subtraction


    • For the subtraction, \(\textcolor{colordef}{(+3)}-\textcolor{colordef}{(+2)}\):
      we remove \(2\) positives from \(3\) positives, leaving us with \(1\) positive.
    • For the addition, \(\textcolor{colordef}{(+3)}+\textcolor{colorprop}{(-2)}\):
      we again remove \(2\) positives from \(3\) positives.
    • Therefore, these two operations are equivalent:
      \(\textcolor{colordef}{(+3)}-\textcolor{colordef}{(+2)}=\textcolor{colordef}{(+3)}+\textcolor{colorprop}{(-2)}\)
      This shows that subtracting a positive number is the same as adding its opposite.
    • For the subtraction, \(\textcolor{colorprop}{(-3)}-\textcolor{colorprop}{(-2)}\):
      we remove \(2\) negatives from \(3\) negatives, leaving us with \(1\) negative.
    • For the addition, \(\textcolor{colorprop}{(-3)}+\textcolor{colordef}{(+2)}\):
      we again remove \(2\) negatives from \(3\) negatives.
    • Therefore, these two operations are equivalent:
      \(\textcolor{colorprop}{(-3)}-\textcolor{colorprop}{(-2)}=\textcolor{colorprop}{(-3)}+\textcolor{colordef}{(+2)}\)
      This shows that subtracting a negative number is the same as adding its opposite.
  • In conclusion, these examples show a fundamental rule in arithmetic: subtracting any number is equivalent to adding the number with its opposite sign.

Definition Subtraction
Subtracting a number means adding its opposite.
Example
Calculate \(\textcolor{colordef}{(+4)}-\textcolor{colorprop}{(-2)}\).

$$\begin{aligned}[t]\textcolor{colordef}{(+4)}-\textcolor{colorprop}{(-2)}&=\textcolor{colordef}{(+4)}+\textcolor{colordef}{(+2)}&&\text{(add the opposite)}\\ &=\textcolor{colordef}{+6}&&\text{(same sign: add the absolute values)}\end{aligned}$$

On the Number Line


  • To show both positive and negative numbers on a number line, we extend the number line in both directions from zero.
  • For each move from left to right by \(1\), the number increases by \(1\): \(0+1=\textcolor{colordef}{+1}\), \(\textcolor{colordef}{+1}+1=\textcolor{colordef}{+2},\dots\)
  • For each move from right to left by \(1\), the number decreases by \(1\): \(0-1=\textcolor{colorprop}{-1}\), \(\textcolor{colorprop}{-1}-1=\textcolor{colorprop}{-2},\dots\)

Definition Number line
A number line is a straight line with markings at equal intervals to denote the numbers.
Example
Find the value of \(x\).

  • So, \(x=\textcolor{colorprop}{-2}\).

Ordering


In the set of integers, the order is defined as:$$\dots\lt\textcolor{colorprop}{-3}\lt\textcolor{colorprop}{-2}\lt\textcolor{colorprop}{-1}\lt 0 \lt \textcolor{colordef}{+1}\lt\textcolor{colordef}{+2}\lt\textcolor{colordef}{+3}\lt\dots $$So, as you move along the number line from left to right, the numbers increase.
  • As \(\textcolor{colordef}{+3}\) is to the right of \(\textcolor{colorprop}{-5}\), \(\textcolor{colorprop}{-5} \lt \textcolor{colordef}{+3}\). So, when one number is positive and the other is negative, the positive number is greater.
  • As \(\textcolor{colorprop}{-2}\) is to the right of \(\textcolor{colorprop}{-4}\), \(\textcolor{colorprop}{-4}\lt \textcolor{colorprop}{-2}\). So, when both numbers are negative, the number closer to zero is greater (the number with the smaller absolute value is greater).
  • As \(\textcolor{colordef}{+6}\) is to the right of \(\textcolor{colordef}{+4}\), \(\textcolor{colordef}{+4}\lt \textcolor{colordef}{+6}\). So, when both numbers are positive, the number further from zero is greater (the number with the greater absolute value is greater).

Method Compare two numbers
  • When one number is positive and the other is negative, the positive number is greater.
  • When both numbers are negative, the number closer to zero is greater (the number with the smaller absolute value is greater).
  • When both numbers are positive, the number further from zero is greater (the number with the greater absolute value is greater).
Example
Compare \(\textcolor{colorprop}{-4}\) and \(\textcolor{colordef}{+3}\).

  • As \(\textcolor{colordef}{+3}\) is positive and \(\textcolor{colorprop}{-4}\) is negative, the positive number is greater than the negative number: \(\textcolor{colorprop}{-4}\lt \textcolor{colordef}{+3}\).

Multiplication


Multiplication of two whole numbers can be seen as repeated addition: \(3\times 2=2+2+2=6\).
We now extend this idea to signed numbers:
  • \(\textcolor{colordef}{(+)}\times \textcolor{colordef}{(+)}\):$$\begin{aligned}[t]\textcolor{colordef}{(+3)}\times \textcolor{colordef}{(+2)}&=3\times \textcolor{colordef}{(+2)}\\ &=\textcolor{colordef}{(+2)} + \textcolor{colordef}{(+2)} + \textcolor{colordef}{(+2)}\\ &=\textcolor{colordef}{+6}\end{aligned}$$
    So, a positive times a positive gives a positive.
  • \(\textcolor{colordef}{(+)}\times \textcolor{colorprop}{(-)}\):$$\begin{aligned}[t]\textcolor{colordef}{(+3)}\times \textcolor{colorprop}{(-2)}&=3\times \textcolor{colorprop}{(-2)}\\ &=\textcolor{colorprop}{(-2)} + \textcolor{colorprop}{(-2)} + \textcolor{colorprop}{(-2)}\\ &=\textcolor{colorprop}{-6}\end{aligned}$$
    So, a positive times a negative gives a negative.
  • \(\textcolor{colorprop}{(-)} \times \textcolor{colordef}{(+)}\): Multiplication is commutative, so the order does not matter.$$\begin{aligned}[t]\textcolor{colorprop}{(-2)} \times \textcolor{colordef}{(+3)} &=\textcolor{colordef}{(+3)}\times \textcolor{colorprop}{(-2)}\\ &=\textcolor{colorprop}{-6}\end{aligned}$$So, a negative times a positive gives a negative.
  • \(\textcolor{colorprop}{(-)} \times \textcolor{colorprop}{(-)}\):We know that \(0\times \textcolor{colorprop}{(-2)}=0\). Also,\(0=\textcolor{colordef}{(+3)} + \textcolor{colorprop}{(-3)}\), so:$$\begin{aligned}[t](\textcolor{colordef}{(+3)} + \textcolor{colorprop}{(-3)})\times \textcolor{colorprop}{(-2)} &= 0\\ \textcolor{colordef}{(+3)}\times\textcolor{colorprop}{(-2)} + \textcolor{colorprop}{(-3)}\times\textcolor{colorprop}{(-2)} &= 0\\ \textcolor{colorprop}{-6} + \big(\textcolor{colorprop}{(-3)}\times\textcolor{colorprop}{(-2)}\big) &= 0\\ \textcolor{colorprop}{(-3)}\times \textcolor{colorprop}{(-2)}&=\textcolor{colordef}{+6}\end{aligned}$$So, a negative times a negative gives a positive.

Definition Multiplication
The product of two signed integers is obtained by:
  • multiplying their absolute values;
  • then deciding the sign using the following rules:
    • \(\textcolor{colordef}{(+)}\times \textcolor{colordef}{(+)} = \textcolor{colordef}{(+)}\): a positive times a positive gives a positive.
    • \(\textcolor{colordef}{(+)}\times \textcolor{colorprop}{(-)} = \textcolor{colorprop}{(-)}\): a positive times a negative gives a negative.
    • \(\textcolor{colorprop}{(-)} \times \textcolor{colordef}{(+)} = \textcolor{colorprop}{(-)}\): a negative times a positive gives a negative.
    • \(\textcolor{colorprop}{(-)} \times \textcolor{colorprop}{(-)} = \textcolor{colordef}{(+)}\): a negative times a negative gives a positive.
Example
Calculate \(\textcolor{colordef}{(+2)}\times \textcolor{colorprop}{(-5)}\).

$$\textcolor{colordef}{(+2)}\times \textcolor{colorprop}{(-5)}= \textcolor{colorprop}{-10}\quad (2\times 5 = 10 \text{ and } \textcolor{colordef}{(+)}\times \textcolor{colorprop}{(-)} = \textcolor{colorprop}{(-)})$$

Division


Multiplication and division are inverse operations:$$\textcolor{black}{3}\times \textcolor{black}{2}=\textcolor{black}{6}, \text{ so }\textcolor{black}{6}\div \textcolor{black}{3}=\textcolor{black}{2}.$$(Here we only divide by non-zero numbers.)
Now, let's look at division with negative numbers:
  • \(\textcolor{colordef}{(+)} \div \textcolor{colordef}{(+)}\):$$\textcolor{colordef}{(+3)} \times \textcolor{colordef}{(+2)}=\textcolor{colordef}{+6}, \text{ so }\textcolor{colordef}{(+6)} \div \textcolor{colordef}{(+3)}=\textcolor{colordef}{(+2)}.$$So, a positive divided by a positive gives a positive.
  • \(\textcolor{colordef}{(+)} \div \textcolor{colorprop}{(-)}\):$$\textcolor{colorprop}{(-3)} \times \textcolor{colorprop}{(-2)}=\textcolor{colordef}{+6}, \text{ so }\textcolor{colordef}{(+6)} \div \textcolor{colorprop}{(-3)}=\textcolor{colorprop}{(-2)}.$$So, a positive divided by a negative gives a negative.
  • \(\textcolor{colorprop}{(-)} \div \textcolor{colordef}{(+)}\):$$\textcolor{colordef}{(+3)} \times \textcolor{colorprop}{(-2)}=\textcolor{colorprop}{-6}, \text{ so }\textcolor{colorprop}{(-6)} \div \textcolor{colordef}{(+3)}=\textcolor{colorprop}{(-2)}.$$So, a negative divided by a positive gives a negative.
  • \(\textcolor{colorprop}{(-)} \div \textcolor{colorprop}{(-)}\):$$\textcolor{colorprop}{(-3)} \times \textcolor{colordef}{(+2)}=\textcolor{colorprop}{-6}, \text{ so }\textcolor{colorprop}{(-6)} \div \textcolor{colorprop}{(-3)}=\textcolor{colordef}{(+2)}.$$So, a negative divided by a negative gives a positive.

Definition Division
The quotient of two integers (with a non-zero divisor) is obtained by:
  • dividing their absolute values;
  • then deciding the sign using the following rules:
    • \(\textcolor{colordef}{(+)}\div \textcolor{colordef}{(+)} = \textcolor{colordef}{(+)}\): a positive divided by a positive gives a positive.
    • \(\textcolor{colordef}{(+)}\div \textcolor{colorprop}{(-)} = \textcolor{colorprop}{(-)}\): a positive divided by a negative gives a negative.
    • \(\textcolor{colorprop}{(-)} \div \textcolor{colordef}{(+)} = \textcolor{colorprop}{(-)}\): a negative divided by a positive gives a negative.
    • \(\textcolor{colorprop}{(-)} \div \textcolor{colorprop}{(-)} = \textcolor{colordef}{(+)}\): a negative divided by a negative gives a positive.
Example
Calculate \(\textcolor{colordef}{(+10)}\div \textcolor{colorprop}{(-5)}\).

$$\textcolor{colordef}{(+10)}\div \textcolor{colorprop}{(-5)}= \textcolor{colorprop}{-2} \quad (10\div 5 = 2 \text{ and } \textcolor{colordef}{(+)}\div \textcolor{colorprop}{(-)} = \textcolor{colorprop}{(-)})$$