Integers

What Are Integers?

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 $\textcolor{colordef}{(+1)}+\textcolor{colorprop}{(-2)}$.

Answer

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 $\textcolor{colorprop}{(-10)}+\textcolor{colordef}{(+3)}$.

Answer

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

Subtraction

Discover

- 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 $\textcolor{colordef}{(+4)}-\textcolor{colorprop}{(-2)}$.

Answer

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

On the Number Line

Discover

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 $x$.

Answer

So, $x=\textcolor{colorprop}{-2}$.

Ordering

Discover

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 $\textcolor{colordef}{+3}$.

Answer

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