Integers and the Number Line
Integers and the Number Line
Up to this point, every number you have worked with has been zero or positive. But the real world is full of quantities that go below zero — temperatures drop, bank accounts are overdrawn, and elevations dip below sea level. Signed numbers, which include both positive and negative integers, let mathematics model all of these situations.
The number line is the fundamental tool for visualizing signed numbers. Every integer has a definite position, and the farther to the right a number sits, the larger it is. Zero serves as the dividing line between positive and negative.
This lesson introduces integers, the number line, absolute value, and the rules for comparing signed numbers — all of which are prerequisites for the operations in the next lesson and for algebra.
The integers are the set of whole numbers and their negatives:
$$\ldots,\; -3,\; -2,\; -1,\; 0,\; 1,\; 2,\; 3,\; \ldots$$
On the number line, numbers increase from left to right. Zero is at the centre. Positive numbers are to the right; negative numbers are to the left.
Absolute Value
The absolute value of a number is its distance from zero — always non-negative.
$$|a| = \begin{cases} a & \text{if } a \ge 0 \\ -a & \text{if } a < 0 \end{cases}$$
Find: $|7|$, $|-7|$, $|0|$
$|7| = 7$, $\;|-7| = 7$, $\;|0| = 0$.
Opposites
The opposite of $a$ is $-a$. They are equidistant from zero on opposite sides.
Find the opposite of $-5$ and $3$.
Opposite of $-5$ is $5$. Opposite of $3$ is $-3$.
Comparing Integers
On the number line, the number farther to the right is greater.
Compare: $-3$ ____ $2$ and $-7$ ____ $-1$.
$-3 < 2$ (−3 is left of 2). $\;-7 < -1$ (−7 is farther left).
Order from least to greatest: $4, -8, 0, -2, 3$.
$$-8 < -2 < 0 < 3 < 4$$
Evaluate: $|{-12}| - |5|$
$12 - 5 = 7$
Practice Problems
Show Answer Key
1. $15$
2. $9$
3. $>$ ($-4$ is right of $-10$)
4. $-7 < -3 < 0 < 2 < 7$
5. $25$
6. $-8$ ($|{-8}| = 8 > 6$)
7. $>$
8. $-4$
9. True
10. $-2 < -1.5 < -1.05 < 0.5$
11. $3$
12. $0$