Integers and the Number Line

What Are Integers?

Integers are whole numbers and their opposites: $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$

• Positive integers: $1, 2, 3, \ldots$ (to the right of 0)
• Negative integers: $-1, -2, -3, \ldots$ (to the left of 0)
• Zero is neither positive nor negative

The Number Line

$\leftarrow \;\; -5 \;\; -4 \;\; -3 \;\; -2 \;\; -1 \;\; 0 \;\; 1 \;\; 2 \;\; 3 \;\; 4 \;\; 5 \;\; \rightarrow$

Numbers increase from left to right.

Comparing Integers

Use $<$, $>$, or $=$:

• $3 > -5$ (3 is to the right of -5)
• $-2 > -7$ (-2 is to the right of -7)
• $-4 < 1$ (-4 is to the left of 1)

Absolute Value

The absolute value $|x|$ is the distance from zero on the number line.

$|5| = 5 \quad |{-5}| = 5 \quad |0| = 0$

Absolute value is always non-negative (≥ 0).

Opposites

Two numbers are opposites if they are the same distance from zero on opposite sides:

• The opposite of $4$ is $-4$
• The opposite of $-7$ is $7$
• The opposite of $0$ is $0$

Ordering Integers

Order from least to greatest: $-8, 3, -1, 5, -3, 0$

Answer: $-8, -3, -1, 0, 3, 5$

Real-World Negative Numbers

• Temperature: $-10°$F (below zero)
• Elevation: $-200$ ft (below sea level)
• Money: -\50$ (debt)

Remember: A negative number is ALWAYS less than a positive number. Among negatives, the number with the bigger absolute value is actually smaller: $-10 < -3$.