Integers & Absolute Value
Whole numbers in both directions — and the measure of how far any number sits from zero.
What are Integers?
In Lesson 4 you saw the number line stretching in both directions from zero. The numbers that sit precisely on the whole-number tick marks — with nothing between them — are called integers.
The symbol ℤ comes from the German word Zahlen, meaning “numbers.” Integers include three distinct groups:
Zero: 0 (neither positive nor negative — its own category)
Negative integers: −1, −2, −3, −4, −5 … (mirror of the positives)
Negative integers are not abstract — they model real quantities in everyday life. Understanding what they represent makes working with them far more intuitive.
| Situation | Positive Integer | Negative Integer |
|---|---|---|
| Temperature | 25°C (above zero) | −8°C (below zero) |
| Bank account | R500 in credit | −R200 (overdrawn) |
| Altitude | 1200m above sea level | −30m (below sea level) |
| Time | 3 years from now | −3 years (3 years ago) |
| Floors in a building | Floor 4 (above ground) | Floor −2 (underground) |
Here is a question: how far is −5 from zero? The answer is 5 — because we’re asking about distance, not direction. Distance is never negative. This is precisely what absolute value measures.
Absolute value is defined with a piecewise rule — two cases depending on whether the number is negative or not:
|x| = −x if x < 0 When x is negative, taking −x flips it to positive. Example: |−7| = −(−7) = 7
The opposite of −7 is +7. So −(−7) = 7. This is a crucial pattern that reappears constantly in algebra.
| Property | Rule | Example |
|---|---|---|
| Non-negativity | |x| ≥ 0 always | |−99| = 99, never −99 |
| Zero | |x| = 0 only when x = 0 | |0| = 0 — the only number with abs value 0 |
| Symmetry | |x| = |−x| | |3| = |−3| = 3 |
| Product | |x × y| = |x| × |y| | |−3 × 4| = |−3| × |4| = 12 |
| Triangle inequality | |x + y| ≤ |x| + |y| | |−3 + 1| = 2 ≤ 3 + 1 = 4 ✓ |
If |x| = 5, what is x? Because both 5 and −5 are exactly 5 units from zero, there are two solutions: x = 5 or x = −5. This two-answer nature is the defining feature of absolute value equations, which we’ll study fully in Lesson 35.
Insert <, >, or = between each pair.
An odometer records how far you’ve driven — always a positive number regardless of whether you drove north or south. A compass records direction.
Absolute value is the odometer. The original signed number is the compass. −5 tells you direction (left/negative) and magnitude (5 units). |−5| = 5 strips away the direction and gives you only the distance.
NOT integers: 0.5, −⅓, 2.1√4 = 2, which is a whole number — so it IS an integer.
|−4| + |10| = 4 + 10 = 14
They are NOT equal (6 ≠ 14).This illustrates the triangle inequality: |x+y| ≤ |x| + |y|. They are only equal when both numbers have the same sign.
(a) |−7| ___ |7| (b) |−3| ___ |−5| (c) −|2| ___ −2
(b) < (3 < 5)
(c) = (−|2| = −2)
The submarine is further from zero — it is 240m away vs the drone’s 180m.
- Integers are all whole numbers: … −3, −2, −1, 0, 1, 2, 3 …
- Zero is an integer — it is neither positive nor negative
- Fractions and decimals are not integers
- Absolute value |x| measures the distance of x from zero — always ≥ 0
- |x| = x when x is positive; |x| = −x when x is negative
- If |x| = a, there are two solutions: x = a or x = −a
- Always evaluate what is inside the absolute value bars before applying the bars