How far from zero?

The idea

The absolute value |x| of a rational number x is its distance from 0 on the number line; it is x itself for positive x and its positive value for negative x, so |x| ≥ 0.

Which is farther?

One house is 2 steps left of the school. Another is 2 steps right. Neither is on the same side, but how far is each from the school at 0?

Which is farther from 0: −2 or 2?

• −2
• 2
• Same distance

Distance from 0

Move the point and look at the orange band. Its length is the distance from 0.

|⁠x| is a distance

The absolute value of a rational number x, written |⁠x|⁠, is its distance from 0 on the number line. It is x itself for positive x and its positive value for negative x, so |⁠x| ≥ 0.

For example |⁠5/3| = 5/3, |⁠−5/3| = 5/3 and |⁠0| = 0.

Notes

The absolute value |⁠x| of a rational number x is its distance from 0. It is x for positive x and the positive value of x for negative x, so |⁠x| ≥ 0.

Check yourself

What is |⁠−7|?

Answer: 7

|⁠−7| = 7. A distance is never negative.

Which is true for every rational number x?

What is |⁠−5/3|?

How many different rational numbers have absolute value 4?

Answer: 2

2: the numbers 4 and −4, both 4 units from 0.

• |⁠x| ≥ 0 — correct. Yes. A distance is never negative.
• |⁠x| < 0. A distance cannot be negative.
• |⁠x| = x. This fails for negative x: |⁠−2| = 2, but x = −2.
• −5/3. An absolute value is never negative.
• 3/5. The absolute value keeps the size 5/3 and drops the sign.
• 5/3 — correct. Yes. −5/3 is 5/3 of a unit from 0.