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Glacius
FoundationsConcept reference

Absolute values

The absolute value of a real number is its nonnegative distance from zero on the number line.

1 min read1 figureSources & further reading
On this page 8 sections
  1. Overview
  2. Absolute value measures distance from zero
  3. Check your understanding
  4. Check your understanding
  5. Key takeaway
  6. Further questions
  7. Sources & further reading
  8. Concept connections

01Absolute value measures distance from zero#

Absolute value measures distance from zero. The point −4-4 sits four units left of zero, so its absolute value is 44. Two vertical bars write this distance as ∣−4∣|-4|.

On a unit-spaced number line, a directed move from -4 to 0 is labeled distance 4.On a unit-spaced number line, a directed move from -4 to 0 is labeled distance 4.
Figure 1On a unit-spaced number line, a directed move from -4 to 0 is labeled distance 4.
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The points −4-4 and 44 have different positions but the same distance from zero. Both absolute values are 44. Distance gives the size without a direction.

∣−4∣=∣4∣=4|-4|=|4|=4

Zero has distance zero: ∣0∣=0|0|=0. Absolute value is never negative. The same idea works between integer ticks: ∣−2.5∣=2.5|-2.5|=2.5.

Check your reasoning

Calculate ∣−9∣|-9|.

Show answer and explanation
9

The point −9 is nine units from zero.

02Check your understanding#

Check your reasoning

Which value equals ∣0∣|0|?

  1. A11
  2. B−1-1
  3. C00
Show answer and explanation
00

There is no distance to travel from zero to itself.

03Check your understanding#

Check your reasoning

A robot is at position −6-6 metres relative to its start. What is its distance from the start, in metres?

Show answer and explanation
6

Its position is negative, but its distance is ∣−6∣=6|-6|=6 metres.

Key takeaway

Absolute value gives distance from zero, so it is never negative.

  • Compute a real number’s distance from zero

Further questions

Does absolute value always make a number positive?
It leaves a positive number unchanged, turns a negative number positive, and leaves zero at zero. Nonnegative is the precise word.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Absolute values.” Math behind ML. /learn/f-absolute