Absolute value means “how far a number is from zero” on the number line, without caring about direction, so the answer is always zero or positive.

Quick Scoop: Core idea

  • The absolute value of a number is its distance from 0 on the number line.
  • Distance is never negative, so absolute values are always positive or zero.
  • We write absolute value using vertical bars: ∣x∣|x|∣x∣. For example, ∣5∣=5|5|=5∣5∣=5 and ∣−5∣=5|-5|=5∣−5∣=5.

Think of it like asking only “How far?” and never “Which way?”.

Simple examples

  • ∣7∣=7|7|=7∣7∣=7 because 7 is 7 units to the right of 0.
  • ∣−7∣=7|-7|=7∣−7∣=7 because −7-7−7 is 7 units to the left of 0.
  • ∣0∣=0|0|=0∣0∣=0 because 0 is zero units from 0.

So positive and negative versions of the same number have the same absolute value.

Formal meaning (a bit more precise)

For any real number aaa:

  • If a≥0a\ge 0a≥0, then ∣a∣=a|a|=a∣a∣=a.
  • If a<0a<0a<0, then ∣a∣=−a|a|=-a∣a∣=−a (you flip the sign to make it positive).

Example:

  • a=−3a=-3a=−3 is negative, so ∣−3∣=−(−3)=3|-3|=-(-3)=3∣−3∣=−(−3)=3.

Number line picture (in words)

Imagine a horizontal line with 0 in the middle, positives to the right, negatives to the left.

  • Absolute value is “how many steps from 0,” whether you’re stepping left or right.
  • That’s why ∣3∣|3|∣3∣ and ∣−3∣|-3|∣−3∣ both give 3: same distance, different direction.

Where it shows up in real life

  • Temperature change : Going from 2°C to −5-5−5°C is a change of ∣2−(−5)∣=∣7∣=7|2-(-5)|=|7|=7∣2−(−5)∣=∣7∣=7 degrees.
  • Games / scoring : If a score moved from 10 to 4, the change is ∣10−4∣=6|10-4|=6∣10−4∣=6 points, regardless of “up” or “down”.
  • Error / difference : When you measure something and compare to the “true” value, absolute value gives the size of the error.

In many problems, absolute value is used to talk about “difference” or “error” without worrying about plus or minus.

Tiny peek beyond real numbers (optional)

  • For more advanced math, absolute value is also called modulus , and it generalizes to complex numbers and vectors as a kind of magnitude or length.

TL;DR:
Absolute value is the distance a number is from zero, so it throws away the sign and keeps only “how big” the number is. ∣5∣=5|5|=5∣5∣=5, ∣−5∣=5|-5|=5∣−5∣=5, and ∣0∣=0|0|=0∣0∣=0.

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