Absolute Value Calculator – |x| with Properties

The absolute value strips a number’s sign, leaving its distance from zero: |−12| = 12, |5| = 5, and for expressions first evaluate inside — |3−9| = |−6| = 6. The calculator applies the piecewise definition, telling you which branch fired, so the habit "absolute value never negative" is reinforced with each run. It also simplifies the classic identities: |x|² = x² (both equal 9 for x = −3), √(x²) = |x|, and |a·b| = |a|·|b| — so |−4×7| = 28. Magnitude comparisons print the distance between two points |a−b|, e.g. |−7−3| = 10, which is the number-line gap. For inputs in scientific or decimal form the result keeps full precision, and the tool notes the special case |0| = 0, the only value whose absolute value equals its negative.

1. 1

   Type a number or expression

   Such as -12, 3-9, or -4×7

2. 2

   Evaluate inside first

   Expressions reduce to a single value before the bars apply

3. 3

   Apply the piecewise rule

   Positive stays, negative flips — the active branch is shown

4. 4

   Review the identities

   x², √(x²) and |a−b| distance forms accompany the result

• 1

  Evaluate the inside completely before dropping the sign

• 2

  |x|² = x² always, but √(x²) = |x|, not x

• 3

  |a−b| is the distance between a and b on the number line

• Answering |3−9| as −6 — inside first, then the bars

• Assuming |x| can output negatives; the minimum is 0

• Writing √(x²) = x, wrong for x = −3 where it equals 3

• Distributing bars over addition: |−3+5| is 2, not 8