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Absolute Value |x|
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Sources & Methodology
Absolute value definition verified against IEEE 754, Khan Academy mathematics curriculum, and standard textbook definitions.
Khan Academy — Absolute Value
Curriculum-aligned definition and examples of absolute value used as reference for step-by-step output and property explanations
National Council of Teachers of Mathematics (NCTM)
Mathematical standards confirming absolute value definition, notation, and curriculum placement
Formula: |x| = x if x ≥ 0; |x| = −x if x < 0. Implementation: if the input number is negative, multiply by −1. If zero or positive, return as-is. Result is always non-negative. The number line distance from 0 to x equals |x| by definition.
⏱ Last reviewed: April 2026
What Is Absolute Value and How Do You Calculate |x|
The absolute value of a number is its distance from zero on the number line, regardless of direction. Written with vertical bars as |x|, it always returns a non-negative result. The concept appears throughout mathematics — from solving equations to defining distances in coordinate geometry and measuring error in statistics.
The Formula
|x| = x if x ≥ 0 |x| = −x if x < 0
Rule 1: If x is positive or zero, |x| = x (unchanged).
Rule 2: If x is negative, |x| = −x (multiply by −1 to flip the sign).
Examples:
|7| = 7 (already positive)
|−7| = −(−7) = 7 (flip the negative)
|0| = 0 (zero stays zero)
|−3.5| = 3.5
Rule 2: If x is negative, |x| = −x (multiply by −1 to flip the sign).
Examples:
|7| = 7 (already positive)
|−7| = −(−7) = 7 (flip the negative)
|0| = 0 (zero stays zero)
|−3.5| = 3.5
Key Properties of Absolute Value
- Non-negativity: |x| ≥ 0 for all real x
- Identity: |x| = 0 if and only if x = 0
- Symmetry: |−x| = |x| (positive and negative have same absolute value)
- Multiplicative: |x × y| = |x| × |y|
- Triangle inequality: |x + y| ≤ |x| + |y|
- Subadditivity: |x − y| ≥ ||x| − |y||
Absolute Value Examples Table
| Number (x) | |x| | Sign | Distance from 0 |
|---|---|---|---|
| −100 | 100 | Negative → flip | 100 units |
| −7 | 7 | Negative → flip | 7 units |
| −0.5 | 0.5 | Negative → flip | 0.5 units |
| 0 | 0 | Zero | 0 units |
| 3.14 | 3.14 | Positive → unchanged | 3.14 units |
| 15 | 15 | Positive → unchanged | 15 units |
| −1,000 | 1,000 | Negative → flip | 1,000 units |
Absolute Value Equations
When solving |x| = a, there are always two solutions for any positive a: x = a and x = −a. This is because both a and −a are the same distance from zero. For example, |x| = 5 means x = 5 or x = −5. When |x| = 0, the only solution is x = 0. When |x| = −a (negative), there is no solution — absolute value cannot be negative.
💡 Memory tip: Think of absolute value as the "unsigned version" of a number — you strip away the positive or negative sign and keep only the magnitude. A temperature of −20°C and +20°C are at the same distance from 0°C; their absolute values are both 20.
Frequently Asked Questions
Absolute value of a number x, written |x|, is its distance from zero on the number line. It is always non-negative. |5| = 5, |−5| = 5, |0| = 0. Absolute value removes the sign and keeps only the magnitude of the number.
The absolute value of −7 is 7. Written |−7| = 7. Since −7 is negative, apply the formula |x| = −x: |−7| = −(−7) = 7. The number −7 is 7 units from zero on the number line, so its absolute value is 7.
The absolute value formula is: |x| = x if x ≥ 0 (positive or zero: keep as-is), and |x| = −x if x < 0 (negative: multiply by −1). This always produces a non-negative result. Example: |−8| = −(−8) = 8.
No. By definition, |x| ≥ 0 for all real numbers. Absolute value is always zero or positive. The only time |x| = 0 is when x = 0. If you see |x| = −5, there is no solution because absolute value can never equal a negative number.
The absolute value of 0 is 0. |0| = 0. Zero is the only number that equals its own absolute value and is also 0 units from zero on the number line. Zero is neither positive nor negative, so the formula |x| = x applies (since 0 ≥ 0).
The absolute value of any negative number −x (where x > 0) is x. Simply remove the negative sign. Examples: |−3| = 3, |−15.7| = 15.7, |−1,000| = 1,000. Formally, |−x| = −(−x) = x.
For |x| = a where a > 0: two solutions exist: x = a and x = −a. Example: |x| = 5 gives x = 5 or x = −5. For |x| = 0: only solution is x = 0. For |x| = −a: no solution (absolute value cannot be negative).
|−15| = 15. Apply the formula: since −15 is negative, |−15| = −(−15) = 15. The number −15 is 15 units from zero on the number line, so its absolute value is 15.
Absolute value |x| applies to one number — it gives the non-negative form of that single number. Absolute difference |A − B| applies to two numbers — it gives the non-negative distance between them. Absolute difference uses absolute value: subtract first, then apply |x|.
Absolute value is used to measure distance (always positive), solve inequalities involving ranges around a point, define metrics in geometry and analysis, compute error magnitudes (MAE, MAD) in statistics, and in programming to get unsigned numeric values. It is fundamental because distance cannot be negative.
|3.5| = 3.5. Since 3.5 is already positive (3.5 ≥ 0), the formula |x| = x applies and the value is unchanged. Absolute value only changes negative numbers — positive numbers and zero are returned as-is.
The same rules apply: |−3/4| = 3/4, |5/8| = 5/8, |−7/2| = 7/2 = 3.5. If the fraction is negative, remove the sign. If it is positive or zero, it stays unchanged. The result is always a non-negative fraction or decimal.
Absolute value models real situations where magnitude matters but direction does not: temperature change (−10° vs +10° are both 10 degrees from 0), distance travelled (always positive), profit/loss magnitude, error in measurements, credit/debit amounts when you just want the size, and speed (as opposed to velocity which has direction).
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