What is "Absolute Value"? — Definition
The absolute value of a number is its distance from zero on the number line, regardless of direction (sign). In mathematical notation it is written as |x|.
Formally: |x| = x if x ≥ 0, and |x| = −x if x < 0. So |−7| = 7 and |7| = 7, while |0| = 0.
Is Absolute Value Useful?
Absolutely! Here are the most common real-world uses:
- Distance. The distance between two points on a number line is always the absolute value of their difference: |a − b|.
- Error analysis. Scientists report measurement error as a positive quantity — that is, the absolute value of the deviation.
- Finance. A loss of $500 and a gain of $500 have the same magnitude (|−500| = |500| = 500). Absolute value lets you compare sizes of changes without worrying about direction.
- Temperature differences. The difference between −10 °C and +10 °C is 20 °, which is |−10 − 10| = 20.
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Programming. Many algorithms need a non-negative magnitude — the
abs()function in every programming language implements this concept.
Absolute Value Functions and Graphs
The simplest absolute value function is f(x) = |x|. Its graph is a V-shape:
- For x ≥ 0, the graph follows the line y = x (slope +1, first quadrant).
- For x < 0, the graph follows y = −x (slope −1, second quadrant).
- The vertex is at the origin (0, 0).
More complex forms like f(x) = |x − h| + k shift the vertex to (h, k). The shape stays V-shaped but moves around the plane.
Absolute Value Equations
An equation such as |x − 3| = 5 has two solutions because both x − 3 = 5 and x − 3 = −5 satisfy it:
- x = 8
- x = −2
General rule: |A| = b (with b ≥ 0) gives A = b or A = −b. If b < 0, there is no solution.
Absolute Value Inequalities
Inequalities with absolute value split into two cases:
- |x| < a (or ≤) → "and" condition: −a < x < a. The solution is a bounded interval.
- |x| > a (or ≥) → "or" condition: x < −a or x > a. The solution is an unbounded set (two rays).
Example: |x − 2| ≤ 3 means −1 ≤ x ≤ 5.
How to Use This Absolute Value Calculator
- Enter any real number — positive, negative, zero, or a decimal like −3.14 or 0.001.
- Choose a unit type if you want the result in a physical context: metric lengths (mm, cm, m, km), imperial lengths (in, ft, yd, mi), metric weight (g, kg, t), imperial weight (oz, lb), or a world currency.
- If you selected Currency, pick your currency from the dropdown — from US Dollar (USD) to Russian Ruble (RUB), Euro, Pound, and many more.
- Click Calculate and instantly see the absolute value, the formula |x| = result, and an interpretation of whether your number was positive, negative, or zero.
The calculator saves your result and generates a shareable link so you can send it to anyone — useful for homework, finance, engineering, or just settling a debate!