Calculate the absolute value |x| of any number or expression instantly.
Absolute value of a number is its distance from zero on the number line, always expressed as a non-negative number. The absolute value of both +7 and -7 is 7, because both are 7 units from zero. Written as |x|, absolute value appears throughout mathematics, physics, computer science, and everyday calculations involving distances and magnitudes.
|x| = x when x ≥ 0. |x| = -x when x < 0 (negating a negative gives positive). |x| ≥ 0 always — absolute value is never negative. |x × y| = |x| × |y|. |x + y| ≤ |x| + |y| (triangle inequality — fundamental in geometry and analysis). |-x| = |x|.
|x| = 5 means x = 5 or x = -5 (two solutions). |x - 3| = 7 means x - 3 = 7 (x = 10) or x - 3 = -7 (x = -4). |2x + 1| = 9 means 2x + 1 = 9 (x = 4) or 2x + 1 = -9 (x = -5). Always check both solutions by substituting back into the original equation.
Absolute value represents magnitude without direction. Temperature differences: Whether it is 5°C above or below normal, the deviation is |5| = 5 degrees. Financial gains/losses: A stock moving ±10% changes by |10%| = 10% in magnitude. GPS distance: Distance is always positive regardless of direction. Error measurement: |Actual - Estimated| gives the magnitude of error.
The abs() function is one of the most commonly used mathematical functions in programming. In Python: abs(-15) returns 15. In JavaScript: Math.abs(-15) returns 15. Used for: finding the difference between two values regardless of order, checking if values are within a tolerance range, normalizing data, and distance calculations in algorithms and game development.
Absolute value represents distance from zero on a number line, regardless of direction, which makes it useful anywhere the sign of a number matters less than its magnitude. In finance, the absolute value of a gain or loss is often used when comparing volatility, since a $500 loss and a $500 gain represent the same magnitude of price movement even though their signs are opposite. In physics, absolute value appears when calculating speed from velocity, since speed is always non-negative even if velocity has a direction (positive or negative).
In programming and data analysis, absolute value functions are used constantly to measure error — the difference between a predicted value and an actual value is often taken as an absolute value so that overestimates and underestimates are treated with equal weight when calculating something like mean absolute error in a forecasting model.
Both absolute value and squaring turn negative numbers positive, but they behave differently for numbers between -1 and 1. Squaring a fraction like -0.5 gives 0.25, while the absolute value gives 0.5 — a significant difference in statistical calculations where the choice between the two methods changes the sensitivity to outliers. Understanding this distinction is useful for anyone comparing standard deviation (which uses squaring) against mean absolute deviation (which uses absolute value).
Solving an equation like |x| = 5 requires recognizing that x could be either 5 or -5, since both values are exactly 5 units from zero. This two-solution property is a common early source of confusion in algebra, since it breaks the usual expectation of a single answer per equation. Inequalities involving absolute value, like |x| < 5, translate into a range (-5 < x < 5), while |x| > 5 translates into two disconnected ranges (x < -5 or x > 5) — a distinction that trips up many students the first time they encounter it.
Absolute value is the distance of a number from zero on the number line, always expressed as a non-negative number. |−5| = 5 and |5| = 5.
Absolute value is represented by two vertical bars around the number, like |x|. For example, |−8| = 8.
No. Absolute value is always zero or positive. It represents distance, which can never be negative.
Absolute value is used in mathematics, physics (distance, speed), programming (error checking), and statistics (mean absolute deviation).
