The absolute value of a number represents its distance from zero on the number line. Distance is always non‑negative, so absolute value is always ≥ 0. For a positive number, |x| = x. For a negative number, |x| = –x (its opposite).
⚠Note: Absolute value is always non‑negative. This tool supports positive integers, negative integers, decimals, and scientific notation. Results are rounded to 6 decimal places.
What Is Absolute Value?
Absolute value is a fundamental concept in mathematics, written as |x| and read as "the absolute value of x." It measures how far a number is from zero on the number line, ignoring whether it is positive or negative.
Distance Concept
Absolute value tells you how far a number is from zero. Because distance can never be negative, absolute value is always ≥ 0.
Notation
|5| means the absolute value of 5, and |-5| means the absolute value of -5. Both equal 5.
Key Rule
Positive numbers stay the same (|5| = 5). Negative numbers become positive (|-5| = 5). Zero stays zero (|0| = 0).
Geometric Meaning
On the number line, |a - b| gives the distance between points a and b. For example, |3 - (-2)| = 5 means 3 and -2 are 5 units apart.
💡 Example: On the number line, both 5 and -5 are exactly 5 units away from zero, so |5| = |-5| = 5.
|x| < a (where a > 0) means -a < x < a. |x| > a means x < -a or x > a. For |ax + b| ≤ c, isolate the absolute value first, then rewrite as two inequalities.
What does the graph of |x| look like?▼
y = |x| creates a V shape with its vertex at the origin (0,0). The left side follows y = -x (slope -1), and the right side follows y = x (slope 1). The graph is continuous everywhere but has a corner at x = 0, where it is not differentiable.
What is the absolute value (modulus) of a complex number?▼
For a complex number a + bi, its modulus is |a + bi| = √(a² + b²). This is the distance from the point (a, b) to the origin on the complex plane. For example, |3 + 4i| = √(9 + 16) = 5.
How do computers compute absolute value efficiently?▼
Computers use bitwise tricks like (x ^ (x >> 31)) - (x >> 31) for signed integers, which completes the calculation in just one or two instructions. Modern CPUs also have a dedicated ABS instruction.
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