Absolute Value Calculator

Absolute value is the distance of a number from zero on the number line, without regard to direction. It is always non-negative. |5| = 5 because 5 is five units from zero. |-5| = 5 because -5 is also five units from zero. The notation |x| reads as the absolute value of x, and the formal definition is |x| = x when x >= 0 and |x| = -x when x < 0.

Drop the negative sign. |-7| = 7. The rule comes from the definition: when x is negative, |x| equals -x, and negating a negative number gives a positive one. So |-7| = -(-7) = 7. This calculator's simple mode shows the sign of the original input (positive, negative, or zero) so you can see at a glance which branch of the definition applied.

It is the distance between x and y on the number line. The subtraction puts the two values into a single signed gap, and the absolute value strips the sign so the answer is a non-negative distance. |5 - 8| = |-3| = 3, and |8 - 5| = |3| = 3, which is why distance is symmetric: it does not matter which value you list first. Use this calculator's difference mode for any number-line distance, time elapsed between two readings, or temperature swing.

For a + bi, the absolute value (also called modulus or magnitude) is sqrt(a^2 + b^2). It is the distance from the origin to the point (a, b) on the complex plane, which is just the Pythagorean theorem applied to the real and imaginary parts. So |3 + 4i| = sqrt(9 + 16) = sqrt(25) = 5, the same 3-4-5 right triangle you know from geometry. |5 + 12i| = 13 by the 5-12-13 triple.

Yes, when applied to a complex number, absolute value and modulus mean the same thing: the length of the vector from the origin to a + bi on the complex plane. Some textbooks use absolute value for real numbers only and reserve modulus for complex numbers, but the two terms describe the same concept. This calculator labels the complex output as both absolute value and modulus to keep both vocabularies covered.