Magnitude Calculator | Find Numerical Distance from Zero

In mathematics, magnitude and absolute value are essentially the same concept for real numbers—both refer to the distance from zero on the number line, disregarding the sign. However, "magnitude" is often used in broader contexts, particularly in physics and engineering. For vectors, magnitude refers to the length of a vector, and for complex numbers, it refers to the distance from the origin in the complex plane (also called modulus). Absolute value typically refers specifically to the mathematical operation on a real number.

The absolute value is never negative because it measures distance from zero on the number line, and distance is always a non-negative quantity. Mathematically, the absolute value function is defined to eliminate any negative sign: for positive numbers, it returns the number unchanged; for negative numbers, it returns the number with its sign reversed (making it positive). This property makes absolute value useful in many applications where only the magnitude matters, not the direction or sign.

Solving equations with absolute values typically requires considering multiple cases due to the piecewise nature of the absolute value function. For an equation like |x| = a (where a > 0), there are two solutions: x = a and x = -a. For more complex equations like |ax + b| = c, you should set up two equations: ax + b = c and ax + b = -c, then solve each separately. Inequalities with absolute values (like |x| < a) represent intervals on the number line (in this case, -a < x < a). Always verify your solutions by substituting them back into the original equation.