Two vertical bars around a number trace back to Weierstrass in 1841
The absolute value strips a number of its sign, so both 3 and minus 3 become 3. Think of it as distance from zero. That simple idea, written with the twin bars Karl Weierstrass introduced in 1841, grows into the notions of distance and magnitude used across mathematics and physics.
For a real number the rule is short: leave it alone if it is zero or positive, flip its sign if it is negative. The result is never negative. An equivalent definition takes the positive square root of the number squared. Geometrically, it measures how far a number sits from zero on the number line, and the absolute value of the difference between two numbers gives the distance between them, the standard way of measuring distance on the real line and the model for more abstract distance functions.
The vocabulary has a history. In 1806 Jean-Robert Argand coined module, French for unit of measure, for the size of a complex number, and English borrowed the Latin form modulus in 1866. The phrase absolute value was in French use by 1806 and English by 1857. Programmers typically write abs(x), a habit dating back to the earliest high-level programming languages. The same bars mean something else in other settings: around a set they denote how many elements it has, around a matrix its determinant.
Four basic properties carry the idea into new territory: it is never negative, it is zero only for zero, the absolute value of a product equals the product of absolute values, and the absolute value of a sum is at most the sum of the absolute values. Because complex numbers cannot be ordered, their absolute value is defined instead as distance from the origin in the complex plane, calculated with Pythagoras's theorem, and it keeps all four properties.
As a function, the real absolute value is continuous everywhere but has a sharp corner at zero, where it cannot be differentiated. It falls on the negative side and rises on the positive, forming a convex V built from two straight lines. Since a number and its negative give the same output, it cannot be inverted, and applying it twice changes nothing.
Source: Absolute value
