Absolute value

The absolute value of a real number is its distance from zero on the number line, expressed as a nonnegative number. It leaves nonnegative numbers unchanged and changes negative numbers to their opposites.

Piecewise function: A function defined by different formulas on different parts of its domain. Absolute value uses one formula for nonnegative inputs and another for negative inputs.
Real number: A number that can be represented by a point on the continuous number line. Absolute value assigns a distance from zero to every real number.
Distance: A nonnegative measure of separation between points or objects. The absolute value of a difference gives distance on the real line.
Square function: The function that maps each number x to x². Both functions remove the sign of real inputs, but squaring also changes magnitude.
Triangle inequality: An inequality stating that the length of one side of a triangle is no greater than the sum of the other two. For real numbers, it gives the bound |x+y| ≤ |x|+|y|.
Number line: A line on which real numbers correspond to points ordered by their numerical values. Absolute value measures the distance between a number’s point and zero.
Metric space: A set equipped with a distance function satisfying nonnegativity, identity, symmetry, and the triangle inequality. Absolute difference defines the standard distance between real numbers.
Square root: A nonnegative number whose square equals a given nonnegative number. For real x, √(x²)=|x|, not always x.
Absolute value equation: An equation in which a variable appears inside an absolute value expression. Solving |x|=a means finding points at distance a from zero.
Opposite number: The additive inverse of a number, which sums with it to zero. A negative number’s absolute value is its opposite.

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