1.27 Absolute Value Definitions

Absolute Value Definitions establishes the vocabulary for the operation that strips a signed number of its direction while preserving its magnitude, grounding that operation in its geometric interpretation as a distance on the number line, and extends the same idea to describe the distance separating two arbitrary numbers rather than a single number from the origin.

Absolute Value

The absolute value of a real number is its magnitude without regard to sign, always expressed as a nonnegative value, denoted by placing vertical bars around the number, such as |−7| = 7 and |7| = 7. Absolute value is formally defined so that the absolute value of a nonnegative number equals the number itself, while the absolute value of a negative number equals its opposite, ensuring the result is never negative.

Distance from Zero

Distance from zero is the geometric interpretation underlying absolute value: the absolute value of a number represents how far that number's position lies from the origin on the number line, measured as a nonnegative length regardless of which direction the number lies in. Since −7 and 7 are each exactly seven units away from the origin, both have an absolute value of 7, even though they lie on opposite sides of it.

Number-Line Distance

Number-line distance is the length separating any two points on the number line, computed by taking the absolute value of the difference between their coordinates, regardless of which point is treated as the starting point. The number-line distance between 3 and 9 is |9 − 3| = 6, and computing it in the opposite order, |3 − 9| = 6, produces the identical result, since absolute value removes any dependence on the order of subtraction.

Absolute Difference

The absolute difference between two numbers is the specific numerical result obtained by subtracting one from the other and taking the absolute value of that result, serving as the algebraic expression of number-line distance. The absolute difference between 3 and 9 is 6, calculated as |9 − 3| or equivalently |3 − 9|, and this value represents both the magnitude of their numerical difference and the physical distance separating them on the number line.

Together, these definitions ground absolute value in its geometric meaning as distance from zero, and extend that meaning through number-line distance and absolute difference to describe the separation between any two arbitrary real numbers, providing the foundation for later work involving absolute value equations and inequalities.