Everyday usage and use in mathematics

The word absolute in everyday English means means "total" or "complete." The Oxford English Dictionary provides many definitions for the word absolute [1] such as "Free from dependency, autonomous; not relative" or "Complete, perfect."

For example, an absolute monarch is a ruler who has total control over their country. The term absolute zero refers to the smallest possible temperature and a complete lack of atomic movement.

The term absolute value is used in two related ways in mathematics. The first meaning of absolute value is the absolute value of a real number. The absolute value of a real number is the piecewise function given by the formula \[|x| = \begin{cases} x & \text{if $x \geq 0$} \\ -x & \text{if $x < 0$.} \end{cases} \] The absolute value captures the size of a number wihtout regard to whether the number is positive or negative. The absolute value $|x|$ of $x$ is the distance from $x$ to zero on a number line.

The term absolute value is also used to refer to the absolute value of a complex number. This is sometimes referred to as the modulus. If $a + bi$ is a complex number, then the absolute value of $a + bi$ is given by the formula \[|a + bi| = \sqrt{a^2 + b^2}.\] The absolute value of a complex number is the distance from the complex number to zero in the complex plane.

Summary

The term absolute value captures the total distance of a number to zero without regard to whether the number is positive, negative, or complex.

Importance in mathematics

The absolute value function is important when describing the size of a change without regard to whether the change is positive or negative. For example, an interval $(a,b)$ is the set of points satisfying the inequality \[ \left|x - \frac{a + b}{2} \right| \leq \frac{b - a}{2} \] For this reason, absolute values are useful in describing bounds on the error in measuring a quantity. In the complex plane, absolute value inequalities are used to describe discs rather than intervals. If $x_0 + i y_0$ is a complex number, then the set of complex numbers $x + i y$ satisfying the inequality \[ |(x + i y) - (x_0 + i y_0)| < r \] is a disc of radius $r$ centered at the point $(x_0, y_0)$.

Exercises

  1. Give a definition of the word absolute. Is there a similar term in your native language?
  2. Compare the use of absolute in absolute value and the use of the word absolute in absolute maximum
  3. Describe the set of real numbers $x$ that satisfy the absolute value inequality $|x - 3| \geq 5$.

References

[1] Oxford English Dictionary