Absolute value

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The absolute value of a real number $x$, denoted $|x|$, is the unsigned portion of $x$. Geometrically, $|x|$ is the distance between $x$ and zero on the real number line.

The absolute value function exists among other contexts as well, including complex numbers.

Real numbers

When $x$ is real, $|x|$ is defined as \[|x| = \begin{cases} x & \text{for } x \ge 0,\\ -x & \text{for } x \le 0.\end{cases}\] For all real numbers $x$ and $y$, we have the following properties:

  • (Alternative definition) $|x| = \sqrt{x^2}$
  • (Non-negativity) $|x| \ge 0$
  • (Positive-definiteness) $|x| = 0 \iff x=0$
  • (Multiplicativeness) $|xy| = |x| |y|$
  • (Triangle Inequality) $|x+y| \le |x|+|y|$
  • (Symmetry) $|x| = |-x|$

Note that

\[|x| \le y \iff -y \le x \le y\]

and

\[|x| \ge y \iff x \ge y \text{ or } x \le -y.\]

Complex numbers

For complex numbers $z$, the absolute value is defined as $|z| = \sqrt{x^2+y^2}$, where $x$ and $y$ are the real and imaginary parts of $z$, respectively. It is equivalent to the distance between $z$ and the origin, and is usually called the complex modulus.

Note that $|z| = |\overline{z}| = \sqrt{z\overline{z}}$, where $\overline{z}$ is the complex conjugate of $z$.

Examples

  1. If $|x|=k$, for some real number $k$, then $x=k$ or $x=-k$.
  2. If $|ax| = k$, for some real numbers $a$, $k$, then $ax = k$ or $ax = -k$, and therefore $x = \frac{k}{a}$ or $x = -\frac{k}{a}$.

Problems

  1. Find all real values of $x$ if $-|x| = x-6$.
  2. Find all real values of $x$ if $5 + 8 \cdot |4x| = 69$.
  3. (AMC 12 2000) If $|x - 2| = p$, where $x < 2$, then find $x - p$.

See Also

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