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'''Absolute value''' is a [[function]] measuring a number's distance from zero. The absolute value of <math>x</math> is written <math>|x|</math>.
*If <math>x</math> is a positive number, then <math>|x| = x</math>.
*If <math>x</math> is a negative number, then <math>|x| = -x</math>.
*If <math>x=0</math> then <math>|x| = 0</math>.
For example, <math>|-3| = 3</math>, and <math>|5| = 5</math>. Notice that <math>|x|</math> is never negative.
Absolute value has several useful properties. One is the ''multiplicative'' property. If <math>x</math> and <math>y</math> are two numbers, then <math>|xy| = |x| \times |y|</math>. Another is the ''triangle inequality'', which is the fact that <math>|x+y| \leq |x| + |y|</math>. For example, if <math>x = 3</math> and <math>y = -5</math>, then <math>|x+y| = |3 + (-5)| = |3 - 5| = |-2| = 2</math>, while <math>|x| + |y| = |-5| + |3| = 5 + 3 = 8</math>. In this case, the triangle inequality is the fact that 2 is not more than 8.
Absolute value has several useful properties. One is the ''multiplicative'' property. If <math>x</math> and <math>y</math> are two numbers, then <math>|xy| = |x| \times |y|</math>. Another is the ''triangle inequality'', which is the fact that <math>|x+y| \leq |x| + |y|</math>. For example, if <math>x = 3</math> and <math>y = -5</math>, then <math>|x+y| = |3 + (-5)| = |3 - 5| = |-2| = 2</math>, while <math>|x| + |y| = |-5| + |3| = 5 + 3 = 8</math>. In this case, the triangle inequality is the fact that 2 is not more than 8.
If <math>x</math> is a [[real number]], then <math>\sqrt{x^2} = |x|</math>. The absolute value is necessary because the principal [[square root]] is, by definition, nonnegative.
[[Complex number]]s also have an absolute value (sometimes called the modulus). If <math>z = x+iy</math> is a complex number with real part <math>x</math> and imaginary part <math>y</math>, then <math>|z| = \sqrt{x^2 + y^2}</math>. If we represent <math>z</math> as a point in the complex plane with coordinates <math>(x,y)</math>, then <math>|z|</math> is the distance from this point to the origin. The absolute value of complex numbers also has the multiplicative property and satisfies the triangle inequality.
[[Complex number]]s also have an absolute value (sometimes called the modulus). If <math>z = x+iy</math> is a complex number with real part <math>x</math> and imaginary part <math>y</math>, then <math>|z| = \sqrt{x^2 + y^2}</math>. If we represent <math>z</math> as a point in the complex plane with coordinates <math>(x,y)</math>, then <math>|z|</math> is the distance from this point to the origin. The absolute value of complex numbers also has the multiplicative property and satisfies the triangle inequality.