A magnitude is a measure of the size of a mathematical entity. For example, the magnitude of a complex number is the distance from the number (graphed on the complex plane) to the origin, a measure of the size of a complex number. The magnitude is generally a nonnegative real number.
Formulaically, the magnitude of a real number is its absolute value , sometimes written . The magnitude of a complex number equals . Both types of magnitude are bound by a form of the Triangle Inequality which states that .
For complex numbers and , we have the identity . Because the absolute value of a real number equals its magnitude when treated as a complex number, the identity also holds for absolute values of real numbers.
Let and be complex numbers.
We have so
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