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Absolute value

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The absolute value (abs(x), or ) of a real number is its distance from 0

where is the sign function and is the Iverson bracket.

On the real number line, this means if is positive, then , while if is negative, then .

For example, the absolute value of the first Stieltjes constant, –0.0728158454836767248..., is 0.0728158454836767248...

Complex norm

The complex norm (or complex modulus) of a complex number is its distance from 0

On the complex plane, all numbers with absolute value can be found by drawing a circle centered at 0 with radius . For example, the following diagram shows the complex numbers with absolute value of 5

Nums wAbsVal 5 Diagram.png

Note that by symmetry there are Gaussian integers with absolute value , i.e.

where is the number of Pythagorean triples that correspond to the hypotenuse

Note that only if corresponds to the hypotenuse in a Pythagorean triple (i.e. has a solution in nonzero rational integers, see A009000). If does not correspond to the hypotenuse in a Pythagorean triple there are only four Gaussian integers (two purely real numbers and two purely imaginary numbers) having the specified absolute value.

The diagram above shows red dots for the twelve Gaussian integers having an absolute value of 5.

See also

  • {{sgn}} (mathematical function template)
  • {{abs}} (mathematical function template)