Absolute value

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

$|x| \equiv \sgn(x) \cdot x = ([x \ge 0] - [x \le 0]) \cdot x,\quad x \in \R, \,$

where $\scriptstyle \sgn(x) \,$ is the sign function and $\scriptstyle [\cdot] \,$ is the Iverson bracket.

On the real number line, this means if $\scriptstyle x \,$ is positive, then $\scriptstyle |x| \,=\, (+1) \, x \,$ , while if $\scriptstyle x \,$ is negative, then $\scriptstyle |x| \,=\, (-1) \, x \,$ .

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

Complex norm

The complex norm (or complex modulus) $\scriptstyle |z| \,$ of a complex number $\scriptstyle z \,=\, x + i y \,$ is its distance from 0

$|z| \equiv \sqrt{x^2 + y^2},\quad z \in \C. \,$

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

Note that by symmetry there are $\scriptstyle 4 + 8k,\, k \,\ge\, 0, \,$ Gaussian integers with absolute value $\scriptstyle c \,$ , i.e.

$\{ c, c i, - c, - c i \} ~ \cup_{j=1}^{k} ~ \{ a_j + b_j i, a_j - b_j i, - a_j + b_j i, - a_j - b_j i, b_j + a_j i, b_j - a_j i, - b_j + a_j i, - b_j - a_j i \}, \,$

where $\scriptstyle k \,$ is the number of Pythagorean triples that correspond to the hypotenuse $\scriptstyle c. \,$

Note that $\scriptstyle k \,>\, 0 \,$ only if $\scriptstyle c \,$ corresponds to the hypotenuse in a Pythagorean triple (i.e. $\scriptstyle c^2 \,=\, a^2 + b^2 \,$ has a solution in nonzero rational integers, see A009000). If $\scriptstyle c \,$ 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.

Absolute value

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Complex norm

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