Modulus (source code)
= Modulus
{title2=$|z|$}
= Moduli
{synonym}
The <modulus> of a <complex number> $z=x+iy$ is
$$
|z|=\sqrt{x^2+y^2}=\sqrt{z\overline z}.
$$
It extends <absolute value> to the <complex plane>: the <modulus> is the <Euclidean distance> from $z$ to zero, or the <Euclidean norm> of its coordinate <vector>. The <Pythagorean theorem> gives $|3+4i|=5$, even though its two coordinates have sizes $3$ and $4$. The <modulus> is always nonnegative, and is zero exactly at zero.
Two essential rules are
$$
|zw|=|z||w|,\qquad |z+w|\leq |z|+|w|.
$$
The second is the <triangle inequality>. For $z=3+4i$ and $w=1+i$, the product has <modulus> $\sqrt{50}=5\sqrt2\approx7.071068$, while the sum has <modulus> $\sqrt{41}\approx6.403124<5+\sqrt2\approx6.414214$. The strict inequality records that the two displacement <vectors> point in different directions.
The <Euclidean distance> between two <complex numbers> $z,w$ is $|z-w|$. Consequently $|z-w|<\varepsilon$ expresses geometric closeness without treating the two coordinates separately. This makes the <modulus> a natural measure of error in <complex analysis>; the <triangle inequality> bounds accumulated error, while the product rule tracks how <multiplication> changes its size. Equal <moduli> do not imply equal <complex numbers>: $5$, $-5$, and $3+4i$ all lie on the same <circle> around zero. The <modulus> records size and deliberately forgets direction; the <argument> supplies the missing information.
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