Mathematics · Ch 10 — Complex Numbers

Modulus of z

10.5.1

Modulus of z

If $z=a+ib$ is a complex number, then the modulus of $z$ , denoted $|z|$ or $r$ , is defined as

|z|=\sqrt{a^2+b^2}

From Fig. 1.3, the point $P(a,b)$ represents the complex number $z=a+ib$ , so

r=|z|=\sqrt{a^2+b^2}=OP

Hence, the modulus of $z$ is the distance of the point $P$ from the origin, where $P$ represents the complex number $z$ in the plane. This follows directly from applying the Pythagorean theorem to the right triangle formed by $P$ , the origin, and the perpendicular feet on the two axes (legs of length $|a|$ and $|b|$ , hypotenuse $OP$ ).

Example. For $z=4+3i$ : $|z|=\sqrt{16+9}=\sqrt{25}=5$ . …

Figure Fig.1.3Fig. 1.3 — modulus as the distance OP

What this figure shows. The point $P(a,b)$ representing $z=a+ib$ is joined to the origin $O$ by a straight segment $OP$ , with dashed perpendiculars from $P$ down to the real axis and across to the imaginary axis forming a right triangle whose legs have lengths $|a|$ and $|b|$ . Applying Pythagoras to that right triangle gives the length of the hypotenuse $OP=\sqrt{a^2+b^2}$ , which is exactly how the modulus $|z|$ is defined …

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