COMPLEX NUMBERS · HL

Complex Numbers 4 — Modulus

The modulus of a complex number — its distance to the origin.

Section 1 of 2

Modulus $|z|$

This is the distance to the origin.

Formula

1.$z = x + yi$

2.$|z| = \sqrt{x^2 + y^2}$

$z = 2 + 3i$, find $|z|$

$x = 2, \quad y = 3$

$|z| = \sqrt{x^2 + y^2}$

$|z| = \sqrt{2^2 + 3^2}$

$= \sqrt{13}$

$w = 5 - 7i$, find $|w|$

$x = 5, \quad y = -7$ ($i$ gets dumped)

$|w| = \sqrt{5^2 + (-7)^2}$

$= \sqrt{25 + 49}$

$= \sqrt{74}$

$z = -5 - 6i$, find $|z|$

$|z| = \sqrt{(-5)^2 + (-6)^2}$

$= \sqrt{25 + 36}$

$= \sqrt{61}$

Section 2 of 2

Modulus of a sum or difference

$z = 2 + 7i, \qquad w = 3 + 2i$. Find:

(i) $|z|$

$|z| = \sqrt{2^2 + 7^2} = \sqrt{53}$

(ii) $|w|$

$|w| = \sqrt{9 + 4} = \sqrt{13}$

(iii) $|z + w|$

Find $z + w$ first.

$2 + 7i + 3 + 2i = 5 + 9i$

Now sub into formula:

$|5 + 9i| = \sqrt{5^2 + 9^2} = \sqrt{25 + 81}$

$= \sqrt{106}$

$z_1 = 3 - 2i, \qquad z_2 = 5 - 7i$. Find:

(i) $|z_1|$

$|z_1| = \sqrt{3^2 + (-2)^2}$

$= \sqrt{9 + 4} = \sqrt{13}$

(ii) $|z_2|$

$|z_2| = \sqrt{5^2 + (-7)^2}$

$= \sqrt{25 + 49} = \sqrt{74}$

(iii) $|z_1 - z_2|$

$z_1 - z_2 = 3 - 2i - 1(5 - 7i)$

$= 3 - 2i - 5 + 7i$

$= -2 + 5i$

$|z_1 - z_2| = \sqrt{(-2)^2 + 5^2}$

$= \sqrt{4 + 25}$

$= \sqrt{29}$

SUM

The lot in one box

Modulus toolkit

1.$|z| = \sqrt{x^2 + y^2}$ — the distance to the origin.

2.The $i$ gets dumped — square the real and imaginary values.

3.For $|z + w|$ or $|z - w|$: combine first, then sub into the formula.

End of lesson

Complex Numbers 4 — Modulus · HL · Mathslive.ie