3.4 Modulus
Definition 3.6 (Modulus). Let \(z=x+iy\) be a complex number. The modulus of \(z\), denoted \(|z|\), is \[|z|=\sqrt {x^{2}+y^{2}} .\]
Find the modulus of each of the following complex number
- (a).
- \(5+12i\)
- (b).
- \(-8+6i\)
- (c).
- \(15i\)
- (d).
- \((1-\sqrt {2})+(\sqrt {2}+2)i\)
Note that \(15i=0+15i\), so its real part is \(0\).
Solution. In each case \(\left |z\right |=\sqrt {x^{2}+y^{2}}\).
(a). \(\left |5+12i\right |=\sqrt {5^{2}+12^{2}}=\sqrt {169}=13\).
(b). \(\left |-8+6i\right |=\sqrt {(-8)^{2}+6^{2}}=\sqrt {100}=10\). The real part is negative but the modulus is not; squaring removes the sign.
(c). \(\left |15i\right |=\sqrt {0^{2}+15^{2}}=15\).
(d). With \(x=1-\sqrt {2}\) and \(y=\sqrt {2}+2\), \begin {align*} x^{2}&=\left (1-\sqrt {2}\right )^{2}=1-2\sqrt {2}+2=3-2\sqrt {2},\\ y^{2}&=\left (\sqrt {2}+2\right )^{2}=2+4\sqrt {2}+4=6+4\sqrt {2}, \end {align*}
so \(x^{2}+y^{2}=9+2\sqrt {2}\) and \[\left |z\right |=\sqrt {9+2\sqrt {2}}\approx 3.439 .\] The surd terms did not cancel here, so the answer stays in surd form. Leaving it as \(\sqrt {9+2\sqrt {2}}\) is exact and preferable to a decimal.
Note 3.8. \(\left |z\right |\) is the distance from \(z\) to the origin in the Argand diagram — it is Pythagoras’ theorem applied to the point \((x,y)\). That is why the modulus is never negative, and is zero only for \(z=0\). It is the same distance reading that made absolute value on the real line easy to handle.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.