1.4 Modulus of a Complex Number

Just as the absolute value of a real number measures its distance from the origin on the real number line, the modulus of a complex number measures the distance from the origin in the complex plane. Notice that the distance from the origin to the point $P(x,y)$ lies along a radial line and forms the hypotenuse of a right triangle, where the horizontal and vertical sides have length $x$ and $y$ respectively.

For example, (i) $\mathrm{\mid }i\mathrm{\mid }=\mathrm{\mid }0+1i\mathrm{\mid }=\sqrt{0^2+1^2}=1$$|i| = |0 + 1i| = \sqrt{0^{2} + 1^{2}} = 1$

​=1∣i∣=∣0+1i∣=02+12

​=1

$\mathrm{\mid }3-5i\mathrm{\mid }=\sqrt{(3{)}^2+(-5{)}^2}=\sqrt{9+25}=\sqrt{34}$$|3 - 5i| = \sqrt{(3)^{2} + (-5)^{2}} = \sqrt{9 + 25} = \sqrt{34}$

​=9+25

​=34

​∣3−5i∣=(3)2+(−5)2

​=9+25

​=34

​

1.4.1 Properties of Modulus of a Complex Number

Property 1