# Modulus of a Complex Number

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Let z = a + ib be a complex number. Then, the modulus of a complex number z, denoted by |z|, is defined to be the non-negative real number. $\sqrt{a^2 + b^2}$
modulus of a complex number z = |z| = $\sqrt{Re(z)^2 + Im(z)^2}$
where Real part of complex number = Re(z) = a and
Imaginary part of complex number =Im(z) =b
|z| = $\sqrt{a^2 + b^2}$ .

Example :
(i) z = 5 + 6i so |z| = $\sqrt{5^2 + 6^2}$ = $\sqrt{25 + 36}$= $\sqrt{61}$

(ii) z = 8 + 5i so |z| = $\sqrt{8^2 + 5^2}$ = $\sqrt{64 + 25}$= $\sqrt{89}$

(iii) z = 3 - i so |z| = $\sqrt{3^2 + (-1)^2}$ = $\sqrt{9 + 1}$= $\sqrt{10}$

(iv) z = 1 + $\sqrt{5}$i so |z| = $\sqrt{1^2 +\sqrt{5}^2}$ = $\sqrt{64 + 25}$= $\sqrt{89}$

(v) -6 + 2i so |z| = $\sqrt{(-6)^2 + 2^2}$ = $\sqrt{36 + 4}$= $\sqrt{40}$

(vi) -8 + 6i so |z| = $\sqrt{(-8)^2 + 6^2}$ = $\sqrt{64 + 36}$= $\sqrt{100}$ = 10

(vii) 12 - 5i so |z| = $\sqrt{12^2 + (-5)^2}$ = $\sqrt{144 + 25}$= $\sqrt{169}$ =13

## Properties of Modulus of a complex Number

Let z be any complex number, then
(I) |-z| = |z |
Example : Let z = 7 + 8i
-z = -( 7 + 8i)
-z = -7 -8i
modulus of (-z) =|-z| =$\sqrt{(-7)^2 + (-8)^2}$=$\sqrt{49 + 64}$ =$\sqrt{113}$
modulus of (z) = |z|=$\sqrt{7^2 + 8^2}$=$\sqrt{49 + 64}$ =$\sqrt{113}$
So from the above we can say that |-z| = |z |

(II) |z| = 0 if, z = 0
Proof : If z = a+ib ⇒ |z| = $\sqrt{a^2 + b^2}$
|z| = 0 ⇒ $\sqrt{a^2 + b^2}$ = 0 ⇒ ${a^2 + b^2}$ = 0
So, $a^2$ = 0 and $b^2$ = 0 ⇒ a = 0 and b = 0
i.e. z = 0 + i0 = 0
So |z| = 0 if, z = 0

(III) The absolute of a product of two complex numbers z1 and z2 is equal to the product of the absolute values of the numbers. i.e
$\left |z1.z2 \right |$= $\left | z1 \right . |$ $\left | z2 \right |$

(IV) The absolute of a quotient of two complex numbers z1 and z2 (≠ 0) is equal to the quotient of the absolute values of the dividend and the divisor.
$\left | \frac{z1}{z2} \right |$= $\frac { \left |z1 \right|} {\left |z2 \right |}$

(V) The absolute of the sum of two conjugate complex numbers z1 and z2 can never exceed the sum of their absolute values, i.e.$\left | z1+z2 \right |$ $\leq$ $\left | z1 \right |$ +$\left |z2 \right |$
This inequality is called triangle inequality.