# Modulus of a Complex Number

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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

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 | $

$\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 |} $

$\left | \frac{z1}{z2} \right |$= $\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.**

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