Complex Numbers, Prove That |z1z2| = |z1||z2|, Division of Two Complex Numbers, Prove That |z1/z2| = |z1|/|z2|

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Prove That |Z1∙Z2|=|Z1|∙|Z2|

Proof:

Let and

Similarly,

Now,

[Since

and]

And argument of

Example:

Find the modulus of the complex number

Solution:

Let

Then

(Since)

But

Division of Two Complex Numbers

  • Division of complex numbers involves multiplying both numerator and denominator with the conjugate of the denominator. We will explain it through an example.

Let and , then.

(Multiplying numerator and denominator with the conjugate of the denominator)

Thus,

Example:

Divide by

Solution:

Multiplying numerator and denominator by the conjugate of we get

Thus,

Prove That |Z1/Z2|=|Z1|/|Z2|

Proof:

Let and

Similarly,

And and

Now,

Thus,

Argument of

Example:

Find the modulus of the complex number

Solution:

Let

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