# What is modulus z complex in z = (1+2i)/(1-i)?

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### 1 Answer

Modulus or magnitude or absolute value of a complex number `z = x + i*y` is given by the following formula, such that:

`|z| = sqrt(x^2 + y^2)`

You also need to remember the properties of absolute value, such that:

`|(z_1)/(z_2)| = |z_1|/|z_2|`

Hence, identifying `z_1 = 1 + 2i` and `z_2 = 1 - i` , you may evaluate the followings:

`|z_1| = sqrt(1^2 + 2^2) => |z_1| = sqrt(1+4) => |z_1| = sqrt5`

`|z_2| = sqrt(1^2 + (-1)^2) => |z_2| = sqrt 2`

Hence, replacing `sqrt5` for `|1 + 2i|` and `sqrt 2` for `|1 - i|` yields:

`|1 + 2i|/|1 - i| = sqrt5/sqrt2 =>|1 + 2i|/|1 - i| = (sqrt5*sqrt2)/2`

`|1 + 2i|/|1 - i| = sqrt10/2`

**Hence, evaluating the modulus of the given complex number `z = (1 + 2i)/(1 - i)` yields `|z| = |1 + 2i|/|1 - i| = sqrt10/2` .**

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