# Complex numbersFind the absolute value for each complex number: l -4 - 10i l l 3- 6i l

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You need to evaluate the aboslute value of complex number `-4 - 10i` using Pythagora's theorem in triangle having horizontal leg |x|, vertical leg |y| and hypotenuse `|-4 - 10i|` , such that:

`|-4 - 10i|^2 = |x|^2 + |y|^2 => |-4 - 10i|^2 = (-4)^2 + (-10)^2`

`|-4 - 10i|^2 = 16 + 100 => |-4 - 10i|^2 = 116 => |-4 - 10i| = sqrt 116`

Hence, evaluating the absolute value of complex number -4 - 10i yields `|-4 - 10i| = sqrt 116.`

You need to evaluate the aboslute value of complex number `3 - 6i` using Pythagora's theorem in triangle having horizontal leg |x|, vertical leg |y| and hypotenuse `|3 - 6i|` , such that:

`|3 - 6i|^2 = |x|^2 + |y|^2 => |3 - 6i|^2 = (3)^2 + (-6)^2`

`|3 - 6i|^2 = 9 + 36 => |3 - 6i|^2 = 45 => |3 - 6i| = sqrt 45 = 3sqrt5`

**Hence, evaluating the absolute value of complex number 3 - 6i yields `|3 - 6i| = 3sqrt5.` **

The absolute value of a complex number is the modulus of the complex number.

z = x + i*y

Re(z) = x

Im(z) = y

|z| =sqrt[Re^2(z) + Im^2(z)]

We'll identify x and y for the first number z.

z = -4 - 10i |z| = l -4 - 10i l Re(z) = -4 Im(z) = -10 |z| =sqrt [(-4)^2 + (-10)^2] |z| = sqrt (16 + 100) |z| = sqrt 116 |z| = 2sqrt29