How do you find $| x + i y |$ $abs( x+iy )$ ?

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Answer 1 · 2016-03-21T22:07:41

$| x + i y | = \sqrt{x^{2} + y^{2}}$ $abs(x+iy) = sqrt(x^2+y^2)$

$| x + i y |$ $abs(x+iy)$ is essentially the distance between $0$ $0$ and $x + i y$ $x+iy$ in the Complex plane.

Using the distance formula which comes from Pythagoras theorem, we have:

$| x + i y | = \sqrt{x^{2} + y^{2}}$ $abs(x+iy) = sqrt(x^2+y^2)$

Notice that $(x + i y) (x - i y) = x^{2} - i^{2} y^{2} = x^{2} + y^{2}$ $(x+iy)(x-iy) = x^2-i^2y^2 = x^2+y^2$

So another way of expressing this is:

$| x + i y | = \sqrt{(x + i y) (x - i y)} = \sqrt{(x + i y) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯ ¯ (x + i y)}$ $abs(x+iy) = sqrt((x+iy)(x-iy)) = sqrt((x+iy)bar((x+iy)))$

So without explicitly splitting a Complex number $z$ $z$ into Real and imaginary parts, we can say:

$| z | = \sqrt{z ¯ z}$ $abs(z) = sqrt(z bar(z))$

How do you find |x+iy|abs( x+iy )?