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

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Answer 1 · 2016-03-21T22:07:41

#abs(x+iy) = sqrt(x^2+y^2)#

#abs(x+iy)# is essentially the distance between #0# and #x+iy# in the Complex plane.

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

#abs(x+iy) = sqrt(x^2+y^2)#

Notice that #(x+iy)(x-iy) = x^2-i^2y^2 = x^2+y^2#

So another way of expressing this is:

#abs(x+iy) = sqrt((x+iy)(x-iy)) = sqrt((x+iy)bar((x+iy)))#

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

#abs(z) = sqrt(z bar(z))#