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Square Root of i


Date: 03/08/99 at 21:21:00
From: Nadine Schiavo
Subject: Square Root of i.

In my Honors Algebra II/Trigonometry class, we just completed a section 
on complex numbers. One of my students asked me the following:

The square root of -1 is i, but what is the square root of i?

Can you help?


Date: 03/09/99 at 08:31:37
From: Doctor Rick
Subject: Re: Square Root of i.

There are two complex numbers which, when squared, equal i. (The same 
holds true for any number. But for real numbers, we arbitrarily say 
that the positive root is THE square root. When the roots have 
imaginary parts, any such choice would be even more arbitrary, and we 
do not bother to choose one.)

The square roots of i are

  (1 + i) * sqrt(2)/2 and
 -(1 + i) * sqrt(2)/2

You can prove this just by squaring each number. To find them in the 
first place, you can use Euler's formula
 
 e^(i*t) = Cos(t)+i*Sin(t)

If t = pi/2, you get

  e^(i*pi/2) = cos(pi/2)+i*sin(pi/2) = 0 + i*1 = i

The square root of this is

  (e^(i*pi/2))^(1/2) = e^(i*pi/4)

and using Euler's formula again, we have

  e^(i*pi/4) = 1/2 + i*1/2

If you set t = 5*pi/2, you again get e^(i*5pi/2) = i. Taking the square 
root of this and using Euler's formula, you get the other root of i.

Euler's formula makes it easy to find powers and roots by working in 
polar coordinates in the complex plane. Any number x + iy can be 
written in terms of a radius r and angle theta (counterclockwise from 
the x axis):

  x + iy = re^(i*theta)
  r = sqrt(x^2 + y^2)
  theta = arctan(y/x)

Then, using Euler's formula,

  (x+iy)^k = r^k e^(i*k*theta)
           = r^k(cos(k*theta) + i*sin(k*theta))

Here are some related pages from our archives:

  Square Roots in Complex Numbers
  http://mathforum.org/dr.math/problems/jurd11.6.97.html   

  Proof of e^(iz) = cos(x) + isin(x)
  http://mathforum.org/dr.math/problems/graf.4.7.97.html   

  Roots of Unity
  http://mathforum.org/dr.math/problems/volante4.18.97.html   

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Imaginary/Complex Numbers

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