The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

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   

  Proof of e^(iz) = cos(x) + isin(x)   

  Roots of Unity   

- Doctor Rick, The Math Forum   
Associated Topics:
High School Imaginary/Complex Numbers

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.