Square Roots of Imaginary Numbers, Using Algebra

So we actually found , the square root of i.  In fact, the number i had two square roots, one half the square root of 2, plus one half the square root of 2 times i and negative one half the square root of 2, minus one half the square root of 2 times i.  So can we find square roots of other imaginary numbers?  Sure.

First example

Let's try to find the square roots of three plus four i.  Since the quantity a plus b i squared equals a squared minus b squared, plus 2 a b i, we need to solve the system of equations

We can't solve this system as quickly as we did when we found the square root of i, but it still can be done.  Solving the second equation for the variable b, we get b equals 2 divided by a.  Substituting this quantity into the first equation, we get a squared minus 4 over a squared equals 3.  Clearing the fractions gives the fourth power of a minus 3 a squared minus 4 equals zero.  This can be solved by factoring, so a squared equals either 4 or negative 1.  But we want real values for the variable a, so we discard negative 1. Therefore, a squared is 4, and the values of a are 2 and negative 2.  Using b equals 2 divided by a, we find the value of b is positive or negative 1.  So the two square roots of three plus four i are 2 plus i and negative 2 minus i.

Second example

Let's try to find the square roots of two plus three i.  Once again, since the quantity a plus b i squared equals a squared minus b squared, plus 2 a b i, we need to solve the system of equations

a squared minus b squared equals 2, and 2 a b equals 3
Solving the second equation for the variable b, we get b equals 3 divided by 2 a .  Substituting this quantity into the first equation, we get a squared minus 9 over 4 a squared equals 2.  Clearing the fractions gives 4 times the fourth power of a, minus 8 a squared, minus 9 equals zero.  This can be solved using the quadratic formula, and we get a squared equals one, plus or minus one half the square root of 13.  But we want real values for the variable a, so we discard the negative sign. Therefore, we get a equals one half the square root of the quantity 4 plus twice the square root of 13.  (Messy, but true!)  Using b equals 3 divided by 2 a , we find the value of b is b equals 3 divided by plus or minus the square root of the quantity 4 plus twice the square root of 13.  So the two square roots of two plus three i are one big messy formula and another big messy formula.

Calculator

A Square Root Calculator is also available.  It gives the square roots of complex numbers in radical form, as discussed on this page.

Epilogue

There is another way to find roots, using trigonometry.  You can read more about this relationship in Imaginary Numbers and Trigonometry.