What is the i-th root of the imaginary number i?

Does it even exist?

For many, the idea of an imaginary number i = √-1 is a bit mind bending. But even more mind bending is the question of roots of i. Probably the most mind-bending question of this nature is to work out the value of the i-th root of i.

The answer to this question is that there are, in fact, infinitely many i-th roots of i, all of them positive real numbers ranging from the infinitesimally small to the infinitely large. Here are a few examples, approximated to a few significant figures: 0.00898, 4.8104, 2575.97, 1379411

What? How?

The idea that the i-th root of i can be such precise real numbers is a bit counterintuitive, right? But let’s look at how we derive the i-th root of i and it will make more sense.

First, let’s move from root to index notation:

Now we rationalize the index by multiplying it by i/i = 1, and hence we derive the following (noting that i² = -1):

Use of Euler’s Formula

Now, taking Euler’s formula, we know that for any given real value x:

So we can conclude from that that when x = π/2 radians, we know that sin(π/2)=1 and cos(π/2)=0, so we have: