What is the ith root of i?

Well as many of you must have observed, complex numbers and geometry for me are the most interesting pieces of mathematics. So far we have looked at Euler’s Identity, the world’s most beautiful equation, and how any polynomial can have infinite complex roots. Now I want to do something more interesting. So today I take you on a journey to find out what the ith root of i is.

This is what I will be attempting to quantify today

Some Basic Rules

This is where I explain some basic rules for simplification. If you already are familiar with basic exponent rules, the rules about integral powers of i (i stands for iota, which is the imaginary unit with the value of i=√-1 and is the solution to x²+1=0), and the Euler form of complex numbers, feel free to skip ahead to where I solve this problem.

So to actually solve this equation, we need to follow some basic rules in exponents, and the complex realm in general.

First off, we will know about the equivalence of roots and fractional powers. This states that the nth root of any number is numerically equivalent to the number raised to the power of 1/n. Mathematically speaking, it states that:¹¹

The basic exponent rule of root-power equivalence.

Another rule we are going to be using is the multiplication of powers rule. It states that if any number raised to a power of n is further raised to the power of m, the entire value is equal to the number raised to the power of the product of m and n.¹²

A basic exponent rule.

And then finally we have the Euler form of complex numbers. It states that any complex number z can be expressed as an exponential function by the virtue of its argument angle in the complex plane and its distance ‘r’ from the origin.¹³

Euler form of complex number Z, where Θ is the argument of Z.

With these basic rules out of the way, we can work out what the ith root of i is.

The Working

Now that we know some basic rules, let us find out what the ith root of i is. First off we apply the root-power equivalence to simplify the expression.

Using the root-power equivalence, we have converted this into fractional form

Now then we shall rationalize the power, to help ease our calculation. To do this, we multiply divide the power by i. So now it becomes:

We have rationalized the powers

Using the fact that i²=-1, we get:

A simplified version of the original problem

Now we shall employ the use of the Euler Form of complex numbers. We know that i = 0+i where the argument of the number ‘i’ is 90° or π/2 radians. The distance ‘r’ of the point (0,i) from the origin is 1 as well. Now we put those into the formula.

So now our expression actually becomes:

Multiplying the exponents, we get:

Using the fact that i²=-1 again, we get:

Which then becomes:

Which is our answer! We have proven that the ith root of i is actually e raised to the power of π/2 which is approximately equal to 4.81. What really surprised me was that we took a purely imaginary number, then took its imaginary root, and then ended up somehow with a purely real number. I personally think that this is pretty cool and shows how beautiful mathematics can really be.

Some interesting things to keep in mind

If you’ve read it this far with a lot of concentration, you must have noticed that there were some number power patterns on some statements. This is where I explain what they mean.

11- Even though the values of an nth root and a fractional power of n of the same number return are equal, the functions are not the same. Let me show you what I mean. If:

And

Then:

This is due to the fact that f(x) and g(x) have different domains or values where the function does not reach an indeterminate form. g(x) cannot accept n=0 for example, as that would give us an indeterminate power. They will give you the same answer to the same question, but you must not think they are the same!

12- This is not to be confused with a power tower. When solving a power tower, you go from the top to the bottom. Let me show you.

A very common mistake made when solving power towers is that you multiply the exponents, instead of solving them. When solving power towers, keep in mind that:

13- Keep in mind that this can be directly proven using Euler’s Formula.

That is it for now. I will try to post something soon related to circles and complex numbers so be sure to check them out as well.

Thank you guys so much for reading. I hope you have a great day ahead!