How To Really Solve i^i (i raised to the power i)?

The problem: i^i (i raised to the power i) seems absurd at first glance. You might even be thinking along the following lines:

“Is it even possible to solve this problem?”

Well, rest assured that it is not only possible to solve this problem, but the solution is fairly straightforward as well. However, we will need to equip ourselves with the following basic tools first:

1. Complex identity fundamentals

2. Logarithms — fundamentals

The real reason why I chose this problem is to introduce a more advanced complex number theory concept. Before I get ahead of myself, let us start with the most important stuff first: the basics.

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Complex Identity Fundamentals

In my previous essay on complex number theory, I demonstrated how to solve 1ˣ = -1. In the same essay, I demonstrated how complex numbers can be represented in both cartesian coordinates as well as polar coordinates. In case you still feel intimidated by the concept of complex numbers and need a super-fast crash course, check this essay out.

In order to compute i^i, we will be using the polar-coordinate representation of a complex number. Below, you can find a quick summary of how a complex number is represented in each coordinate system.

Logarithms — Fundamentals

A logarithm of any given base asks the following question:

To what power must the base be raised to get a particular result.

This ‘particular result’ is just the number to which the logarithmic function is applied. It is perhaps easier to understand if you take a look at a couple of examples:

In other words, the logarithmic function could be considered the inverse of the exponential function. Apart from this knowledge, there are two other useful properties of logarithms that we will be needing.

The first property is the fact that the logarithm of a product is equal to the sum of the logarithms of each number being multiplied.

The second property of logarithms that we need allows us to express the logarithm of a number raised to an exponent as the product of the exponent and the logarithm of the number.

At this point, we have almost all the tools necessary to tackle our main challenge.

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How to Solve i^i (i raised to the power i)?

i^i looks intimidating because we seem to have an exponent that we are not used to. Usually, when I’m faced with problems like these, my intuition is to somehow ‘dethrone’ the exponent.

In order to do this, we could use the logarithmic function. But in mathematics, we cannot just introduce a function out of thin air. No matter what we introduce, we have to make sure that the original expression’s meaning/value does not change.

Remember that I told you that the logarithmic function is just the inverse of the exponential function. This is where we are going to employ this knowledge. The trick is to introduce an exponential function first and then apply the logarithmic function to the expression.

As the next step, we can just apply the exponential property of logarithms to dethrone the i at the top. The result we get is conveniently in the general polar-coordinate form for complex numbers.

I have chosen to use the natural logarithm here, which is nothing but the logarithmic function with Euler’s number (e) as its base.

At this point, we need to compute ln(i). How are we supposed to do that? Right, it is now time to introduce the advanced complex number concept that I mentioned at the beginning of the essay.

The Logarithm of a Complex Number

Inorder to solve this problem further, we will need to extend our understanding of logarithms to the realm of complex numbers as well. To do this, let us first consider the general polar-coordinate form of complex numbers.

We could now go ahead and apply the product rule of logarithms to the expression we have. Since the natural logarithm is the inverse of the exponential function, the two functions undo each other. Therefore, the logarithm is eliminated from the imaginary part of the expression:

Let us now turn our attention to the unknown in the main expression we are trying to compute: ln(i). To compute this logarithm, we need to first express i in polar coordinates. To do this, consider the complex plane below.

i is located at a distance of 1 unit from the origin and encloses an angle of π/2 radians (or 90°) with the positive real axis in the anticlockwise direction. Consequently, we get the polar expression for i as a complex number.

Now, we just need to apply the knowledge we have acquired by taking the logarithm of i:

Now that we have the value of the natural logarithm of i, we have everything we need to solve the main expression.

The Solution to i^i (i raised to the power i)

We had initially started solving i^i and arrived at the following point.

Now, if we just plug in the value of ln(i) we have calculated, we obtain the following result:

There we go! That is the solution to i^i. The only other point we have to note here is that this approach is valid for all angles of unit vector rotation that end up at i: That is, ψ = π/2, (π/2 + 2 π), (π/2 + 4 π), etc. We could avoid countless solutions by defining our domain (for example, between — π and + π).

To sum up, we started by applying our basic understanding of complex numbers and finally expanded the concept of logarithms to complex numbers to arrive at the solution. The solution, as it turns out, is not just a real number but also a transcendental number. Mathematics never ceases to fascinate!

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Further reading that might interest you:  How To Really Calculate The Square Root Of i? and How To Intuitively Understand Euler’s Identity?