What Is i Raised To The Power i?

Table of Contents

The imaginary unit represents the square root of -1, such that Imaginary numbers live in a world of their own; the numbers are counted on an entirely different plane or axis that is solely devised for them. However, imaginary numbers have acquired a somewhat nefarious reputation, considering that their discovery has compounded the difficulty of problems that math was already replete with. I mean, as if the numbers we already had weren’t enough?

Our problem, however, combines not just one, but two of the many haunting aspects of mathematics. If  itself is so difficult to comprehend, what could  be? You might be surprised to know that unlike , the value of  is a real number! How is that possible?

First, we must revise one of the most common, but least understood, mathematical operations — logarithms. Let me remind you how they work: If , then . For instance, given that . Here, b is called the base of the logarithmic operation.

Logarithmic ruler on a wooden table(Saim Tokacoglu)S

Log scale (Photo Credit : Saim Tokacoglu/Shutterstock)


Recommended Video for you:


To solve the problem at hand, the base of our operation is , or Euler’s number, with a value of 2.71828… If , then . The logarithm with the base  is known as a “natural logarithm”. It is imperative to understand that the exponential and logarithmic functions are inverse functions. This is very important for our calculation.

                                              

The second thing to recall is that  can be written as a complex number , which can also be written as . However, according to Euler’s formula, . Therefore,  is equal to .

Now, let’s say . Applying logarithm on both sides of the equation, we get:

Remember that logarithmic and exponential functions are inverse functions, such that . Thus, from the above expression, one can discern: Applying the exponential on both sides of the equation: Again, we encounter the inverse functions together. They neutralize to give the outcome:

Thus, an imaginary number raised to an imaginary number turns out to be real. However, while learning complex analysis, one learns that an exponential with respect to  does not have a single, fixed value. Rather, the function is multi-valued — the value we arrived at in our calculation is just one of many values. This is because the value of  isn’t particularly , but  and every odd multiple of . Therefore, the value of  also varies from  to  and so on.

Suggested Reading

Was this article helpful?
YesNo
Help us make this article better

Follow ScienceABC on Social Media:

About the Author

Akash Peshin is an Electronic Engineer from the University of Mumbai, India and a science writer at ScienceABC. Enamored with science ever since discovering a picture book about Saturn at the age of 7, he believes that what fundamentally fuels this passion is his curiosity and appetite for wonder.

.
Science ABC YouTube Videos

  1. Gut Microbiome Explained in Simple WordsGut Microbiome Explained in Simple Words
  2. Particle accelerators: What are they, how do they work and why are they important to us?Particle accelerators: What are they, how do they work and why are they important to us?
  3. How Do Neurons Work?How Do Neurons Work?
  4. How Scientifically Accurate Is The HBO Miniseries Chernobyl?How Scientifically Accurate Is The HBO Miniseries Chernobyl?
  5. Cellular Respiration: How Do Cell Get Energy?Cellular Respiration: How Do Cell Get Energy?
  6. Multiverse Theory Explained: Does the Multiverse Really Exist? Truth of Multiple RealitiesMultiverse Theory Explained: Does the Multiverse Really Exist? Truth of Multiple Realities
  7. What Exactly is Spacetime? Explained in Ridiculously Simple WordsWhat Exactly is Spacetime? Explained in Ridiculously Simple Words
  8. What Are The Different Atomic Models? Dalton, Rutherford, Bohr and Heisenberg Models ExplainedWhat Are The Different Atomic Models? Dalton, Rutherford, Bohr and Heisenberg Models Explained