What Is i Raised To The Power i?

The imaginary unit , What Is i Raised To The Power i?, Science ABC, Science ABC represents the square root of -1, such that , What Is i Raised To The Power i?, Science ABC, Science ABCImaginary 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 , What Is i Raised To The Power i?, Science ABC, Science ABC itself is so difficult to comprehend, what could , What Is i Raised To The Power i?, Science ABC, Science ABC be? You might be surprised to know that unlike , What Is i Raised To The Power i?, Science ABC, Science ABC, the value of , What Is i Raised To The Power i?, Science ABC, Science ABC 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 , What Is i Raised To The Power i?, Science ABC, Science ABC, then , What Is i Raised To The Power i?, Science ABC, Science ABC. For instance, given that , What Is i Raised To The Power i?, Science ABC, Science ABC, What Is i Raised To The Power i?, Science ABC, Science ABC. 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)

To solve the problem at hand, the base of our operation is , What Is i Raised To The Power i?, Science ABC, Science ABC, or Euler’s number, with a value of 2.71828… If , What Is i Raised To The Power i?, Science ABC, Science ABC, then , What Is i Raised To The Power i?, Science ABC, Science ABC. The logarithm with the base , What Is i Raised To The Power i?, Science ABC, Science ABC 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.

, What Is i Raised To The Power i?, Science ABC, Science ABC                     , What Is i Raised To The Power i?, Science ABC, Science ABC

, What Is i Raised To The Power i?, Science ABC, Science ABC, What Is i Raised To The Power i?, Science ABC, Science ABC                     , What Is i Raised To The Power i?, Science ABC, Science ABC , What Is i Raised To The Power i?, Science ABC, Science ABC  , What Is i Raised To The Power i?, Science ABC, Science ABC

The second thing to recall is that , What Is i Raised To The Power i?, Science ABC, Science ABC can be written as a complex number , What Is i Raised To The Power i?, Science ABC, Science ABC, which can also be written as , What Is i Raised To The Power i?, Science ABC, Science ABC. However, according to Euler’s formula, , What Is i Raised To The Power i?, Science ABC, Science ABC. Therefore, , What Is i Raised To The Power i?, Science ABC, Science ABC is equal to , What Is i Raised To The Power i?, Science ABC, Science ABC., What Is i Raised To The Power i?, Science ABC, Science ABC

Now, let’s say , What Is i Raised To The Power i?, Science ABC, Science ABC. Applying logarithm on both sides of the equation, we get:

, What Is i Raised To The Power i?, Science ABC, Science ABC

, What Is i Raised To The Power i?, Science ABC, Science ABC

, What Is i Raised To The Power i?, Science ABC, Science ABC

, What Is i Raised To The Power i?, Science ABC, Science ABC

Remember that logarithmic and exponential functions are inverse functions, such that , What Is i Raised To The Power i?, Science ABC, Science ABC. Thus, from the above expression, one can discern:

, What Is i Raised To The Power i?, Science ABC, Science ABC, What Is i Raised To The Power i?, Science ABC, Science ABC

, What Is i Raised To The Power i?, Science ABC, Science ABC

Applying the exponential on both sides of the equation:

, What Is i Raised To The Power i?, Science ABC, Science ABC

Again, we encounter the inverse functions together. They neutralize to give the outcome:

, What Is i Raised To The Power i?, Science ABC, Science ABC

, What Is i Raised To The Power i?, Science ABC, Science ABC

, What Is i Raised To The Power i?, Science ABC, Science ABC

, What Is i Raised To The Power i?, Science ABC, Science ABC

, What Is i Raised To The Power i?, Science ABC, Science ABC

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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 , What Is i Raised To The Power i?, Science ABC, Science ABC 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 , What Is i Raised To The Power i?, Science ABC, Science ABC isn’t particularly , What Is i Raised To The Power i?, Science ABC, Science ABC, but , What Is i Raised To The Power i?, Science ABC, Science ABC, What Is i Raised To The Power i?, Science ABC, Science ABC and every odd multiple of , What Is i Raised To The Power i?, Science ABC, Science ABC. Therefore, the value of , What Is i Raised To The Power i?, Science ABC, Science ABC also varies from , What Is i Raised To The Power i?, Science ABC, Science ABC to , What Is i Raised To The Power i?, Science ABC, Science ABC, What Is i Raised To The Power i?, Science ABC, Science ABC and so on.

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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.

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