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

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.

References

  1. Harvey Mudd College, Claremont, California
  2. Brilliant.org
  3. University of Toronto
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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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