Stumbling through mathematics: strange powers

You may have been told by your teacher that square roots of negative numbers do not exist. But what if they actually did? It turns out that the square roots of negative numbers are surprisingly useful in mathematics and engineering.

High school and college students may remember this strange creature from their math classes: √(-1). If you try putting this number into your calculator, you will likely get an error message. So what is it? Mathematicians call this number an imaginary number, and denote it using the symbol i = √(-1).

It turns out that this mysterious number i has some interesting (as well as useful) properties. For example: what happens when we raise i to the second power, the third, etc.?

i¹ = i
i² = (√(-1))² = i² = -1
i³ = (√(-1))²⋅√(-1) = i²⋅i = -1⋅i = -i
i⁴ = (√(-1))³⋅√(-1) = -i⋅i = -i² = -(-1) = 1
i⁵ = (√(-1))⁴⋅√(-1) = 1⋅i = i
i⁶ = (√(-1))⁵⋅√(-1) = i⋅i = i² = -1
i⁷ = (√(-1))⁶⋅√(-1) = i⁶⋅i = -1⋅i = -i
i⁸ = (√(-1))⁷⋅√(-1) = -i⋅i = -i² = -(-1) = 1


...and so on. Notice how the pattern of i, -1, -i, 1 keeps repeating itself? This repetition is one property that makes imaginary numbers so useful.

But what happens if we raise i to an imaginary power, say, i ⁱ ? What happens then?

To see what i ⁱ is, we must use Euler's equation (an equation familiar to physicists and electrical engineers as well as mathematicians). Let e be the base of the natural logarithm (approximately equal to 2.7182....), and cos(x) and sin(x) be the cosine and sine functions of x:

e^ix = cos(x) + isin(x)

A complex number is a number made up of a real number and an imaginary number. Complex numbers have the form a + bi or a - bi, where a and b are real numbers. Any real number can be written as a complex number. An example would be the real number 3, which can be written in two different (but equally valid) ways: 3 + 0i or 3 - 0i. All your life you have been using complex numbers - and perhaps did not even know it!

Therefore, i can be written as the complex number 0 + 1⋅i. One value of x that would make cos(x) = 0 and sin(x) = 1 is x = π/2. Then Euler's equation becomes:

e^iπ/2 = 0 + 1i = i

Now raise both sides on this equation to the i power:

(e^iπ/2)ⁱ = e^i2π/2 = i ⁱ

Since i ² = -1:

e^-π/2 = i ⁱ

On the right side of this equation we have an imaginary number being raised to an imaginary power. On the left side we have a real number with no imaginary part whatsoever (the value of e^-π/2 is an irrational number and is approximately equal to 0.2078795764...). This result is counter intuitive, to say the least!

But that is not the end of the story. The value x = π/2 is only one possible result. Because cos(x) and sin(x) are periodic functions, there are many values of x that make cos(x) = 0 and sin(x) = 1. In fact, there are an infinite number of values of x! We can re-write the cosine and sine terms to reflect this: cos(π/2 ± 2πn) = 0 and sin(π/2 ± 2πn) = 1, where n is any non-negative integer such as 0, 1, 2, 3, .... The equation can now be re-written as:

e^{-π/2 ±2πn} = i ⁱ

e^-π/2⋅e^±2πn = i ⁱ

So, i ⁱ results in an infinite number of real solutions. There are many strange results in mathematics, but I would be hard-pressed to find one stranger than this. It is why mathematics is the best science ever!