I remember the time when I was still studying at *lukio *(equivalent school is high school in the US, *gymnasium* in Sweden, *Gymnasium* in Germany) when there were exercises to determine positive integer powers of imaginary unit *i.* I invented at that time a method to easily determine the power without using the taught method of grouping the exponent. Later on my university times I invented a formula to determine arbitrary real number power of imaginary unit *i.*

**The method**

Complex number *z* is of form *z = a + bi*, where the previous is the real part and the latter the imaginary part. Now we’re interested only in imaginary unit *i*. So we examine complex number *z = i*, the magnitude of *i *= 1.

The argument (arg) of imaginary unit *i *is 90°. From this fact it follows that multiplying with *i* corresponds 90° rotation on the complex plane. Thus, if the exponent is a positive integer, there’s exactly 4 possibilities for the power of *i*: 1, -1, *i*, *-i*.

The power can be determined by dividing the exponent by 4 and examining the decimal part of the division. There are now 4 possibilities: .0; .25; .5; .75. These tell how many percent of unit circle on the complex plane has been rotated.

Based on this the following table can be presented:

The first line is the decimal part of division by 4 of the exponent, the second line tells place in the unit circle of the complex plane, the third line tells the the power of imaginary unit *i*.

For example to determine *i*^{12345678}* *, we first divide the exponent 12345678 by 4, the result is 3086419.5. The decimal part is .5, so we see from the table above, that the power is -1.

**The formula for arbitrary real number exponent**

I invented this formula myself some time in the year 1997 when I was studying for the second year at the university of Jyväskylä. I thought my formula was too simple, so I didn’t show it to the maths department staff. Anyway, the formula is as follows:

Many years ago, I asked on local science magazine’s net forum, that does this kind of formula already exist. The existing formula was a bit different. In the same forum one reader presented a proof to this formula.

Many years ago when I was studying for the second year in the university of Jyväskylä in one sleepless night I somehow invented how to determine logarithm of negative real number, although I had only basic knowledge of complex numbers. And yes, the logarithm of negative real number is, of course, a complex number.

Now to the formula…

*x*< 0 and

*k*> 1. Now

I was able to prove this formula in less than 30 minutes, about one A4 paper of proof. Next morning I showed my proof to one person at staff of department of mathematics at university of Jyväskylä (Finland). He didn’t find anything wrong in my proof. A tricky part was a situation where there was two variables in one equation in my proof, but the other varibale had as coefficient sin π that equals 0, so the other variable was eliminated from the equation. I don’t have the proof anymore and I don’t really remember any relevant details of my proof other than I used Euler’s formula (*) in some part of the proof that I had become familiar with in course of differential equations.

(*) As far I can see, the formula probably was the following:

e^{x}* ^{ + iy }= e^{x}*(cos

*y*+

*i*sin

*y*)