Thread: Sequence of powers of i and Pi gives 1

This is probably a trivial problem for an experienced mathematician...

I was playing with the Google calculater and got this (for me) surprising result. All other combinations produce much 'uglier' outcomes.

i^(pi^(i^(pi^(i^(pi^(i^pi)))))) = 1

Why is this sequence of powers of Pi and i equal to 1 ?

Thanks!

It's actually not. The problem is calculators only give approximations. If you calculate:

i^(π^(i^(π^(i^π)))

You get:

-4.96013383 × 10⁴² - 1.54189139 × 10⁴² i

This is a complex number whose real part is negative and very large in magnitude. So since:

|e^x+iy| = e^x

π to the above number is a very small complex number. So small, in fact, that Google calculates it to be 0. Then anything to the 0th power is defined to be 1.

Thanks. Got it. Thought in that direction as well.

I also found:

i^(pi^(i^(pi^(pi^(i^pi))))) = 0

Assume this is the same problem, but you now raise a number close to zero to the power pi.

It's the same thing. In this case, it's easier to see this can't be right: a^b ≠ 0 unless a = 0.