perplexus.info :: Just Math : The Powers that Be

The Powers that Be (Posted on 2003-11-25)

Is it possible for two complex numbers to have a real exponentiation?

In other words, if x and y are complex (each have the form a+bi), show that x^y can have a real value, or prove that it is impossible.

Note: i is the imaginary value defined as the number that yields -1 when squared. a and b are any real numbers, but b is not 0.

See The Solution Submitted by DJ

Rating: 4.4444 (9 votes)

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Puzzle Thoughts

Comment 21 of 21 |

Let x=1+i

y = (1+i) = √2 e^iπ/4

Then,

x^y= e^((log(√2))^2 - (iπ/4)^2 = (log(√2))^2 + (π^2)/16), which is obviously a real number.

Edited on January 9, 2024, 11:26 pm

Posted by K Sengupta on 2024-01-09 23:24:41

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