Mathematics Teaser
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Sneakattack
Posts: 1056

Posted: 12:25PM Feb 25, 2012 

I just submitted a teaser involving pi, i, and e. But I have a feeling that I made an error. Here is my teaser (if I didn't make an error, great, it was a pleasure sharing it):
If a number is raised to a power with a complex number in it, and that the entire number (after being raised to the power) is multiplied by 1, the outcome is about 19333.7.
Why is this so?
Answer:
i is a complex number, representing the square root of 1.
e^(pi * i) = 1.
The properties of exponents states that two equal numbers, when multiplied, have their exponents added. So, 3^3 * 3^2 = 3^5.
So,
e^(pi * i) * e^(pi * (1 / i)) equals
e^(pi^2 * (i / i)) or
e^((pi^2 * i) / i)
The i's cancel in the exponential fraction, leaving
e^(pi^2)
pi^2 = 9.869604401
e^(pi^2) = 19333.68907.
What are you looking at? 
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spikethru4
Posts: 17

Posted: 10:55PM Dec 10, 2012 

There is a major flaw in your logic. You correctly state that exponents are added when powers of the same number are multiplied, but in your reasoning you multiplied the exponents.
e^(pi * i) * e^(pi * 1/i) = e^(pi * (i + 1/i))
Now, i^2 = 1 and i^4 = 1, so 1/i = i^4/i = i^3 = i^2 * i = i
The i's do cancel off, but to zero, leaving you with e^(pi * i) * e^(pi * 1/i) = e^(pi * 0) = e^0 = 1

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JQPublic
Posts: 1811

Posted: 02:33AM Dec 11, 2012 

Hi spikethrough. Do complex indices work by the same law as real ones? Thanks.
Otherwise, I under stand what sneak wrote and spike's correction.
'An idea, like a ghost, must be spoken to a little before it will explain itself.'  Charles Dickens 
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spikethru4
Posts: 17

Posted: 08:47PM Dec 11, 2012 

JQPublic wrote:
Hi spikethrough. Do complex indices work by the same law as real ones? Thanks.
(a^x) * (a^y) = a^(x+y) holds for all real (or complex?) a and real or complex x and y.
However, (a^x)^y = a^(x*y) does not generally hold for complex x and y. A simple counterexample is (e^(2*pi*i))^i, which gives 1^i = 1 if you evaluate the powers in order and e^(2*pi) = 0.0018674 if you multiply the powers first.

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JQPublic
Posts: 1811

Posted: 03:34AM Dec 12, 2012 

I see spikethru, thanks!
'An idea, like a ghost, must be spoken to a little before it will explain itself.'  Charles Dickens 
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