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2016-12-06, 08:57   #2
only_human

"Gang aft agley"
Sep 2002

2×1,877 Posts

Quote:
 Originally Posted by Nick 47. Calculate $$(\frac{1}{\sqrt{2}}(1+i))^{300}$$ without using a computer.
Let me take a stab at this one.

At the end of Proposition 44 is
Quote:
 $$|1+i|=\sqrt{2}$$
So I can replace the radical in the denominator but I'd rather have the norm instead of the absolute value so I distribute a square from the outside exponent:
$$(\frac{1}{1 + i}(1+i)^{2})^{150}$$
Then since 1 + i is not equal to zero, the expression is meaningful and I can cancel.
So I think that I get (1 + i)150

I'm not sure I am doing this right at all but I'd like to end with an apocryphal story that I like:
https://en.wikipedia.org/wiki/Hippasus
Quote:
 Some modern scholars prefer to credit Hippasus with the discovery of the irrationality of √2. Plato in his Theaetetus,[24] describes how Theodorus of Cyrene (c. 400 BC) proved the irrationality of √3, √5, etc. up to √17, which implies that an earlier mathematician had already proved the irrationality of √2.[25] A simple proof of the irrationality of √2 is indicated by Aristotle, and it is set out in the proposition interpolated at the end of Euclid's Book X,[26] which suggests that the proof was certainly ancient.[27] The proof is one of reductio ad absurdum, and the method is to show that, if the diagonal of a square is commensurable with the side, then the same number must be both odd and even.[27] In the hands of modern writers this combination of vague ancient reports and modern guesswork has sometimes evolved into a much more emphatic and colourful tale. Some writers have Hippasus making his discovery while on board a ship, as a result of which his Pythagorean shipmates toss him overboard;[28] while one writer even has Pythagoras himself "to his eternal shame" sentencing Hippasus to death by drowning, for showing "that √2 is an irrational number."[29]

Last fiddled with by only_human on 2016-12-06 at 09:05 Reason: s/151/150/ exponent 150 not 151