Saturday, 31 July 2010

An 'i' for an 'i' !!!

I'm learning about complex and imaginary numbers right now. Now, the previous sentence probably throws my less mathematically inclined readers into a state of utter confusion. I'm afraid it just can't be helped because this little mathematical curiosity is so interesting that I find myself unable to delay the moment of revelation even to give a short overview about what complex and imaginary numbers are. If you are very keen on understanding this post I suggest you brush up by clicking here and here.

As is usual when I start learning about a topic I started to fiddle around with the concept. A little bit of manipulation here and there goes a long way.  Very soon I stumbled across a mystery which I managed to explain to myself. We all know that
i 2 = −1
i^{4n} = 1\,
i^{4n+1} = i\,
i^{4n+2} = -1\,
i^{4n+3} = -i.\,

So I took this simple problem: i^5 and decided that it could be represented as (i^4)^(5/4). Since i^4 = 1 the expression can be simplified to 1^(5/4). 1^(5/4) = 1.

I was mystified. I could use this technique to prove that any power of i is equal to one. Had I discovered some disrepancy, some fallacy in mathematics? Not likely, I thought. So I set out to analyse my results. 

I wish to prove that i^(5) is i and not 1.

Very soon I found the error in my calculations. I was just beginning to understand how quirky imaginary numbers can be. My mistake lay in assuming that 1^(5/4) = 1. After some thinking, I figured out that it was not. Here's what I did. I reasoned that 

1^(5/4) = (1^5)^(1/4)
           = (1^(1/2))^(1/2)                      Now since 1^(1/2) = +1 or -1
           = 1^(1/2) or -1^(1/2)
           = -1 or 1 or i or -i

So in the end I concluded that by breaking up the exponent of 5 to 4 and 5/4 I was actually introducing 3 new solutions into the problem. It is similar to a situation in which you have a linear equation and by squaring the equation you introduce new spurious solutions which were never part of the original problem. Since (i^4)^(5/4) is not equal to i^5, i^5 remains equal to i and only i. Q.E.D

Interesting isn't it?

Tuesday, 13 July 2010

Weapons of Math Instruction

I found this online. It's Hilarious!

A public school teacher was arrested today at JFK INTL Airport as he attempted to board a flight while in possession of a ruler, a protractor, a set square, a slide rule and a calculator.
At a morning press conference, Attorney General Alberto Gonzales said he believes the man is a member of the notorious Al-gebra movement. He did not identify the man, who has been charged by the FBI with carrying weapons of math instruction.
“Al-gebra is a problem for us,” Gonzales said. “They desire solutions by means and extremes, and sometimes go off on tangents in a search of absolute value. They use secret code names like ‘x’ and ‘y’ and refer to themselves as unknowns’, but we have determined they belong to a common denominator of the axis of medieval with coordinates in every country. As the Greek philanderer Isosceles used to say, ‘There are 3 sides to every triangle’.”
When asked to comment on the arrest, President Bush said, “If God had wanted us to have better weapons of math instruction, He would have given us more fingers and toes.” White House aides told reporters they could not recall a more intelligent or profound statement by the president.