Friday, May 30, 2014

Math & Stuff:

Some infinities are bigger than other infinities. A writer we used to like taught us that. There are days, many of them, when I resent the size of my unbounded set. I want more numbers than I'm likely to get, and.... I cannot tell you how thankful I am for our little infinity. I wouldn't trade it for the world. You gave me a forever within the numbered days, and I'm grateful.”
―John Green, The Fault in Our Stars

The proof:


5 comments:

  1. Cute story, but the video is wrong. There is no end to the counting numbers, therefore one cannot run out of them.

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    Replies
    1. How about this proof? Consider the function y=x, it's a straight line passing through the origin at a 45 degree angle. Let's use calculus to find the area under the line from 0 to 1. We will approximate the area by summing an infinite number of small rectangles, each rectangle having an area = xdx and sum these up from 0 to 1. The integral of x is (x^2)/2 and the area is 1/2 under the line from 0 to 1. Now we do the same thing from 0 to 2 instead and find that the area is 2. Since 2>1, the infinity of rectangles from 0 to 2 must greater than the infinity between 0 and 1.

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  2. By its definition infinity cannot be measured. How then can one infinity be greater than another?

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  3. It's not measuring, it's one being larger than another. What if you you have y=x divided by y=x^2, and determine the value as x approaches infinity? It goes to zero, because the denominator becomes larger so much faster. Isn't one one infinity greater than the other?

    See Some Infinities Are Larger Than Others: The Tragic Story of a Math Heretic at http://www.wired.com/2007/07/some-infinities/

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