Problems pp. 156-161, #2, 6, 7, 8, 13, 20, 37. Due Mon, 4/5.
Today: we begin chapter 3: To Infinity, and Beyond!
What do you think of when you hear the word "infinity"?
What do you already know about infinity?
Just as in the case of high dimensions, it might be wise to creep
up on infinity via the finite numbers. Let's begin, therefore,
with a brief history of numbers:
Let's start small: 1 -- it's the loneliest number
What came next? 2!
Then what? "Rationals"
Then what? "Irrationals"
Then what? "Negatives" (and hence the Integers)
Then what? Finally 0 made it in....
Then what? "Imaginary"
Infinity - so mysterious.... Is it a number?
Q: Are mathematicians trying to take all the fun out of it by
studying it?! A: Of course not! We relish the mind-blowing mystery. And
there's plenty of fun to be had....
William Blake (from Auguries of Innocence):
To see a World in a Grain of Sand
And a Heaven in a Wild Flower,
Hold Infinity in the palm of your hand
And Eternity in an hour.
Let's get back to the idea of number. What does "2" mean? If we
can't answer that, then how on earth are we supposed to
answer the same question about infinity?
So what is infinity: is it a number? Is it just one thing?
Some things are infinite in size. Maybe it's better to
focus on the adjective than on the noun....
What is a one-to-one correspondence, and how is it useful?
Let's look at some problems:
II.9
III.16
II.6
IV.22
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