What follows after 0, 1, 2, … , once you’ve managed to list every counting number?
Around 1875, Georg Cantor created — or discovered if you like — the transfinite ordinals : the list continues 0, 1, 2, …, then ω , ω + 1, ω + 2, etc, for quite a long long way. John H. Conway tells us about his Surreal Numbers , which add in such gems as
1 / √ ω
Check out Knuth’s Surreal Numbers, Conway & Guy’s Book of Numbers , or for more advanced users, Conway’s On Numbers and Games.
Niclas Hedell, a listener, poses a problem from his days in the Swedish military: given two trees in the forest, and a rope twice as long as the distance between the trees, how do you find a third tree so that all three make a right triangle.
And we explain how the Stork can catch the Frog.
Amusingly, this problem has exactly the same solution as the proof that there are as many rational numbers as there are counting numbers. And the proof generalizes: one stork can catch three frogs, or ten or fifty.
Here are some bonus problems:
A contestant for our Million-Dollar-Give-Away sent in Rayo’s Number, hitherto the largest number ever used for any real purpose: to wit, winning the
Check out the article by Scot Aaronson that inspired them to duke it out! And this thread on the math forum is quite interesting as well.
With enough time and patience and bananas, can we go as far as we please?
A classic puzzle reveals why rockets require so much fuel, even for wee payloads.
There are vastly many more real numbers than fractions!
How to do infinitely many things in a few minutes!
Are there more fractions than counting numbers?
Are there only half as many even numbers as counting numbers?
The Math Factor in iTunes
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