Quick interviews with folks here at the Gathering For Gardner, including Stephen Wolfram, Will Shortz, Dale Seymour, John Conway and many others.
Archive for infinity
Just why does e appear in so many guises?
That the worm falls off the end of the rope depends on the fact that the incredible
1 + 1/2 + 1/3 + 1/4 + . . .
diverges to infinity, growing as large as you please!
Dana Richards, editor of Martin Gardner’s Colossal Book of Short Puzzles and Problems explains why the worm makes it, in only about 15,092,688,622,113,788,323,693,563,264,538,101,449,859,497 steps! (Give or take a few.) This incredible fact depends on the mysterious Harmonic Series, discussed a little more in our next post.
Dana Richards, editor of The Colossal Book of Short Puzzles and Problems discusses the amazing Martin Gardner and his legacy!
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
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:
- The stork can catch the frog even if it can start at any rational number and hop any fixed rational distance each step.
- However, if the frog can start at any real number or hop any real distance, the stork has no strategy that guarantees a catch. This is, in effect, the same as proving that the real numbers are not countable.
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
With enough time and patience and bananas, can we go as far as we please?