Books and Articles

"Hidden Harmonies: The Lives and Times of the Pythagorean Theorem"
by Robert Kaplan and Ellen Kaplan
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Bozo Sapiens: Why to Err is Human
by Michael and Ellen Kaplan


"Out of the Labyrinth: Setting Mathematics Free"
by Robert Kaplan and Ellen Kaplan


"Chances Are . . . : Adventures in Probability"
by Ellen Kaplan and Michael Kaplan


"The Art of the Infinite: The Pleasures of Mathematics"
by Robert and Ellen Kaplan


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Disturbed by the poor quality and low level of math education in the country, three of us began The Math Circle in September 1994...

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Quiz (with answers) based on The Nothing That Is: A Natural History of Zero, by Robert Kaplan, Ellen Kaplan (Illustrator).

  1. Can you divide by 0? Why, or why not?
  2. What is 13 raised to the 0 power? What is 0 raised to the 0 power?
  3. If you add 1/2 + 1/4 + 1/8 + 1/16 + ... will you ever get to 1?
  4. Why does zero matter in solving equations?
  5. How do zero and calculus combine to tell us when the waves in our affairs crest or crash?
  6. Extra Credit: Are there more counting numbers or even numbers?


1. Answer: Alas, you can't - unless you're willing to have all numbers turn out to be the same. The reason is given in the book, pp. 72-74.

2. Answer: 13 raised to the 0 power is 1 (see pp. 117-118). The question about 0 raised to the 0 power leads us into hideous tangles along one route, but along another we can agree to assign this symbol the value 1 (see pp. 163-169).

3. Answer: Not if you stop at any finite resting-place: you'll always fall short of 1 by 1 over the last denominator in your series. But the notion of limit, in which zero plays so crucial a part, lets us say that the infinite series sums to 1 (see pg. 158).

4. Answer: This was John Napier's great insight. It is explained on pp. 132-136 (see too the notes to this section on the website).

5. Answer: This wonderful story - the story underlying all our science and engineering - is told on pp. 169-172.

6. Answer: Although there are, remarkably enough, different sizes of infinity, the size of the set of counting numbers and that of the even numbers is the same - although you would be be in good company if you thought there were twice as many counting numbers as even numbers (after all, the evens make up half of the counting numbers). But we can pair up the members of the two sets.

Think of the counting numbers lined up and walking through a door, and the even numbers lined up too. Let them hold hands as they walk through: the counting number 1 with the even number 2, counting number 2 with even number 4, 3 with 6, 4 with 8 and so on. In this way they're all paired up and no number of either sort is left without a partner. We've made a one-to-one correspondence between these two sets, so they must be the same 'size' of infinity. Mind-boggling but true.

For the story of the infinite, and the even more startling revelations about sets with more elements than the counting numbers, see the Kaplans' Art of the Infinite. Here's a bonus question from it, due to the ninth century Indian mathematician Mahavira:

One night, in a month of the spring season, a certain young lady was lovingly happy with her husband in a big mansion, white as the moon, set in a pleasure garden with trees bent down with flowers and fruits, and resonant with the sweet sounds of parrots, cuckoos and bees which were all intoxicated with the honey of the flowers. Then, on a love-quarrel arising between husband and wife, her pearl necklace was broken. One third of the pearls were collected by the maid-servant, one sixth fell on the bed - then half of what remained and half of what remained thereafter and again one half of what remained thereafter and so on, six times in all, fell scattered everywhere. 1,161 pearls were still left on the string; how many pearls had there been in the necklace?