Erdös: The Man Who Loved Only Numbers
Assignments for Class #5
• Read TWO chapters, Chapter 3, Einstein vs. Dostoyevski , pages 131 - 144, and Chapter π, pages 145 - 158.
• Remember this question about Chapter e? What is e, anyhow?
Here is the exuberant, the irrepressible, the unstoppable James Grime to tell you how e first appeared in mathematics, and how it simplifies the mathematics of growth and decay (think population growth, compound interest, radioactive decay...).
• On page 132 of our text, the author recounts a story about Erdos challenging a young boy with a math problem. The technique the boy uses to solve the problem is know to mathematicians as the “pigeonhole principle” (although it is not called that in our text). Watch this video about the pigeonhole principle:
• Watch this one-minute video about the “traveling salesman problem” (discussed on page 168 of our text):
• If you like to dance, maybe you can Walk Like an Egyptian. Can you do math like an Egyptian? This video gets a bit heavy by the end, but in the beginning, it's a good explanation of those Egyptian fractions, discussed in our text, starting on page 153.
Questions to Think About
• Read the second full paragraph on page 157, beginning with "Erdös's 1932 paper...".
In the paragraph, what does it mean to say that Erdös's 1932 paper illustrates "... how specific results are made increasingly general."? How does the progress on Egyptian fractions (pages 153-157) illustrate this?
• Are you surprised that there is still no best solution known for the traveling salesman problem? What makes it so difficult?
Other Resources
• If you are interested in seeing more applications of the pigeonhole principle read this paper. It includes a very short and remarkable proof by Erdos of the the theorem (see our text pp 54-55) that any sequence of more than n2 distinct numbers has a monotone sub-sequence of length more than n. (Monotone sub-sequence means that no matter how you arrange the numbers, you cannot avoid having a sequence of n + 1 integers that form either and increasing or a decreasing sequence.)
• For you pencil-and-paper mathematicians, here are the details of converting the fraction 2/7 into 1/4 + 1/28 (page 153):
1) First, the slightly-smaller-unit-fraction trick: Find the largest unit fraction slightly less than 2/7: Invert 2/7 -> 7/2 = 3.5; round 3.5 to next integer, 4, invert 4 to 1/4, so 1/4 < 2/7.
2) this means that 2/7 = 1/4 + x; but what is x?
3) x = 2/7 - 1/4 = ? To do this subtraction, you need to express 2/7 and 1/4 with same denominator: 28.
4) 2/7 - 1/4 = 8/28 - 7/28 = 1/28. DONE, because this is a unit fraction.
5) So 2/7 = 1/4 + 1/28.
• For fun, here are "Mathematical" Limericks.
• And from Steve, an MIT cheer:
‘E to the U du dx, / E to the X dx. / Cosine, secant, tangent, sine, 3.14159. / Integral radical mu dv / Slipstick, sliderule, MIT. / Go Tech!’
What? Sports at MIT? Well, yes -- plenty: see MIT Engineers at Wikipedia.
