Pythagorean Triples


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For those who are interested, I have added a page to my website explaining how to find all the Pythagorean triples where a leg is a particular number. If you don’t know what a Pythagorean triple is, you can read about it here.

fun mathematics

Going Green


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My daughter, Libby, just bought a new car, a green Honda Civic. She uses it to drive back and forth to work. She works at a daycare in our church, so she makes the same trip at least six times a week, sometimes more often. One of the main reasons she wanted to buy a Honda was the good gas mileage they reputedly get.

Libby recently graduated from college and has thousands of dollars in student loans to repay, so she has become extremely cost conscious.

Our family has been making the trip to our church for many years, and we have basically two routes we always follow. Both routes usually take the same amount of time. My wife prefers the highway route. I prefer the back-road route. We’ve gone back and forth about the merits of our favorite route over the years. She likes the sense of getting where she’s going fast on the highway route and doesn’t like the shabby industrial buildings along the back-road route. I like the sense of taking the shortest way, and—why fight it?—I like the shabby industrial look.

Newly cost-conscious Libby was not content with our impressionistic reasons for preferring one route to another. She wanted hard data, so she measured how long it took and how many miles she drove on both routes. She found that both routes take about the same amount of time. Confirming my impression, however, she found that the back-road route was about 3 miles shorter. (Google maps makes it 2.3 miles shorter). Since she travels the same route at least 12 times every week, choosing the shorter route could actually save a considerable amount of gas.

This is just the sort of calculation people all over the world are doing now. They are finding ways to reduce dependence on fossil fuels and making the calculation part of an overall strategy to cut costs.

So how much will Libby save? It’s not really easy to say. Since the trip takes the same amount of time regardless of route, the Honda’s engine probably consumes the same amount of gas. Moreover, the longer route has more highway miles, which tend to boost fuel efficiency. Just for the fun of it, however, let’s assume that the Honda’s average gas mileage of 35 mpg is constant regardless of route. Libby’s car will drive 2.3 × 12 × 52 =  1432.5 fewer miles in a year,  requiring 1432.5 ÷ 35 ≈ 41 fewer gallons of gas. At $2.80 per gallon, Libby will save 2.80 × 41 ≈ $114.80 in a year. By using the savings to pay down principle on her loans, she could end up saving far more. Go, Libby!

about me mathematics

Math Is Fun


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In grade school I hated arithmetic. More accurately, I hated repetitive drills in arithmetic. I never had trouble remembering facts, and by second grade, I had discovered that if I remembered what my teachers said and what I read, I could do well in school. I had no trouble learning the basic facts of addition, subtraction, multiplication, and division. It wasn’t enough to know the facts, however, I had to demonstrate that knowledge over and over again with mind-numbing pages of arithmetic problems. I remember doing timed tests of long division problems where we were supposed to complete 50 problems in 15 minutes. I was so busy silently fuming at the injustice of it all that I would finish only 8 or 9 problems. It wasn’t that I couldn’t do them; it was that they presented no challenge.

One day, my older sister came home and showed me what she was learning about. She showed me how 3 could be written as 11 in binary. It was like switching on a light. Suddenly, math was fun. I started tinkering with different bases on my own, writing familiar numbers in unfamiliar ways:

1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, … 44, 100, …

I made multiplication tables for different number systems:

× 0 1 2 3 4
0 0 0 0 0 0
1 0 1 2 3 4
2 0 2 4 11 13
3 0 3 11 14 22
4 0 4 13 22 31

I created new arithmetics using the new bases. I even found ways to convert familiar constants such as π to different bases.

π(10) = 3.14159265…

π(5) = 3.03232214303343…

What I liked best about math, though, was the comfort it gave me. Mathematics gave me access to an abstract world, a world where the rules were absolute, where absolute certainty was not only possible but necessary. I could divide the 21 abstract cookies in my math problems among 7 abstract friends because the cookies were all the same, and the friends all wanted an equal number. I didn’t have to deal with the odd, misshapen cookie or with Paul, who wanted an extra cookie for being my best friend.

Measurements were always exact. Squares and circles and prisms and spheres were always perfect, and cylindrical tanks could always be filled exactly full. You never had to worry about cutting a 20-foot board into five 4-foot lengths only to find that the last length was actually only 477/8” because the saw took out 1/32” with every cut. The mathematical world was pristine and pure, beautiful and symmetrical. Every operation had its inverse. Every number had its own unique qualities. Every theorem had its own peculiar applicability.

While other boys were discovering girls, I was discovering the five platonic solids. While they tinkered with their cars, I taught myself how to use a slide rule and how to solve problems involving logarithms. I fell in love with the ideal and had neither time nor concern for the real. The ideal world was consistent, logical, and true. The interior angles of an n-sided polygon always summed to n−2 straight lines. I could trust the ideal world.

Math is still fun. The ideal world turned out to be less ideal than I had thought. Kurt Gödel saw to that. But the real world turned out to have in it more beauty than I had at first noticed: fractal beauty.