Proof Practice
February 24, 2004. This morning I visited one of the many Javascript tutorial sites on the web. The web author was creating a web page with Javascript as a learning exercise. I too have advocated the "learn by writing" approach, especially in my books.
For practice doing mathematical proofs (and to have an addition to this site), I'll try the following exercise, which I found near the beginning of Calculus with Analytic Geometry, by Howard Anton: "Prove the following results about sums of rational and irrational numbers: (a) rational + rational = rational; (b) rational + irrational = irrational."
The book does not give the solution; I can think of three possible reasons:
- 1. The author wants you to bear down and work it out yourself as practice;
2. The book is thick enough already;
3. There is more than one correct solution.
I'll give it a try here. If I overlooked something, I welcome constructive criticism.
Let a and c = any integers. Let b and d = any integers except 0. Now let a/b and c/d represent two rational numbers formed from a, b, c, and d.
a/b + c/d = (ad + bc)/bd. Let e = (ad + bc). Let f = bd.
Given that a and d are integers, ad is an integer. Given that b and c are integers, bc is an integer. Since the sum of two integers is an integer, (ad + bc) is an integer. Substitute e for (ad + bc).
Given that b and d are integers, bd is an integer. Substitute f for bd.
Since e and f are integers, e/f is a rational number.
Over 35 years ago, when Mrs. Yoyo (see Brain Bombardment was my geometry teacher, I had to prepare for the New York State Math 10 Regents. As part of my curriculum, I had to memorize proofs and reproduce them on a blank sheet of paper during tests.
This method of learning was merely an exercise in short-term memory training. It also reminds me of a child who can recite the Gettysburg Address, line by line, without understanding it.
To me it is far more important to learn how to do a proof; to be able to find the correct steps instead of regurgitating them.
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