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History of Mathematics : Square and Triangular Numbers and the Pythagorean Theorem

Figurative Numbers & Pythagorean Theorm. See attached file for full problem description.

9) Which basic trigonometric identity is actually a statement of the Pythagorean Theorem? Justify your answer.
5) From very early on, mathematicians were interested in finding right triangles whose sides had integer length. By the Pythagorean Theorem, this boils down to the problem of finding three whole numbers a, b, and c such that a2 + b2 = c2. One solution is well known: a = 3, b = 4, c = 5. Triples of integers (a, b, c) such that a2 + 2 = c2 are usually called Pythagorean triples. This question is about finding more of them.
a. Check that (6,8, 10) is also a solution. How is it related to (3,4,5)?
b. Check that (5, 12, 13) is a solution. Solutions like this one, in which the numbers have no common factors, are called primitive.
c. The Pythagoreans discovered that if you add consecutive odd numbers beginning with 1, the sum will always be a square. So:
1 + 3 + 5 + 7 = 16 = 42, and
1+3+5+7+9+11+13+15+17+19=100= 102.
Can you explain why this is true? Can you prove it? (The Pythagoreans probably did this by laying out pebbles in square arrays. Compare two such squares.)
d. If the last number added is itself a square, this yields a Pythagore triple. For example,
gives the (3,4, 5) triple. Use this idea to generate more Pythagorean
- triples. Can you get all such triples this way?
About two centuries separated the Pythagoreans (c. 500 B.C.) and
Euclid (c. 300 a.c.). These centuries were rich, eventful times in Ancient Greece. Classify each of the following events as "before the Pythagoreans,""between the Pythagoreans and Euclid," or "after Euclid."
a. Socrates and Plato became prominent Greek philosophers.
b. The Athenians defeated the Persians at the battle of Marathon.
c. Draco created the first formal code of laws for Athens.
d. Hippocrates began the study of medicine as an empirical science.
e. Athenian dramatist Euripides wrote Medea.
f. Plutarch wrote his Parallel Lives of Greek and Roman leaders.
g. Homer wrote the Iliad and the Odyssey.
h. Herodotus wrote a history of the Persian Wars.


Solution Preview

Here are the answers to your questions:

A triangular number is t_n = n(n+1)/2.
The observation by Plutarch is

8t_n+1 = 4n(n+1)+1 = 4n^2 +4n+1 = (2n+1)^2

Just derive it:
t_(9n+4) - t_{3n+1) = (9n+4)(9n+4+1)/2 - (3n+1) (3n+1+1)/2
= (9n+4)(9n+5)/2 - (3n+1) (3n+2)/2
= (81n^2+81n+20)/2 - (9n^2+9n+2)/2
= (72n^2+72n+18)/2
= 9(4n^2+4n+1) = 9(2n+1)^2 = ...

Solution Summary

Square and Triangular Numbers and the Pythagorean Theorem are investigated. The mathematicl parts are shown with full working. Web links are provided for the historical parts.