Download Pythagorean Triples Historical Context: Suggested Readings

Pythagorean Triples
Historical Context:
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When: ca. 500 B.C.
Where: Samos, Greek Ionia
Who: Pythagoras
Mathematics focus: Investigation of numerical relations connected to right
triangles.
Suggested Readings:
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Pythagoras and his contributions to mathematics and philosophy:
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Pythagoras.html
P. Davis’ “Number” (Scientific American, September 1964) or Chapter 13 in M.
Kline’s (ed.) Mathematics in the Modern World (1968), pp. 89-97
NCTM’s Historical Topics for the Mathematics Classroom (1969): “Number
beliefs of the Pythagoreans” (pp. 51-52) and “Figurate numbers” (pp. 53-57).
Key search words/phrases: Pythagoras, Number mysticism, Pythagoreans,
right triangle, number theory, Pythagorean triples
Problem to Explore:
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Investigation integral solutions to the equation a2 + b2 = c2, where these
solutions also are the sides of a right triangle.
Why This Problem is Important:
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Pythagorean triples are a direct consequence of the Pythagorean Theorem
and are part of its rich history.
Introduce the mathematical area known as number theory.
Problem Solving Experiences:
To Pythagoras and his followers, numbers were the quintessential model, being "the
principle, source, and root of all things." In turn, Ore (1988, p. 166) suggests that “in
view of the particular interest of the Pythagoreans in relations that could be
expressed in whole numbers, it would appear natural that they should have
investigated the problem of finding right triangles with integral sides.”
The problem to explore: Given a right triangle with hypotenuse h and legs m and n,
what integral triples (m, m, h) satisfy the equation a2 + b2 = c2 from the Pythagorean
Theorem? Some quick examples are (3, 4, 5), (5, 12, 13), and (8, 15, 17). But, how
can one approach the question systematically to explore the world of all possible
integral triples (a, b, c)?
As a starting point, the Pythagoreans investigated properties of special numbers
known as triangular numbers, square numbers, etc. Consider this array showing the
connection between 42 and 52:
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From the array, we can make two Pythagorean-type observations:
• 42 + (4 + 4 + 1) = 52, which suggests the generalization n2 + (2n + 1) = (n+1)2,
an expansion of the trinomial (n+1)2
• The difference between successive squares of integers will be the odd
numbers in sequence (e.g. 32 + 7 = 42, 42 + 9 = 52, 52 + 11 = 62)
Furthermore, the equation n2 + (2n + 1) = (n+1)2 can be used to generate
Pythagorean triples, but the term (2n+1) must be viewed as a perfect square.
1. Let m2 = (2n + 1), for n a positive integer. Rewrite the expression n2 + (2n + 1)
= (n+1)2 in terms of m, then use it to produce Pythagorean triples for values of
m = 3, 5, …, 13. Why does m have to be odd? Does this process generate all
Pythagorean triples (a, b, c)?
Proclus, an ancient Greek philosopher and commentator on Euclid’s Elements,
claimed that the method used by Pythagoras “starts from odd numbers. For it makes
the odd number the smaller of the sides about the right angle; then it takes the
square of it, subtracts unity, and makes half the difference the greater of the sides
about the right angle; lastly it adds unity to this and so forms the remaining side, the
hypotenuse.” (Heath, 1956, p. 356)
2. Letting m be an odd number, show that the method used by Pythagoras is the
same method as that generated in Problem #1.
After describing Pythagoras’ approach, Proclus continues by describing the approach
used by Plato, which “argues from even numbers. For it takes the given even number
and makes it one of the sides about the right angle; then, bisecting this number and
squaring the half of it, it adds unity to the square to form the hypotenuse, and
subtracts unity from the square to form the other side about the right angle.” (Heath,
1956, p. 356)
3. Letting p be an even number, represent Plato’s method by a formula, then use
it to produce Pythagorean triples for values of p = 4, 6, …14. Does it produce
the same set of Pythagorean triples as Pythagoras’ method starting with an
odd number? Also, examine other differences about some of the triples
generated?
Plato’s method introduces Pythagorean triples such as (6, 8, 10), which is a multiple
of the Pythagorean triple (3, 4, 5). The latter is called a “primitive” solution, which
occurs whenever the triple (a, b, c) has no common factors.
4. Explain why Pythagoras’ method will generate only primitive Pythagorean
triples, while Plato’s method often generates non-primitive triples.
Extension and Reflection Questions:
Extension 1: A standard set of expressions for generating Pythagorean triples (a,b,c)
is a = p2 – q2, b = 2pq, and c = p2 + q2, when p > q > 0.
• Show that these expressions satisfy the Pythagorean Theorem.
• When generating possible triples with these expressions, why ignore cases
when p and q are not relatively prime?
• Use these expressions to produce Pythagorean triples for values of p = 2, 3, 4,
5, 6, and 7, where p and q are relatively prime.
• Do these expressions generate any non-primitive Pythagorean triples?
• Do these expressions pick up all of the Pythagorean triples produced by both
Pythagoras’ and Plato’s methods?
• Are any new Pythagorean triples produced that were not generated by either
Pythagoras’ or Plato’s methods?
Extension 2: The approach and expressions used to generate all Pythagorean triples
(a,b,c) in Extension Problem #1 is similar to Euclid’s approach in Lemma 1 to
Proposition 28 (Book X): To find two square numbers such that their sum is also
square. His argument for generating a triple (a,b,c) is that one can start with two
different composite numbers p and q such that (1) both are odd or both are even with
p>q, and (2) pq = a2 for an integer a. Then, b = (p-q)/2 and c = (p+q)/2. Use
examples, starting with x = 9 or x = 25, to show that Euclid’s method seems to work.
Then, prove that Euclid’s method will always produce Pythagorean triples.
Extension 3: The approach taken by Pythagoras in Problem #1 leads to another
interesting relationship. Recall that all triangles generated by his method had an odd
number as its smallest side and its other two sides differed by one. Prove that the
circle inscribed in these triangles have integral radii. Remember, the center of the
inscribed circle is formed by the common intersection of the angle bisectors.
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Open-ended Exploration: Start with the four consecutive Fibonacci numbers 3, 5, 8,
and 13. Multiply the two extremes (3x13), then double the product of the two inner
numbers (2x5x8). The products 39 and 80 serve as the legs of a right triangle, whose
hypotenuse 89 (e.g.
39 2 + 80 2 = 89 ) also happens to be a Fibonacci number.
Finally, note that the product of the original four numbers 3x5x8x13 also equals 1560,
which is the area of the triangle formed. Some exploration questions:
• Will this work for any sequence of four consecutive Fibonacci numbers? Can
you prove it using recursive notation for Fibonacci numbers?
• Does a special relationship exist between the hypotenuse and the initial four
numbers (i.e. given the four numbers, could you predict the hypotenuse)?
• Explore and formulate a similar relationship starting with five consecutive
Fibonacci numbers?