Pythagorean triples

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Pythagorean triples
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Who was Pythagoras?

• Pythagoras was a bully!

He lived in Greece from about 580 BC to 500 BC
He is most famous for his theorem about the
lengths of the sides in right angled triangles.
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What are Pythagorean Triples?
Pythagorean Triples are sets of 3 numbers which
follow the rule for right angled triangles.
a²b²c²
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What we are investigating

• We are going to find as many Pythagorean triples
  as we can.
• We are going to find a way to generate them.

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Some Pythagorean triples that we know

• 3,4,5
• 5,12,13
• 7,24,25
• 6,8,10

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What have we noticed?

• 3²9 and 459
• 5²25 and 121325
• 7²49 and 242549
• We got 6,8,10 from doubling the first Pythagorean
  triple.
• For the odd numbered triples, we square the small
  number and halve that answer rounding up and down
  to give the other 2 numbers in the triple.

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Where did Pythagorean triples originate from?

• The name for these triples may give a hint as to
  who thought of them first, but is it actually the
  truth?
• The answer is ..NO. Actually the Babylonians
  were the first to discover these numbers
• This following formula was found on a tablet made
  by the Babylonians almost 1500 years before
  Pythagoras was born!!

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The Babylonians formula

• If we have 3 numbers written as
• 2pq
• p² - q²
• and p² q²
• then we can make Pythagorean triples by changing
  the values of p and q (qltp)
• e.g. p 3 q 2
• 2pq 12
• p² - q² 5
• p² q² 13
• This is the Pythagorean triple 5, 12, 13

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Using Fibonacci numbers to make Pythagorean
triples.

• We will look at these Fibonacci numbers
• 1 2 3 5 8
• Starting with 1, 2, 3 and 5
• Multiply the inner numbers 2x36
• Double the result 2x612
• Multiply the outer numbers 1x55

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• 4. The third side is found by adding together the
  squares of the inner 2 numbers (2² 4 and 3² 9
  and
• 4 9 13)
• We have generated the 5, 12, 13 triple!!
• 5. Using 2, 3, 5 and 8 gave us the Pythagorean
  triple 16, 30, 34
• (Note, this is 2 times 8, 15, 17)

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Some fascinating facts!!

• The 5, 12, 13 and the 6,8,10 triangles are the
  only two Pythagorean triangles whose areas are
  equal to their perimeters.
• The 3,4,5 triangle is the only one whose sides
  are consecutive whole numbers and whose perimeter
  is double its area (12 2 x 6)

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More fascinating facts!!

• The probability that a Pythagorean triangle will
  have an area ending with the digit 6 is 1/6,
  ending in a 4 a probability of 1/6 and ending in
  a 0 a probability of 2/3.
• The primitive triangle whose sides are 693, 1924
  and 2045 has an area of 666 666 square units!!!
• Where is the devil?!!