Pythagorean triples. Who was Pythagoras? He lived in Greece from about 580 BC to 500 BC He is most famous for his theorem about the lengths of the sides.

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Presentation transcript:

Pythagorean triples

Who was Pythagoras? 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.

Pythagorean Triples are sets of 3 numbers which follow the rule for right angled triangles. a²+b²=c²

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

Some Pythagorean triples that we know 3,4,5 3,4,5 5,12,13 5,12,13 7,24,25 7,24,25 6,8,10 6,8,10

What have we noticed? 3²=9 and 4+5=9 3²=9 and 4+5=9 5²=25 and 12+13=25 5²=25 and 12+13=25 7²=49 and 24+25=49 7²=49 and 24+25=49 We got 6,8,10 from doubling the first Pythagorean triple. 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. 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.

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 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 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!! This following formula was found on a tablet made by the Babylonians almost 1500 years before Pythagoras was born!!

The Babylonians’ formula If we have 3 numbers written as If we have 3 numbers written as2pq p² - q² and p² + q² and p² + q² then we can make Pythagorean triples by changing the values of p and q (q<p) e.g. p = 3 q = 2 2pq = 12 p² - q² =5 p² + q² = 13 This is the Pythagorean triple 5, 12, 13

Using Fibonacci numbers to make Pythagorean triples. We will look at these Fibonacci numbers: We will look at these Fibonacci numbers: Starting with 1, 2, 3 and 5 Starting with 1, 2, 3 and 5 1. Multiply the inner numbers 2x3=6 2. Double the result 2x6=12 3. Multiply the outer numbers 1x5=5

4. The third side is found by adding together the squares of the inner 2 numbers (2² = 4 and 3² = 9 and =13) =13) We have generated the 5, 12, 13 triple!! 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) (Note, this is 2 times 8, 15, 17)

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 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) 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)

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 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 square units!!! The primitive triangle whose sides are 693, 1924 and 2045 has an area of square units!!!