Pythagoras Πυθαγόρας. Tom Button

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2 Pythagoras Πυθαγόρας Tom Button

3 Brief biography - background Born c570 BC Samos Died c495 BC Metapontum Much of what we know is based on 2 or 3 accounts written years after he died Many things attributed to Pythagoras may be have been developed by other Pythagoreans Samos

4 Early life Landowning Mother and merchant Father Travelled to Alexandria and Babylon Returned to Samos but left for Croton about age 40

5 Philosophers: lovers of wisdom Secretive Significant role in Croton: governing and education High status for women Strict rules on diet The Pythagoreans

6 Tuning stringed instruments Lyre Monochord

7 Why do some notes sound good together? 2:2 ratio 4:2 ratio 3:2 ratio No simple ratio

8 Ratios of lengths of strings Octave: 2 freq. Perfect 5 th 1½ freq. 325mm 433mm 650mm

9 Using Maths to make a scale

10 A new note in the scale = =

Repeating the process 295 443 262 332 393 524 498 Multiply the

11 Repeating the process Multiply the current note by 1.5 If the note is outside the octave ( ) divide by 2

12 Filling in the gap ? 393 Hz is the 5 th for 262 Hz 262 Hz is the 5 th for Hz

13 The Major Scale in C C 262 Do D 295 Re E 332 Mi F 349 Fa G 393 So A 443 La B 498 Te C 524 Do

14 Continuing a Pythagorean Tuning

15 Equal temperament r 12 r A 440 A# 466 B 494 C 523 C# 554 D 587 D# 622 E 659 F 698 F# 740 G 784 G# 831 A

16 All is number

17 Pythagoras Theorem: A brief history C1900BC Babylon Triples C1400BC Egypt Use of triangle in construction (possibly known much earlier) C600BC India Triples, statement of the theorem (and proof?) C500BC Greece Algebraic methods to construct triples (Pythagoras) C300BC Greece Formal proof (Euclid) C100BC China Proof of the theorem (possibly based on much older texts)

18 Proofs of Pythgoras Theorem How many proofs do you know? a² + b² = c²

19 Similar triangles proof

20 You can use any similar shapes

21 Similar triangles proof

22 Dissection proof

23 Did Pythagoras prove it? Thales c BC is the first evidence of deductive reasoning Pythagoras c BC Plato c BC references the theorem of Pythagoras Euclid c BC

24 Hippasus

25 Pythagorean triples All primitive Pythagorean triples can be constructed using: m² n², 2mn, m² + n² m > n m,n coprime m or n even m n m² n² 2mn m² + n²

26 The legacy of Pythagoras Plato and Euclid Roman mathematics Islamic mathematics Descartes and Leibniz Bertrand Russell All is number

Further information Pythagoras: His Lives and the Legacy of a Rational Universe Kitty Ferguson Pythagoras and the Pythagoreans: A Brief History Paperback Charles H.

27 Further information Pythagoras: His Lives and the Legacy of a Rational Universe Kitty Ferguson Pythagoras and the Pythagoreans: A Brief History Paperback Charles H. Kahn MacTutor History of Mathematics Pythagoras Biography www-history.mcs.stand.ac.uk/biographies/pythagoras.html Cut the Knot Proofs of the Pythagorean Theorem In Our Time (Radio 4) Pythagoras

28 About MEI Registered charity committed to improving mathematics education Independent UK curriculum development body We offer continuing professional development courses, provide specialist tuition for students and work with industry to enhance mathematical skills in the workplace We also pioneer the development of innovative teaching and learning resources

Further Mathematics Support Programme Maths and Music Recap In the Maths and Music session you learnt the two basic rules for whether notes of different frequencies sound good together: Notes with

29 Further Mathematics Support Programme Maths and Music Recap In the Maths and Music session you learnt the two basic rules for whether notes of different frequencies sound good together: Notes with frequencies in the ratio 2:1 sound the same but higher (an octave) Notes with frequencies in the ration 3:2 go well together (a perfect fifth) This gives us two mathematical rules for creating a scale: 1.5 to get a new note. 2 if it is outside the octave. Starting with middle C at 262Hz explain how you would obtain the frequencies of the following notes in the range Hz: This is known as a Pythagorean tuning. C D E F G A B C Equal Temperament Most modern, western music uses a 12-note tuning system called Equal Temperament. This is a method of constructing a scale based on using an equal multiple for each note moved up the scale. To find the multiplier so that after 12 notes you are an octave higher, or at twice the frequency, you would use: Copy and complete this table for the frequencies of notes in 12-tone equal temperament: C C# D D# E F F# G G# A A# B C Compare the frequencies of the notes in the Pythagorean tuning to the notes in 12-tone equal temperament. Further Investigation Find out more about Pythagorean tunings (and other Just Intonations) and Equal Temperament. Can you hear the difference between them? Apply the rule for generating the Pythagorean tuning 12 times. Do you get back to where you started?

30 Pythagorean triples problems Find the radius of the largest circle that can be inscribed in a triangle. The circle x 2 + y 2 = 5 2 has 12 points with integer co-ordinates, as does the circle x 2 + y 2 = Investigate this for other Pythagorean triples. To find a circle with more than 12 points with integer co-ordinates multiply 5 and 13 to obtain x 2 + y 2 = 65 (65 can be written as the sum of two distinct squares in two different ways). Does this result generalise: can the product of the largest values in two Pythagorean triples always be written as the sum of two distinct squares in two different ways? A Pythagorean triple is primitive if there isn't a common factor that divides a, b and c. (3,4,5) and (5,12,13) are primitive Pythagorean triples but (6,8,10) isn t. Is the smallest number in a primitive Pythagorean triple always odd? and (8, 5, 17) is a Pythagorean triple. Add the reciprocals of any two consecutive odd numbers. Will the resulting fraction, x y, always generate an integer Pythagorean triple, (x, y, z)? Is the largest number in a primitive Pythagorean triple always odd?

31 MEI Further Pure with Technology June This question concerns Pythagorean triples: positive integers a, b and c such that a b c. The integer n is defined by c b n. (i) Create a program that will find all such triples for a given value of n, where both a and b are less than or equal to a maximum value, m. You should write out your program in full. For the case n = 1, find all the triples with 1 a 100 and 1 b 100. For the case n = 3, find all the triples with 1 a 200 and 1 b 200. [9] (ii) For the case n = 1, prove that there is a triple for every odd value of a where a > 1. [4] (iii) For the case n = p, where p is prime, show that a must be a multiple of p. [3] (iv) For the case n = b, determine whether there are any triples. [4] (v) Edit your program from part (i) so that it will only find values of a and b where b is not a multiple of n. Indicate clearly all the changes to your program. Use the edited program to find all such triples for the case n = 2 with 1 a 100 and 1 b 100. [4]

Vector and scalar quantities

Vector and scalar quantities A scalar quantity is defined only by its magnitude (or size) for example: distance, speed, time. It is easy to combine two or more scalar quantities e.g. 2 metres + 3 metres