pythagoras Πυθαγόρας - meimei.org.uk/files/conference16/c3-pythagoras.pdfbrief biography -...

Pythagoras

Πυθαγόρας

Tom Button

tom.button@mei.org.uk

Brief biography - background

• Born c570 BC Samos

• Died c495 BC Metapontum

• Much of what we know is

based on 2 or 3 accounts

written 150-200 years after

he died

• Many things attributed to

Pythagoras may be have

been developed by other

Pythagoreans

Samos

Early life • Landowning Mother and merchant Father

• Travelled to Alexandria and Babylon

• Returned to Samos but left for Croton about age 40

The Pythagoreans

• Philosophers:

“lovers of wisdom”

• Secretive

• Significant role in

Croton: governing

and education

• High status for

women

• Strict rules on diet

Tuning stringed instruments

Lyre

Monochord

Why do some notes sound good

together?

• 2:2 ratio

• 4:2 ratio

• 3:2 ratio

• No simple ratio

Ratios of lengths of strings

650mm

325mm

433mm

Octave:

2×freq.

Perfect 5th

1½×freq.

Using Maths to make a scale

262 393 524

A new note in the scale

262 393 524

295

393×1.5 = 589.5

589.5÷2 = 294.75

Repeating the process

262 332 393 524

295 443

498

• Multiply the current

note by 1.5

• If the note is outside

the octave (262-524)

divide by 2

Filling in the gap

262 332 393 524

295 443

498 ?

• 393 Hz is the 5th for 262 Hz

• 262 Hz is the 5th for ___ Hz

The Major Scale in C

262 332 393 524

295 443

349 498

C 262 Do

D 295 Re

E 332 Mi

F 349 Fa

G 393 So

A 443 La

B 498 Te

C’ 524 Do

Continuing a

Pythagorean

Tuning

262

1 393

2 295

3 442

4 332

5 497

6 373

7 280

8 420

9 315

10 472

11 354

12 266

Equal

temperament

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 880

×1.059

×1.059

×1.059

×1.059

×1.059

×1.059

×1.059

×1.059

×1.059

×1.059

×1.059

×1.059

12

12

2

2

1.059...

r

r

All is number

Pythagoras’ Theorem: A brief history

• C1900BC – Babylon – Triples

• C1400BC – Egypt – Use of 3-4-5 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)

Proofs of Pythgoras’ Theorem

How many proofs do you know?

a² + b² = c²

Similar triangles proof

You can use any similar shapes

Similar triangles proof

Dissection proof

www.geogebra.org/m/d3cXJkds

Did Pythagoras prove it?

• Thales c. 624 – 546 BC is

the first evidence of

deductive reasoning

• Pythagoras c. 570 – 495 BC

• Plato c. 427 – 347 BC

references the theorem of

Pythagoras

• Euclid c. 350 – 250 BC

Hippasus

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

2 1 3 4 5

4 1 15 8 17

6 1 35 12 37

3 2 5 12 13

5 2 21 20 29

4 3 7 24 25

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. Kahn

• MacTutor History of Mathematics – Pythagoras

Biography www-history.mcs.st-

and.ac.uk/Biographies/Pythagoras.html

• Cut the Knot – Proofs of the Pythagorean Theorem

www.cut-the-knot.org/pythagoras/

• In Our Time (Radio 4) – Pythagoras www.bbc.co.uk/programmes/b00p693b

About MEI

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and work with industry to enhance mathematical

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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 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 262-524Hz:

C D E F G A B C’

262 295 332 349 393 442 497 524

This is known as a Pythagorean tuning. 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:

12 2 1.05946309...

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’

262 277.6 524

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?

Pythagorean triples problems

Find the radius of the largest circle that can be inscribed in a 3-4-5 triangle.

Investigate this for other Pythagorean triples.

The circle x2 + y2 = 52 has 12 points with integer

co-ordinates, as does the circle x2 + y2 = 132.

To find a circle with more than 12 points with

integer co-ordinates multiply 5 and 13 to

obtain x2 + y2 = 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?

Is the largest number in a primitive

Pythagorean triple always odd?

1 1 8

3 5 15 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)?

MEI Further Pure with Technology June 2014

3 This question concerns Pythagorean triples: positive integers a, b and c such that 2 2 2a 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 100a and 1 100b .

For the case n = 3, find all the triples with 1 200a and 1 200b .

[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 100a and 1 100b .

[4]