Pythagoras
Πυθαγόρας
Tom Button
[email protected]
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
Monochord
Lyre
Why do some notes sound good
together?
•
•
•
•
2:2 ratio
4:2 ratio
3:2 ratio
No simple ratio
Ratios of lengths of strings
Octave:
2×freq.
325mm
433mm
650mm
Perfect 5th
1½×freq.
Using Maths to make a scale
262
393
524
A new note in the scale
295
262
393
524
393×1.5 = 589.5
589.5÷2 = 294.75
Repeating the process
295
262 332
443
393
524
498
• Multiply the current
note by 1.5
• If the note is outside
the octave (262-524)
divide by 2
Filling in the gap
295
262 332
443
393
524
498
?
• 393 Hz is the 5th for 262 Hz
• 262 Hz is the 5th for ___ Hz
The Major Scale in C
295
443
262 332 393
524
349
498
C
D
262 Do
295 Re
E
F
G
332 Mi
349 Fa
393 So
A
B
443
498
C’
524 Do
La
Te
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
r 2
12
r 2
12
 1.059...
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
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
m² − n² 2mn m² + n²
m
n
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.stand.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
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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
262 277.6
C’
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.
The circle x2 + y2 = 52 has 12 points with integer
co-ordinates, as does the circle x2 + y2 = 132.
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 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
and (8, 5, 17) is a Pythagorean
 
3 5 15
triple.
Add the reciprocals of any two consecutive odd
numbers. Will the resulting fraction,
x
, always
y
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 a 2  b2  c 2 .
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]