28/07/2011
Greeks & Rational Numbers
?
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Maths in Context
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Hippasus asked Pythagoras about the length of this diagonal.
Ben Sparks – Freelance Mathematics Speaker
ben@bensparks.co.uk
www.bensparks.co.uk
Rational?
If we assume √2 is rational then it can
be written?as a fraction:
1
a
(where a and b have no common factors)
b
2
a
2= 2 1
b
2
2
(Is this enough?)
So a² is an even number 2b = a
2=
so a is an even number
a = 2c
and a 2 = 4c 2
so
2b 2 = 4c 2
b 2 = 2c 2 So b² is an even number
so b is an even number
so we can substitute 4c²
But if a and b are even they BOTH have a common factor of 2
for the a² in this equation
So we have a contradiction
√2 ≠
a
b
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28/07/2011
1 unit
Most irrational?
•Irrationality
•Manipulating Surds
•Ratio
•Perimeter
•Pythagoras
•Proof
•Congruence
•Kinaesthetic stuff
•Display work
•…
Irrationals can be approximated by rational
fractions… but some better than others.
 What has this got to do with flowers…?

√2 (1.414…) units
Phyllotaxis!
 (Geogebra demo)
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Roots of –ve numbers
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There are many other types of equation (in this
case ‘polynomials’)
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Cubics:
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Quartics:
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Quintics:
ax 3  bx 2  cx  d  0
ax 4  bx 3  cx 2  dx  e  0
Roots of –ve numbers
Gerolamo Cardano (1501-1576)
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Niccolò Fontana Tartaglia (1500-1557)
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And so on…
Solved the ‘depressed’ cubic even earlier than all this
Cardano saw del Ferro’s notebook sometime before1545, prompting him to
publish all his work (including Tartaglia’s ideas)
Ludovico Ferrari (1522-1565)
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Had won ‘cubic competitions’ in Bologna as early as 1535
Told Cardano how to solve the ‘depressed’
cubic in 1539 (having sworn him to secrecy)
BUT…
Scipione del Ferro (1465-1526)

ax 5  bx 4  cx 3  dx 2  ex  f  0
Published Ars Magna in 1545
Detailed how to solve general cubics and quartics, crediting various other people
BUT…


Student of Cardano
Solved the Quartic, but relied on Tartaglia’s method
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28/07/2011
Roots of –ve numbers
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Tartaglia was (understandably?) upset
Decade long exchange of insults
Eventually challenged by Ferrari (Cardano’s student) to a
final ‘cubic showdown’ in 1548
Ferrari outshone him in the first day of competition
Tartaglia fled overnight, leaving victory, fame and fortune to
Ferrari.
Tartaglia was discredited, unemployable and died a poor man
Ferrari became famous, retired early…
…and was then poisoned by his sister…
Roots of –ve numbers
If ax 3  bx 2  cx  d  0
Roots of –ve numbers
Cardano’s first actual mention of complex numbers was in a
quadratic though:
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“Find two numbers whose sum is equal to 10 and whose product is
equal to 40”
The answer is 5 + √(−15) and 5 − √(−15)
Cardano called this “sophistic”
But wrote “nevertheless we will operate”
He then says this answer is “as subtle as it is useless”.
Roots of –ve numbers

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This nasty looking formula was the final result
of lots of work from lots of people, and it
worked.
Sometimes, however a problem occurred.
With for example a cubic like this
x3  7 x  6  0

(and this is only really a part of it…)
The formula needed to calculate the roots of
this equation – which are very nice and easy:
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28/07/2011
Roots of –ve numbers
Roots of –ve numbers

They knew the roots were -3,1 and 2, but the
formula produced a result involving √(-3).

This was ‘obviously’ nonsense, but Cardano
was reluctant to accept his formula didn’t work.
Eventually in his calculations the negative
square roots cancelled out, and he showed his
formula produced the correct results.

-3
1
2
Roots of –ve numbers

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This was the first time mathematicians began
to suspect that these negative square roots
might need to be looked at.
What if, despite not existing on the Real
number line, these were numbers of some sort.
Roots of –ve numbers

So if it is a number what is it like?
let i   1

(Imaginary bit)
i 2  1
(square both sides)
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i 
??i
Since i³=i² x i
i 4  1??
Since i4=i³ x i
i5  i
And off we go again…
This “i” has some funny properties… but it does
follow the normal laws of algebra.
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