Transcript Mathematical Ideas that Shaped the World

Mathematical Ideas that
Shaped the World
By Julia Collins and Pamela Docherty
Who are we?
Who are we?
Email and website
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To get in touch, email!
[email protected]
[email protected]
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For information, look on the website
www.maths.ed.ac.uk/~jcollins/OpenStudies/
Who are you?
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Get into pairs and chat for a few minutes:
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Introduce yourself
Why have you come to this course? What do you
hope to get out of it? Which class are you most
looking forward to?
What is your favourite number, and why?
Mathematical Ideas that
Shaped the World
Inventing numbers
Plan for this class
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How and why were numbers invented?
Which number system is the best one?
Why don’t computers count like we do?
What different sorts of numbers caused
problems for our ancestors?
Are imaginary numbers imaginary?
What does the number line really look like?
But first...
Some mathematical mind
reading!
Where do numbers come from?
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Early societies did not have many number
words.
There are still tribes which only have words for
one, two and many.
They count objects by pairing them with other
objects.
www.youtube.com/watch?v=asM39tfblMQ
How good is your number sense?
In the beginning there was…time?
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The earliest mathematical objects were trying
to measure the passage of time.
The
Ishango
bone,
estimated
to be
20,000
years old.
And then there was trade
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The Sumerians had clay pieces of different
shapes to represent different goods.
Envelopes and tokens
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They placed the tokens in an envelope when
they traded.
Number symbols
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They then had to indicate on the outside what
was in the envelope.
Number symbols
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Eventually they realised that with the number
symbols, the tokens were unnecessary.
In doing so, they made the important step of
thinking of numbers as abstract objects.
=
How to write numbers?
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When inventing a number system, how do you
decide which numbers to give symbols to?
The Babylonians had symbols for only 1 and 10.
Babylonian numbers
Place value
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The Babylonians worked in a place value system
with base 60.
Each column signified a power of 60.
E.g. 71 =
What about other cultures?
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The Greeks, Egyptians and Romans did not have
a place value system.
They counted in fives and tens, with symbols
for the powers of ten.
Place value vs additive systems
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The Mayans, Chinese and Indians also used a
place value system.
What do you think are the advantages of a
place value system?
What are the disadvantages?
What things do you need for a place value
system to be effective?
Binary
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The easiest place value system uses 2 as its
base.
The only numbers you need in this system are 0
and 1.
This makes it perfect for computers: they have
switches where 0 = off, 1 = on.
Zero
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Zero was invented as a symbol before it was
discovered as a number.
The Indian word sunya means empty, which
became the Arabic word zifr.
Zero
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We still think of zero differently from the other
numbers.
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Where is zero on a computer keyboard?
Where is zero in our calendar system?
To the Greeks it was a paradox.
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How can nothing be something?
Calculating with zero
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The Indian mathematician Brahmagupta wrote
down the first rules for using zero as a number,
in 628AD.
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0+x=x
0+0=0
0·x=0
0/x = 0
x/0 = ?
0/0 = ?
Negative numbers
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The early Chinese and Indians (200BC – 400AD)
used negative numbers in problems of debt.
Brahmagupta wrote down rules for using
negative numbers:
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x+(-x) = 0
Positive times negative is negative
Negative times negative is positive
The integers
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The number line with positive and negative
whole numbers and zero is called the integers.
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The positive numbers are called the natural
numbers.
Negative numbers
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Society didn’t fully accept negative numbers for
a long time.
An equation like 4x+20=0 was considered
“absurd” in ancient Greece.
Negative roots of equations were not
considered. The Indian mathematician Bhaskara
said people do not approve of negative roots“.
Negative numbers
In 1759 AD Francis Maseres, an English
mathematician, wrote that negative numbers
“darken the very whole doctrines of the equations
and make dark of the things which are in their
nature excessively obvious and simple”
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Fractions
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The Babylonians
produced tables of
reciprocals, i.e. 1/x for
values of x.
What is missing from this
table?
It was written that “7
does not divide”.
Infinite fractions did not
exist.
2
3
4
5
6
8
9
10
12
15
30
20
15
12
10
7,30
6,40
6
5
4
More fractions
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The Greeks thought of fractions as ratios of
lengths.
Without a place value system, there were no
infinities to worry about.
2=
7
2
5
The Pythagoreans
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The Pythagoreans preached that all numbers
could be expressed as ratios.
One day a man called Hippasus declared that
the square root of 2 was irrational, i.e. not a
fraction.
1
√2
1
A deadly secret…
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Hippasus, not long afterwards, drowned while
at sea. Some said Pythagoras sentenced him to
death; others that it was a punishment from the
gods.
Irrational fears
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Irrational numbers are not only problematic
conceptually, but practically.
The decimal expansion of an irrational number
never ever stops or repeats.
This means that irrational numbers can never
be written down exactly.
√2 =1.414213562373095048801688724209698078
56967187537694807317667973799....
Numbers as solutions of equations
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In the Middle Ages, Islamic
mathematicians developed
algebra (al-jabr = restoration):
methods to solve equations.
In this system, an irrational
number like √2 was simply
seen as the solution of x2 = 2.
Motto: a number is what it does.
How do you solve
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2
x
= -1?
There is no number on the number line that
solves this equation.
Descartes called such numbers imaginary.
Solving the cubic
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Surprisingly, imaginary numbers became
important in the race to solve the cubic, not in
the solution of quadratics.
Cardano had an equation for solving cubics. In
1572 a man called Bombelli realised that if he
treated √-1 as a normal (real) number then he
got sensible solutions.
Complex numbers
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The great mathematician
Leonhard Euler invented the
notation i for √-1.
He also proved the amazing
formula:
Numbers of the form a + ib
are called complex numbers.
Another dimension
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Having complex numbers is like introducing
another dimension to the number line.
Multiplying by i is
equivalent to rotating by
90 degrees.
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Complex numbers unite arithmetic and geometry.
Applications
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Modern life wouldn’t be possible without
complex numbers!
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Computer graphics
Electricity and circuits
Magnetism
Signal processing
Quantum mechanics
Fluid dynamics
Resonance
Any more weird numbers to come?
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How do we know that there aren’t more new
numbers to be found? No more equations to be
solved?
For example, we needed a new number to be
able to solve x2 = -1, so surely we need another
new number to solve x4 = -1?
Thankfully not!
The fundamental theorem of algebra
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In 1608 it was conjectured that every
polynomial equation of degree n has n
solutions.
For example, the solutions to x4 = 1 are 1, -1, i
and –i.
The solutions to x4 = -1 are equally pretty!
What did we learn?
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That in mathematics, your imagination is the
limit!
Whenever you see something that looks
impossible, ask yourself
...but what if I could do that?
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That in maths you shouldn’t worry about what
objects are, but only what they do.