By Julia Collins and Pamela Docherty

Who are we?

Who are we?

Email and website
 To get in touch, email!
Julia.Collins@ed.ac.uk
P.Docherty@ed.ac.uk
 For information, look on the website www.maths.ed.ac.uk/~jcollins/OpenStudies/

Who are you?
 Get into pairs and chat for a few minutes:

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?

Inventing numbers

Plan for this class
 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...

Where do numbers come from?
 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?
 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
 The Sumerians had clay pieces of different shapes to represent different goods.

Envelopes and tokens
 They placed the tokens in an envelope when they traded.

Number symbols
 They then had to indicate on the outside what was in the envelope.

Number symbols
 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?
 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
 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?
 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
 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
 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
 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
 We still think of zero differently from the other numbers.

Where is zero on a computer keyboard?

Where is zero in our calendar system?
 To the Greeks it was a paradox.

How can nothing be something?

Calculating with zero
 The Indian mathematician Brahmagupta wrote down the first rules for using zero as a number, in 628AD.

0 + x = x

0 + 0 = 0

0 · x = 0

0/x = 0
 x/0 = ?

0/0 = ?

Negative numbers
 The early Chinese and Indians (200BC – 400AD) used negative numbers in problems of debt.
 Brahmagupta wrote down rules for using negative numbers:
 x+(-x) = 0

Positive times negative is negative

Negative times negative is positive

The integers
 The number line with positive and negative whole numbers and zero is called the integers.
 The positive numbers are called the natural
numbers.

Negative numbers
 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”

Fractions




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.
8
9
10
12
15
4
5
2
3
6
30
20
15
12
10
7,30
6,40
6
5
4

More fractions
 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
 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…
 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



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
 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 x 2 = 2.
Motto: a number is what it does .

How do you solve x 2 = -1?
 There is no number on the number line that solves this equation.
 Descartes called such numbers imaginary.

Solving the cubic
 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


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
 Having complex numbers is like introducing another dimension to the number line.
Multiplying by i is equivalent to rotating by
90 degrees.
 Complex numbers unite arithmetic and geometry.

Applications
 Modern life wouldn’t be possible without complex numbers!

Computer graphics

Electricity and circuits

Magnetism

Signal processing

Quantum mechanics

Fluid dynamics

Resonance

Any more weird numbers to come?
 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 x 2 = -1, so surely we need another new number to solve x 4 = -1?
 Thankfully not!

The fundamental theorem of algebra



In 1608 it was conjectured that every polynomial equation of degree n has n solutions.
For example, the solutions to x 4 = 1 are 1, -1, i and –i.
The solutions to x 4 = -1 are equally pretty!

What did we learn?
 That in mathematics, your imagination is the limit!
 Whenever you see something that looks impossible, ask yourself
...but what if I could do that?
 That in maths you shouldn’t worry about what objects are, but only what they do.