Mathematical Ideas that
Shaped the World
Calculus
Plan for this class
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What were Zeno’s paradoxes?
What happens when we try to add up
infinitely many things?
Does 0.999… = 1?
What is calculus?
How has it been important?
Who invented it: Newton or Leibniz?
Zeno of Elea
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Lived 490 – 430BC
Contemporary of Socrates
Written about in a dialogue
of Plato called Parmenides
Believed in one indivisible
entity and in unchanging
reality
His paradoxes are proofs
by reductio ad absurdum
against the contrary
opinions of divisibility and
change
Achilles and the tortoise
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Achilles runs at 10m/s, the
tortoise only goes at 1m/s.
The race is 1km long, and
the tortoise gets a 100m
head start.
Hypothesis: Achilles can
never catch the tortoise.
Proof: When Achilles gets
to where the tortoise was,
the tortoise has always
moved forward.
Motion is impossible (1)
Hypothesis:
Motion can never begin to happen.
Motion is impossible (1)
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Proof: Suppose you have to run for the bus.
Before you can reach the bus stop, you must
first get halfway there.
But before you can get halfway there, you
must first get a quarter of the way there.
But before you can get a quarter of the way
there, you must first get 1/8 of the way
there…
Motion is impossible (2)
Hypothesis:
Everything is at rest.
Motion is impossible (2)
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Proof: time is composed of a series of
moments of ‘now’.
Consider an arrow flying. At each moment of
time, it is not moving. It can travel no
distance in that moment.
But if it doesn’t travel in any
one moment, then when
does it move?
Argument against divisibility
Hypothesis:
If an object is divisible, it
cannot actually exist.
Argument against divisibility
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Proof: Suppose a body is completely
divisible. Divide it in half. Divide those halves
into half, and repeat.
Eventually you will have divided your body
into parts which have no size.
But a sum of parts of zero size is surely a
body which is also of zero size!
A modern paradox
Do you think that
0.9999…. = 1?
The evidence
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Arguments for
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We believe that
⅓ = 0.3333…
so 3 times this number
should equal 1.
What is 1-0.9999…?
If x=0.9999…. then
10x = 9.9999…
and so
10x – x = 9x = 9,
so x = 1.
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Arguments against
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It’s a number that’s
always less than 1. It
approaches 1 but never
reaches it.
There shouldn’t be two
ways to write down the
same number.
How to decide?
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To answer the question, we need to really
define what 0.99999… is.
We will do this using the theory of limits.
The definition was first made
by the French mathematician
Cauchy (1789 – 1857).
Born in Paris in the year of
the French revolution!
The theory of infinite sums
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Given a sequence of numbers x1, x2, x3, …, the
limit of this sequence is L if:
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For any distance  the terms of the sequence
eventually become within a distance  away from
L.
In other words, the sequence eventually gets
as close to the limit as you want it to get.
Examples
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The sequence 1, ½, ¼, ⅛, etc has limit 0.
The sequence 0.9, 0.99, 0.999, etc has limit 1.
When we write 0.9999…, what we mean is
“the limit as the number of 9’s goes to
infinity”.
So 0.9999… = 1.
Achilles & the tortoise revisited
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For Achilles to catch the tortoise, he must
travel an infinite number of distances:
100m + 10m + 1m + 0.1m + 0.01m + …
Could this infinite sum actually have a finite
value, just like the infinite sequences had a
limit?
Infinite sums
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Conclusions
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You can always run to the bus stop!
Achilles can always catch the tortoise!
Λοσερ!*
*Loser!
Question
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What condition do you need to put on a
sequence in order for the infinite sum to have
a finite value?
Answer: the elements in the sum need to be
getting closer and closer to zero.
Is the converse true?
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Creating the maximum overhang
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Stacking bricks according to the harmonic
series gives you an overhang however big
you want!
Prime reciprocals?
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Do you think this series is finite or infinite?
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It is infinite! This was proved by Euler in 1737.
Surprising equations
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Sometimes infinite sums can be very
beautiful.
For example, can you guess what these sums
are?
Totally weird
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Infinite sums with negative numbers are very
counterintuitive.
The same infinite sum may sometimes have a
finite value, and sometimes be infinite
depending on how you add it up.
Example:
Homework problem
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Suppose there is a rubber band 1m long.
A worm is on the band crawling at 1cm per
minute.
After each minute, the band is stretched by
1m.
Can the worm ever reach the end of the
band?
Calculus
…or how to add up infinitely
many tiny things and how
to divide by infinitely small
things
What is calculus?
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Differentiation – calculating how much a
quantity changes in response to changes in
another quantity.
E.g. the speed of a cannonball, or the profits
of an airline in relation to the temperature.
What is calculus?
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Integration – finding areas and volumes of
curved shapes.
E.g. a deep-sea diver needs to know the
amount of air in their oxygen tank and the
force of the water on their head.
Archimedes (287- 212 BC)
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Once upon a time, a man
called Archimedes wanted to
find the area of a circle.
How do you work out the
exact area of a curved shape?
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Area of a circle
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Idea: cut up circle into wedges:
As the number of wedges goes to infinity, the
approximation gets more accurate.
Finding π
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Archimedes used the method of exhaustion
to find a good approximation for π.
He found the areas of polygons in and
around the circle.
His best estimate used a 96-sided polygon!
Modern integration
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Our current method of integration was
developed by Newton & Leibniz, and made
rigorous by Riemann.
Idea: split the area under a graph into tiny
rectangles.
How to integrate
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We want the area under a curve f from x=a to x=b.
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If Δx is the width of
each rectangle, the area
is the sum of
f(xn) Δx
for each xn between a
and b.
How to integrate
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As Δx gets smaller, the approximation gets
better.
When Δx is infinitesimally small, the
calculation of the area is exact!
Leibniz used the notation
∫ f(x) dx
for this exact value, where ∫ means ‘sum’.
Speed at an instant
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Suppose a car is accelerating from 30mph to
50mph.
At some point it hits the speed of 40mph, but
when?
Speed = (distance travelled)/(time passed)
How is it possible to define the speed at a
single point of time?
(We’re back to Zeno’s paradox!)
Approximations
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Idea: find distance/time for smaller and smaller
time intervals.
Here f shows how the
distance is changing
with the time, x.
The time difference is
h while the distance
travelled is
f(x+h) – f(x).
The derivative
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Our approximation of the speed is
As h gets smaller, the approximation of the
speed gets better.
When h is infinitesimally small, the
calculation of the speed is exact.
Leibniz used the notation dy/dx for the exact
speed.
Example
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f(x) = x2
f(x+h) – f(x) = (x + h)2 – x2
= (x2 + 2xh + h2) - x2
= 2xh + h2
Speed = (2xh + h2)/h
= 2x + h
When h is infinitesimally small, this is just 2x.
Philosophical problem
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We treat h as a normal quantity in the
formula.
In particular, we are allowed to divide by it.
Then at the end we decide that it is zero!
In 1734 the philosopher Bishop Berkeley
wrote “a discourse addressed to the infidel
mathematician” attacking the calculus.
He objected to the use of ghosts of
departed quantities.
Resolving the paradox
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The resolution of this is that we don’t actually
set h equal to zero, but take the limit.
h approaches zero but never reaches it.
It was not until the work of Cauchy and
others in the following century that the
foundations of calculus were made rigorous.
Newton and Leibniz
Newton: 1643 – 1727
Leibniz: 1646 - 1716
Biscuits!
Fundamental theorem of calculus
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Newton and Leibniz are credited with
inventing calculus, but the main ideas were
developed long before them.
Their main contribution was to discover the
fundamental theorem of calculus:
Differentiation and integration are the
opposites of each other.
The great dispute
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Newton and Leibniz also did a lot to unify
and make precise the work of others.
Unfortunately their work was clouded in a
bitter dispute over who wrote down the ideas
first and whether there had
been plagiarism.
Let’s take a look at how
things happened…
Calculus timeline
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1669 Newton writes first manuscript that
mentions his ‘theory of fluxions’.
Disseminated among many in England and
Europe through John Collins.
1672 Newton writes a treatise on fluxions,
but it remains unpublished until 1736.
1673 Leibniz visits London, reads some of
Newton’s work on optics but doesn’t mention
his work on fluxions.
Calculus timeline
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1673-1675 Leibniz develops methods and
notation of calculus using the theory of
infinitesimals. He is aware of Newton’s work
on infinite sums.
1676 Leibniz is in London again. Meets
Collins and definitely sees Newton’s first
manuscript on fluxions. Newton writes to
Leibniz telling him about fluxions – but in an
anagram!
Calculus timeline
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1677 Leibniz writes to Newton, clearly
explaining his methods and applications of
calculus. (Doesn’t mention having seen
Newton’s 1669 manuscript…)
1683 Collins dies.
1684 Leibniz publishes the first paper on
calculus. (Why the delay?)
1687 Newton publishes the Principia on
gravitation. Uses calculus but does not
explain the underlying principles.
Calculus timeline
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1693 & 1699 On two occasions Leibniz gives
the impression that he was the first to invent
the calculus. Newton takes no notice.
1704 Newton publishes his Optiks which
explains the method of fluxions. Leibniz
writes an anonymous review which implicitly
accuses Newton of plagiarism.
1711 Newton’s friend Keill counter-accuses
Leibniz of plagiarism.
Calculus timeline
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1712 A committee of the Royal Society is set
up to investigate the matter. The committee
is mostly formed of Newton’s friends.
1713 Report published (written by Newton!)
asserting that Newton was the first inventor
of the calculus. Does not accuse Leibniz of
plagiarism but implies that he is capable of it.
1716 Leibniz dies and the quarrel gradually
subsides.
The verdict
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Clearly Newton can claim to be the first
inventor of calculus, but was Leibniz an
independent inventor?
Let’s look at the evidence for and against…
The evidence for plagiarism
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For
Leibniz clearly had
access to Newton’s
work while developing
his own ideas.
Why did he not publish
until after Collins died?
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Against
Leibniz’s methods were
quite different to
Newton’s.
His notation was
completely different.
He had already done
work on infinitesimals
before coming to
London in 1673.
Consequences
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As a result of the quarrel, English
mathematicians became isolated from the
continent for a century.
Their refusal to use Leibniz’s notation and the
new methods of analysis held back British
mathematical advances for a long time.
What did we learn?
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That good mathematics can come out of
crazy Greek philosophy.
That infinity can be weird and beautiful.
That infinity can also be very useful if we deal
with it properly.
That choosing good symbols for your maths
is very important if you want people to use it
in the future.