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Plan for this class • 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 • 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 • 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) • 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) • 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 • 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?
Arguments for 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. Arguments against 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. The evidence
How to decide? • 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 • Given a sequence of numbers x1, x2, x3, …, the limit of this sequence is L if: • 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 • 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 • 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?
Conclusions • You can always run to the bus stop! • Achilles can always catch the tortoise! Λοσερ!* *Loser!
Question • 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.
Creating the maximum overhang • Stacking bricks according to the harmonic series gives you an overhang however big you want!
Prime reciprocals? • Do you think this series is finite or infinite? • It is infinite! This was proved by Euler in 1737.
Surprising equations • Sometimes infinite sums can be very beautiful. • For example, can you guess what these sums are?
Totally weird • 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 • 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? • 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? • 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) • 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? ?
Area of a circle • Idea: cut up circle into wedges: • As the number of wedges goes to infinity, the approximation gets more accurate.
Finding π • 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 • 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 • We want the area under a curve f from x=a to x=b. • 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 • 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 • 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 • 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 • 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 • 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 • 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 • 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
Fundamental theorem of calculus • 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 • 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 • 1669 Newton writes first manuscript that mentions his ‘theory of fluxions’. Disseminated among many in England and Europe through John Collins. • 1672Newton 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 • 1673-1675Leibniz 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 • 1677Leibniz writes to Newton, clearly explaining his methods and applications of calculus. (Doesn’t mention having seen Newton’s 1669 manuscript…) • 1683 Collins dies. • 1684Leibniz 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 • 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 • 1712 A committee of the Royal Society is set up to investigate the matter. The committee is mostly formed of Newton’s friends. • 1713Report 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.