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##### Mathematical Ideas that Shaped the World

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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.