From 0 to Infinity in 26 Centuries

by Chris Waring

  a2 - b2 = (a + b)(a - b)

  Not So Perfect

  A perfect number is a number whose factors (not including the number itself) add up to make the number.

  For example, 6 is a perfect number (the first one, in fact) because the factors of 6 are 1, 2 and 3 and 1 + 2 + 3 = 6

  Perfect numbers are rare – the next one is 28, followed by 496 and then 8128. The fifth perfect number is 33,550,336.

  Numbers whose factors add up to less than the number are called deficient numbers. For example, 8’s factors are 1, 2 and 4, which have a sum of 7, so 8 is deficient.

  Numbers whose factors add up to more than the number are called abundant numbers, e.g. the sum of 12’s factors are: 1 + 2 + 3 + 4 + 6 = 16.

  In the case of some abundant numbers, no combination of its factors will make up the number. These numbers are called weird numbers. For example, 24 is not a weird number because we can add its factors together (2, 4, 6 and 12) to make 24. The first weird number is 70 – we cannot add any combination of its factors 1, 2, 5, 7, 10, 14 and 35 to get a total of 70.

  For example, if a = 8 and b = 5:

  82 - 52 = (8 + 5)(8 - 5)

  64 - 25 = 13 × 3

  39 = 39

  Fermat needed to test odd numbers (because 2 is the only even prime) to see if they were prime. He made the number he was testing, n, equal to the difference of two squares:

  n = a2 - b2

  which means that:

  n = (a + b)(a - b)

  This shows that n is two numbers multiplied together, in which case n cannot be a prime number unless (a + b) = n and (a - b) = 1.

  Fermat took the first statement and rearranged it:

  a2 - n = b2

  This meant that he could pick a starting value for a, square it and subtract n and see if he was left with a perfect square which is easily identifiable. If b2 was not a perfect square he would increase his starting value by one and try again until he either found numbers for a and b that worked or got to a point where a × b was larger than n.

  BLAISE PASCAL (1623–62)

  Educated by his father, Blaise Pascal was a French prodigy who worked in the fields of mathematics, physics and religion. His precocious talent saw his first mathematical paper published at the tender age of sixteen.

  Speeding things up

  Pascal’s father was a tax collector during a time of war in Europe, which made his job a somewhat onerous endeavour. Pascal sought to help his father by developing the first mechanical calculator – a machine known as a ‘Pascaline’, designed to add and subtract numbers. After he’d created a number of prototypes, Pascal’s finished product comprised a box with a series of numbered dials on its front and with a digit displayed above each dial. Numbers to be added were ‘dialled’ into the machine and the result would be displayed.

  Unfortunately, the Pascaline was very expensive to make and was therefore seen as more of a deluxe executive toy than a useful mathematical device. But Pascal’s contribution to mathematics should not be underestimated – he paved the way for Leibniz and others to develop more effective mechanical calculators and, eventually, modern computing.

  In all probability

  Pascal was also interested in games of chance and gambling. His work with his acquaintance Pierre de Fermat (see here) led to the field of mathematics we now call probability. In probability, we talk about an event (e.g. rolling a die) having a certain number of outcomes (rolling a 1, 2, 3, 4, 5 or 6 has six outcomes). Each outcome has a probability – for our die, each outcome is equally likely – which is expressed as a fraction (1/6), and the sum of the probabilities of all the events must add up to one. Probability is part of the branch of mathematics called statistics, which has a wide variety of applications in science and economics.

  Under pressure

  As a scientist, Pascal was fascinated with the notion of a vacuum. At the time, many scientists conformed to the view expounded by Aristotle: vacuums cannot exist; you cannot have emptiness because ‘nature abhors a vacuum’. However, Pascal noticed that if you place a glass beaker upside down in liquid (he used mercury) and then pull it out, there is a small gap at the top of the up-ended beaker that somehow holds up the column of liquid below it. He reasoned that this could only be a vacuum and that it must provide some sort of suction force to hold up the liquid.

  Pascal went on to conduct more experiments on pressure within liquids and as a result the unit of pressure is called the Pascal (Pa) in his honour.

  Absolute Proof

  In 1654 Pascal had a profound religious experience and it changed the course of his life. He subsequently devoted himself to an ascetic existence and focused on writing theological commentaries. He used his knowledge of probability to expound a reason for assuming God exists, now known as Pascal’s wager:

  You cannot tell whether God does or does not exist through logic.

  If you believe that he exists and he does not, you lose nothing.

  If you believe that he does not exist and he does, you lose an eternal afterlife.

  Therefore, there is nothing to lose and possibly infinite reward to be had from believing in God and nothing to gain from not believing in him.

  So, on balance, you may as well believe God exists.

  ISAAC NEWTON (1642–1727)

  One of the greatest scientists of his era, Isaac Newton hailed from Lincolnshire and wrote one of the most important books ever to be written: The Mathematical Principles of Natural Philosophy, often known by a shortened version of its Latin name Principia Mathematica. In the book Newton essentially rewrites the laws of physics that govern the way objects move and react to forces exerted on them. With his laws, Newton was able to explain the motion of the planets and prove conclusively that the sun sits at the centre of the solar system.

  Scientifically, Newton’s insight was to recognize gravity as a force that is caused by the earth’s mass, and his ability to understand intuitively how objects would behave when the earth’s gravity was not present. Newton’s critics saw gravity, which acts invisibly and at a distance, as some kind of demonic force and that Newton, an alchemist, was obviously in league with such forces. However, Newton’s Law of Universal Gravitation and his Equations of Motion were perfectly sufficient to allow us to send men to the moon three hundred years later.

  Change afoot

  Mathematically, Newton’s greatest achievement is calculus, which was also developed independently by Gottfried Leibniz at approximately the same time. Calculus is a tool used today in a range of different disciplines to describe and predict change. Building on the work of Descartes (see here), calculus can be split into two main branches.

  1. Differentiation involves finding the gradient of the line of an equation. Straight lines have a constant gradient that can be easily measured on a graph.

  We can see that for every square the line moves to the right (the positive direction) the line goes up two squares. The gradient of the line, therefore, is 2.

  However, not all equations give straight lines. Any equation with x2, x3 or higher powers produces a curved line:

  The gradient of this line constantly changes. However, the gradient of the tangent – the line that meets the curve at a point – is the same as the gradient of the line at that point:

  Differentiation lets us find a formula for the gradient of the line at any point so we no longer need to draw the tangent, which eliminates an area prone to error.

  If we have the formula for the gradient at any point, we can find the places where the gradient is zero. These are called the turning points of the equation and finding them can be very helpful. Many problems in mathematics, banking and business involve finding the maximum or minimum values of an equation – differentiation lets us find these points. Scientists have also found that many phenomena are governed by differential equations. For example, Newton’s Second Law, force = mass x acceleration, is derived by differentiating momentum.

  2. The other branch of calculus is integration, whi
ch is concerned with finding the area between a curve and the x axis:

  Again, drawing the graph might enable us to estimate the area under the curve, and there are various numerical methods that allow us to calculate an approximation of the area. One method would be to divide the area into thin rectangular strips and add each area together:

  As you increase the number of rectangles you get closer to the actual value. Newton and Leibniz took this one stage further. They imagined the rectangles became infinitely thin, in which case you get the true value of the area.

  You can use this method if you want to calculate the precise area of the shape under a curve. As with differentiation, there are many scientific laws that rely on integration.

  Back to the Beginning

  It turns out that integrating and differentiation are the inverse of each other, which means that integrating an equation and then differentiating it again takes you back to the original equation. This is known as the Fundamental Theorem of Calculus. Mathematicians use it to help them perform calculus on more difficult equations.

  GOTTFRIED LEIBNIZ (1646–1716)

  A mathematician and philosopher from Saxony, in the present-day state of Germany, Gottfried Leibniz was the son of a philosophy professor who died when Leibniz was just six years old. Leibniz inherited his father’s extensive library, through which he gained much knowledge, after first having taught himself Latin so he could read the books. Leibniz began his working life as a lawyer and diplomat, and, while on secondment in Paris, he met a Dutch astronomer called Christiaan Huygens, who assisted him in his learning of science and mathematics.

  Things turn ugly

  Leibniz is important for several reasons, although, perhaps unfortunately, he is mainly remembered for the bitter dispute he had with Isaac Newton over the invention of calculus. Newton was based in Cambridge and Leibniz in Paris, where both men devised the concept of calculus independently of each other. Newton began work on calculus as early as 1664 but he failed to publish his findings. That responsibility was left to Leibniz, who published his first paper on the subject in 1684.

  The argument centred on whether or not Leibniz had been exposed to Newton’s prior work. No evidence exists to prove whether or not Leibniz did have access to Newton’s work, and there is no reason to assume that Leibniz could not have come up with calculus independently. However, he died with the matter still unsettled.

  Leibniz and Newton developed different notations for calculus and both are used in different areas of mathematics. Leibniz’s is perhaps more commonly used.

  To evaluate the area shown on the graph you would use Leibniz’s notation to write:

  Which is shorthand for: ‘integrate x2 between x=1 and x=2 with respect to x’.

  If you wanted to find the gradient of the line at a point you would need to use differentiation, for which the notation is:

  A new dawn

  Leibniz was also instrumental in a new field of mathematics that was emerging at the time: computing. In our Hindu-Arabic numeral system, each column in a number is 10 times larger than the one on its right – a decimal system. Leibniz was interested in a way of writing numbers in which each column is twice the value of the column on its right – a binary system. The binary system required only the digits 0 and 1, and the columns have values of 1, 2, 4, 8, 16, etc., doubling each time. So the decimal number 13 would be written as 1101:

  Column: 8 4 2 1

  Digit: 1 1 0 1 because 8 + 4 + 1 = 13

  This system seems quite peculiar but it has the advantage of using only two digits, which makes calculations easier. The binary system would later become very important for electronics and computers.

  Number Punching

  Leibniz also pioneered the ‘Stepped Reckoner’, one of the first mechanical calculators that could perform multiplication and division. It was a very intricate machine and the complex system of gears could be unreliable, but as manufacturing technology improved over time the Stepped Reckoner went from strength to strength, and Leibniz’s ideas were used for hundreds of years, well into the twentieth century.

  JOHANN BERNOULLI (1667–1748) AND JACOB BERNOULLI (1654–1705)

  The Bernoulli brothers were Swiss mathematicians. Although they both pursued alternative professions – Johann was trained in medicine and Jacob as a minister – both siblings loved mathematics, especially the calculus of Leibniz. The Bernoulli brothers were able to push on with the fledgling field of calculus and became proficient in its use, turning it from an intellectual and political curiosity into a useful mathematical tool. They were also fiercely competitive with each other, which spurred on their discoveries even more.

  Arch rivalry

  One example of the Bernoulli brothers’ rivalry stemmed from the problem of the catenary curve: the shape produced by a rope or chain when it hangs from both ends. A mathematical equation of this shape had eluded mathematicians up to this point. Jacob proposed the problem in 1691; Johann, with some assistance from Leibniz and the Dutch mathematician Christiaan Huygens, then went on to solve it. Catenary curves have important applications in bridge building and in architecture because arches that follow an upside-down catenary curve are the strongest.

  In everyone’s interest

  Jacob also discovered something interesting when he looked at a problem involving compound interest. Jacob noticed that if you had £100 in a bank account that paid 10% interest per annum, the way the interest is paid throughout the year affects the total money you will have at the end of the year. Compound interest payments add to the principal sum of money in a bank account, which increases the interest you earn year after year:

  Interest

  Interest Calculation

  Total

  10% paid at end of year

  100 × 1.1

  £110

  5% every six months

  (100 × 1.05) × 1.05

  £110.25

  2.5% every 3 months

  ((((100 × 1.025) × 1.025) × 1.025) ×1.205)

  £110.38

  Daily interest

  £110.51

  Admittedly, the changes are not making a vast difference to your balance, a fact that most banks rely on. Of greater significance was Jacob’s investigation into instances when interest is paid continuously, in tiny amounts, over the entire year. He discovered that if your interest rate is x (e.g. x=0.045 for a rate of 4.5%) at the end of a year you would have 2.718281x times what you started with. I have rounded the 2.718281 – it is in fact an irrational number, like π, that goes on for ever without repeating.

  Napier made reference to this number in his work with logarithms, and it also plays a very important role in calculus, as exemplified by our next mathematician.

  LEONHARD EULER (1707–83)

  Euler (rhymes with ‘boiler’ rather than ‘ruler’) was a Swiss mathematician who had originally intended on becoming a priest. However, while at university he met Johann Bernoulli, who recognized Euler’s extraordinary mathematical talent and managed to persuade Euler senior to allow his son to transfer to studying maths.

  An increase in power

  Euler’s contributions to mathematics and science were far-reaching. The number 2.718281..., discovered by Jacob Bernoulli in relation to his compound interest problem, also turned up in Euler’s work on calculus.

  When you integrate to find the area under a graph you need to increase the power of x by one. For example, if your graph is y=x2, the integral is 1/3 x3 – the power of x has gone up by one. If you’re faced with something slightly more tricky, let’s say y = 1/x4, there is a handy rule of powers that can help you:

  1/xn = x-n

  So y=1/x4 becomes y=x-4, and when you increase the power by one your answer will be something to do with x-3, which is 1/x3.

  But what happens if your graph is y = 1/x?

  This is the same as x-1, but if you increase the power by one you get x0. Anything to the power of zero is 1, implying that, no matter which section of the graph you look at,
the area will be the same. This doesn’t make sense!

  Well, it turns out, through a complicated system of algebra developed by Euler, that the area is equal to the natural logarithm of x. A natural logarithm is similar to a normal logarithm, but its base is the number 2.7818281... There is a sizeable family of equations that can only be integrated or differentiated using natural logs, and the 2.7818281 was known as ‘e’ for Euler’s number.

  Returning to the area under the graph, if you wanted to know the area between x=1 and x=4, you would need to work out:

  area = loge 4 - loge 1

  As ‘loge’ turns up so often in calculus, it is denoted by ln and you will find this button on all good scientific calculators.

  area = ln 4 - ln 1 = 1.386 (to 3 decimal places)

  A bridge too far

  Euler’s work on ‘The Seven Bridges of Königsberg’ contributed to methods of simplifying maps. Königsberg was the old Prussian name for the city of Kaliningrad in that strange bit of Russia that sits between Poland and Lithuania. The city is centred on an island, which straddles a river. Seven bridges connect the two sides of the island at various locations: