From 0 to Infinity in 26 Centuries

by Chris Waring

  A popular Sunday afternoon pastime for the residents of Königsberg was to attempt to walk over all seven bridges and return to their starting place without having to use the same bridge twice. Not one person ever managed it, but Euler was the first to tackle the problem mathematically. He redrew the map as a network:

  Euler counted the number of routes into and out of each node or intersection on his network. He then reasoned that you would have to walk into and out of each node as you made your way around the map, so you needed each node to have an even number of routes. All the nodes in Königsberg have an odd number, so it is impossible to complete the challenge. Networks where all the nodes have even numbers are known as eulerian. A semi-eulerian network is one that possesses two nodes with an odd number; if you start at one odd node and finish at the other you can complete the network without repeating yourself, as with the following famous example:

  In My Imagination

  Euler was also interested in what are known as complex numbers. These are numbers that are made up of two parts, one real (i.e. any number between plus and minus infinity) and one imaginary.

  Diophantus had the first inklings of imaginary numbers (see here), but it wasn’t until the sixteenth century and the arrival of two Italian mathematicians, Niccolò Fontana Tartaglia and Gerolamo Cardano, that the study of imaginary numbers really took off. Tartaglia and Cardano discovered that some equations only generate an answer if you are prepared to allow negative numbers to have square roots – which can’t happen with real numbers, because a negative multiplied by a negative gives a positive.

  Descartes coined the term ‘imaginary’ – even though such numbers don’t exist, you can permit them to exist in your imagination in order to find answers to previously unsolvable equations.

  The letter i is used to denote the square root of minus 1: √-1

  This allows you to reference the square root of any negative number:

  √ -49 = √(49 × -1) = √49 × √-1 = 7i

  Despite these numbers being imaginary, they have many practical applications, especially in electronics and electrical engineering.

  This way of simplifying and thinking about maps and routes had implications for cartography. It’s also an important part of decision mathematics, the branch of mathematics that many businesses rely on for planning delivery routes and other logistical operations.

  The many faces of geometry

  Euler worked in three-dimensional geometry too. He discovered there is a relationship between the number of corners (or vertices), edges and faces of a polyhedron (a three-dimensional shape with flat faces, like a cube or a pyramid):

  vertices – edges + faces = 2

  Mathematical Perfection

  Euler devised an equation now known as Euler’s identity (an identity is an equation that is always true no matter what value you use for the unknown), which is said to be the most beautiful and elegant mathematical equation. It relates to complex numbers, but unfortunately its meaning is beyond the scope of this book. Here is the equation in all its glory:

  eiπ + 1 = 0

  Its subjective mathematical beauty arises from the fact that it uses five of the most important numbers in mathematics: e, i, π, 1 and 0.

  A cube has 8 vertices, 12 edges and 6 faces: 8 – 12 + 6 = 2

  A tetrahedron (triangular-based pyramid) has 4 vertices, 6 edges and 4 faces: 4 – 6 + 4 = 2

  A dodecahedron (12-sided polyhedron) has 20 vertices, 30 edges and 12 faces: 20 – 30 + 12 = 2

  A truncated icosahedron (the combination of hexagons and pentagons used to make a football) has 60 vertices, 90 edges and 32 faces: 60 – 90 + 32 = 2

  Euler also teamed up with Daniel Bernoulli (1700–82, son of Johann, nephew of Jacob) to work in applied mathematics. They considered the forces acting on beams in buildings and how the forces would make the beams bend – a very useful tool in engineering applications.

  True or False?

  Mathematician Christian Goldbach (1690–1764), in a letter to Euler about the nature of prime numbers, wrote what has become known as Goldbach’s conjecture:

  ‘Every even whole number greater than 2 can be written as the sum of two prime numbers.’

  For example, 10 can be made up from 5 + 5 and 28 is 11 + 17.

  In mathematics, ideas are split into three categories:

  1. Propositions are statements that may or may not be true. Euclid proposed many in his Elements that he then showed to be true.

  2. When a proposition has been shown to be true in all possible cases, it is said to be a theorem, like Pythagoras’ – it works for all right-angled triangles.

  3. A conjecture is a proposition that holds the middle ground – mathematicians believe it to be true but have not yet been able to prove it is always true.

  Although Goldbach’s conjecture has been checked as far as 4,000,000,000,000,000,000 without finding a counter-example, it is still only a conjecture rather than a theorem. Very picky, these mathematicians!

  CARL GAUSS (1777–1855)

  Carl Gauss was born into a poor family in Germany in 1777 but it soon became apparent that he possessed an extraordinary intellect and a special ability in mathematics in particular.

  According to legend, when Gauss was at school he continually annoyed his maths teacher by completing his work far faster than the rest of his class. Exasperated, Gauss’ teacher finally told him to add together all the numbers from 1 to 100, thinking that might give him some peace.

  Gauss immediately stated the correct answer: 5050.

  Gauss was not a lightning calculator; he had instantaneously seen a shortcut. If you repeat the series, but backward, it can be seen that all the terms add up to 101:

  Gauss then quickly worked out that 100 terms of 101 gives 10100, but as this is twice the sum we actually want, he halved it to give the answer 5050.

  Downing Tools

  When Gauss was at university he was interested in the classical geometry of the Ancient Greeks, but new developments in mathematics meant it was now possible to prove geometric theories using algebra, rather than graphically. Gauss proved that it was possible to draw a regular polygon (all sides of equal length and all interior angles of equal size) with 5, 17 or 257 sides using only a pair of compasses and a straight edge.

  Back to the beginning

  Gauss furthered Euler’s initial work in a branch of number theory called modular arithmetic, in which numbers are allowed up to a certain value, after which they wrap-around and start again. The twenty-four-hour clock is an example of modular arithmetic – after 23:59 we start again at 00:00.

  Much like normal arithmetic, in modular arithmetic you need to define what your highest number can be. In normal arithmetic we work in tens, but our highest digit is one less than this: 9. If we are working in modulo 8 we can only use the digits from 0 to 7. This means that 8 would be 0 in modulo 8 because we start again from zero after we reach 7. Likewise, 15 would be 7 in modulo 8 because 15 = 8 + 7 but the 8 counts as 0. Mathematicians would write this as:

  15 ≡ 7 (mod 8)

  In certain circumstances, if you divide two numbers you may be more interested in the remainder (what’s left over) than the quotient (the answer to the division). This is where modular arithmetic can be useful, because a number’s value in a particular modulo is the same as the remainder if you were dividing it. For example:

  75 ÷ 8 = 9 remainder 3

  75 ≡ 3 (mod 8)

  If you wanted to check whether a number was prime, you could see whether the number was ever equal to zero in successive modulos, which is something that computers are good at.

  A magnificent spread

  Gauss’ work naturally moved into prime numbers, which remain one of the greatest mysteries in mathematics. Gauss made a conjecture, now called the prime number theorem (it is a theorem because it has since been proven, (see here), about the way in which prime numbers are spread out. Although we do not have a formula for making prime numbers, Gauss noticed that the h
igher up through the numbers you go, on average the more spread out the prime numbers become. He wrote:

  number of primes less than x ≈ x / lnx

  The symbol ≈ means ‘is roughly equal to’ and the symbol ln means ‘natural logarithm’. Therefore:

  number of primes less than 1000 ≈ 1000 / ln 1000 ≈ 145

  number of primes less than 10000 ≈ 10000 / ln 10000 ≈ 1086

  This shows that, although we made x ten times larger, there are fewer than eight times as many primes. This trend continues as we make x bigger, so primes become fewer the higher we count.

  Uneven distribution

  Gauss also made an important contribution to statistics by being the first person to introduce the normal distribution. This bell curve applies to all manner of real-world situations such as animals’ heights and weights, marks in examinations, measurements made in scientific experiments and so on.

  If you measured the height of every thirteen-year-old boy in the country, you could work out the average or mean height (worked out by adding up all of the data and dividing by how many data there were). You could then look at the percentage of the boys in a certain height bracket and you would find that most of them were within a certain distance from this mean. As you move away from the mean, either higher or lower, you find that there are fewer and fewer boys. Thinking in terms of percentages like this is the same as thinking in terms of probabilities, and so the normal distribution is said to be a probability density function:

  The shape of the graph shows what we know to be true. Think about your own friends. Unless you hang out with professional basketball players, most of your friends are clustered around an average or normal height for their gender, and you probably know far fewer very tall and very short people.

  The idea of IQ (intelligence quotient) is an example of a score that has been standardized using the normal distribution. An IQ score of 100 is the mean, and a 15-mark interval is called the standard deviation, which is a measure of how spread out the marks are. As a result of the equation of the line used for the normal distribution, it turns out that over 68% of scores are within one standard deviation of the mean, so nearly 70% of people will have an IQ of between 85 and 115. Over 95% of people are within two standard deviations, scores of 70 to 130. Over 99.7% of people are within three standard deviations, scores of 55 to 145. Mensa, a society for people with high IQs, has an entrance test intended to select people who have an IQ higher than 98% of the population, which corresponds to an IQ of just under 131.

  A Question of Identity

  An equation (such as y + 3 = 10) is something we can try to solve in order to find out if there are any values that satisfy the equation. Linear equations, where the unknown has an index of 1 (i.e. it is not squared or cubed), have only one answer. Equations that incorporate squares, or cubes, or higher can have more than one answer, but equally may have no answer. For example, there is no real number x that works in x2 = -6.

  In formulae (such as Einstein’s E = mc2 or average speed = distance ÷ time) we can substitute values for the letters in order to solve the equation. For example, if you travelled 200 kilometres in 4 hours, you would get the following:

  Average speed = 200 ÷ 4 = 50 km / hour

  An identity is something that is always true for any value of the unknowns. Gauss invented the triple-bar symbol, ≡, to show this. For example:

  (y + 2)(y - 3) ≡ y2 - y - 6

  This is an identity because it works for any value of y. Say I make y = 7:

  (7 + 2)(7 - 3) = 72 - 7 - 6

  9 × 4 = 49 - 13

  36 = 36

  The Digital Age

  Although they didn’t become commonplace in homes until the 1980s, one can’t now imagine a life without computers. The history of the computer is intimately linked with the story of mathematics. We have seen already that pioneers like Leibniz and Pascal blazed (sorry...) a trail with mechanical calculators, but the need for faster and more flexible machines sparked off the digital revolution.

  CHARLES BABBAGE (1791–1871)

  The English mathematician Charles Babbage was the son of a banker, and is best known as the ‘Father of the Computer’ because of his work in mechanical computing. Babbage read mathematics at Cambridge University but found his studies unfulfilling, which led him to found the Analytical Society in 1812 along with a group of fellow students. Apparently, Babbage’s moment of inspiration came when he was one day poring over a group of logarithm tables that were known to contain many errors. Working out the entries for log tables is a very tricky process and inevitably human ‘computers’ – as they called people who calculated sums in those days – made mistakes. Babbage maintained that the tedious calculations, which require little thought, only accuracy, could be performed by an elaborate calculating machine.

  Machine takes over

  Babbage secured government funding and produced plans for his Difference Engine. However, such was the complexity and size of the machine that it was never built in his lifetime. (A Difference Engine was built in the 1980s and today resides in London’s Science Museum. At just over 2 metres high and 3.5 metres long it’s a fairly sizeable machine.)

  As interest in the Difference Engine project began to wane, Babbage started work on the Analytical Engine using the knowledge he had gleaned while designing his first computer. This machine was intended to be much more like a computer as we know it. It could be programmed to perform particular combinations of mathematical functions. One set of punched cards would be used to programme the engine, another set would introduce data to the engine and then the machine itself could punch blank cards with the results, effectively saving them in its memory for future use.

  Babbage was still working on the Analytical Engine when he died. Because of the immense cost and complexity involved, a working model has never been built, although, at the time of writing, British programmer and mathematician John Graham-Cumming is leading a project to build one for the first time.

  ADA LOVELACE (1815–52)

  One of the few notable pre-twentieth-century female mathematicians, Ada Lovelace was the daughter of the famous poet and rake Lord Byron and Anne Isabella Milbanke, to whom Byron was briefly married. Milbanke was convinced Byron’s excesses had been the result of a kind of insanity and she sought to shield her daughter from a similar fate by encouraging her to pursue a persistent education, especially in the logical discipline of mathematics.

  Get with the programme

  Despite being unable to enter university, Lovelace was privately tutored and she continued with mathematics throughout her life. She encountered Charles Babbage, who asked her to translate into English an article on his Analytical Engine that had been written by an Italian mathematician. Lovelace duly did so, adding extensive notes on the machine, including a set of instructions that would have had the machine produce the Bernoulli numbers, a sequence of numbers named in honour of Jacob Bernoulli. For this, Lovelace is considered to be the first ever computer programmer. Sadly she died from cancer at the age of thirty-six.

  Ockham’s Razor

  When Ada Lovelace married William King in 1835 she moved to her husband’s estate in Ockham, Surrey, which is believed to have been the birthplace of a monk known as William of Ockham in the late thirteenth century. William of Ockham was a natural philosopher who first coined the principle known as Ockham’s razor: the simplest explanation for a phenomenon more often than not turns out to be true. This principle has been adopted in all scientific fields – when researchers look to explain what they have observed, they try to use existing theories and laws rather than fabricate new ones to fit what they have seen.

  GEORGE BOOLE (1816–64)

  An Englishman who became a mathematics professor in Ireland, George Boole published major works on differential equations (equations that involve derivatives of a function), but he is most remembered for his work on logic.

  Boole sought to set up a system in which logical statements could be defined mathematica
lly and then used to perform mathematical operations on the statements; the results would be generated without the need to think through the problem intellectually. Boole’s system aimed to take a raft of logical propositions and see how they combined together with the aid of maths rather than philosophy.

  A logical step

  In order to set up the system, Boole developed what became known as Boolean algebra: letters defined either as logical statements or as groups of things. The letters can have a value of 1, meaning ‘true’, and 0, meaning ‘false’.

  So imagine you are considering dogs as a group of things, and you let x represent shaggy dogs and y represent yappy dogs. You can then make a table for the dogs using values for x and y:

  Boole then defined three simple mathematical operations we could conduct with the results – AND, OR and NOT. AND is defined as the multiple of the two values. As the table below shows, if x and y are ‘true’, we expect the answer ‘true’ or 1, and everything else to be 0 or ‘false’:

  This seems rather self-evident – a dog can only be shaggy AND yappy if the dog is shaggy and yappy, but the example is useful because it shows you can come to the same conclusion using very simple arithmetic.

  There are a host of other Boolean operations that can be reduced to arithmetic.

  In the 1800s Boole’s work had implications for mathematical logic and set theory, but it was not until the twentieth century that a more practical use was found. An electronics researcher called Claude Shannon discovered that he could use Boolean operations in electrical circuits – he could generate a simple electrical circuit to take logical steps and therefore make decisions based on them. As Boole’s work uses only values of 0 or 1, ‘true’ or ‘false’, or ‘on’ or ‘off’ it is known as a digital method. In essence, Boole paved the way for the first electronic digital computers.