Unlike the humanities, which are in a permanent state of reinvention, as new ideas or fashions replace old ones, and unlike applied science, where theories are undergoing continual refinement, mathematics does not age. The theorems of Pythagoras and Euclid are as valid now as they always were.
A bold and encouraging introduction to the book. Especially because Euclid’s theorems are investigated in this very book and are found to be true only under certain conditions.
Tens of thousands of years ago, well before the arrival of numbers, however, our ancestors must have had certain sensibilities about amounts. They would have been able to distinguish one mammoth from two mammoths, and to recognize that one night is different from two nights. The intellectual leap from the concrete idea of two things to the invention of a symbol or word for the abstract idea of ‘two’, however, will have taken many ages to come about. This occurrence, in fact, is as far as some communities in the Amazon have come. There are tribes whose only number words are ‘one’, ‘two’ and ‘many’. The Munduruku, who go all the way up to five, are a relatively sophisticated bunch.
Explaining how counting more than 5 was not a really important skill for the ancient humans.
When I say that I have a big family, I am telling you that I don’t know [how many members it has]. Where does my family stop and where does the others’ family begin? I don’t know. Nobody ever told me that.
A great question! We understand the concept of family quite intuitively. But have we ever wondered how big our families are?
In almost all Western European languages, number words do not follow a regular pattern. In English we say twenty-one, twenty-two, twenty-three. But we don’t say tenty-one, tenty-two, tenty-three – we say eleven, twelve, thirteen. Eleven and twelve are unique constructions, and even though thirteen is a combination of three and ten, the three part comes before the ten part – unlike twenty-three, when the three part comes after the twenty part. Between ten and twenty, English is a mess.
Explaining the development of number-words and the apparently arbitrary manner in which it happened.
Hah! The glory of discovery! Brought a smile to my face. Bhaskara was talking about the proof for Pythagoras’ theorem.
Since we can always find an infinite number of rational numbers between any two rational numbers, it might be thought that the rational numbers cover every number. Certainly, this is what Pythagoras had hoped. His metaphysics was based on the belief that the world was made up of numbers and the harmonic proportions between them. The existence of a number that could not be described as a ratio diminished his position, at the very least, if it did not contradict it outright. Yet unfortunately for Pythagoras, there are numbers that cannot be expressed in terms of fractions, and – rather embarrassingly for him – it is his own theorem that leads us to one. If you have a square where each side has length 1, then the length of the diagonal is the square root of two, which cannot be written as a fraction.
One small embarrassment for Pythagoras and one giant leap for humanity.
So, what’s the point of logarithms? Logarithms turn the more difficult operation of multiplication into the simpler process of addition. More precisely, the multiplication of two numbers is equivalent to the addition of their logs. If X × Y = Z, then log X + log Y = log Z.
Yes, we’ve learned about this in school. But this still appears to be so ridiculously convenient when you suddenly come across it! I loathed logarithms in school, but it was slowly and surely, making my life better.
Place one grain of wheat on the corner square of a chessboard. Place two grains on the adjacent square, and then start filling up the rest of the board by doubling the grains of wheat per square. How much wheat would you need to fill the final square? A few truckloads, or a container, maybe? There are 64 squares on a chessboard, so we have doubled up 63 times, meaning that the number is 2 multiplied by itself 63 times, or 263. In grains, this number is about 100 times more than the world’s current annual wheat production. Or, to consider it another way, if you started counting a grain of wheat per second at the very moment of the Big Bang 13 billion or so years ago, then you would not even have counted up to a tenth of 263 by now.
Yes. Be prepared to be mind-mangled by the sheer magnitude of numbers in this book.
The most playful of the great mathematicians was Leonhard Euler, who, in order to crack an eighteenth-century brainteaser, invented a whole new branch of mathematics.
Wright is a juggler, which ‘seemed like the obvious thing to do after I learned to ride a unicycle’, he said. He also helped develop a mathematical notation for juggling, which might not sound like much, but has electrified the international juggling community. It turns out that with a language, jugglers have been able to discover tricks that had eluded them for thousands of years. ‘Once you have a language to talk about a problem, it aids your thought process,’ said Wright, as he took out some bean balls to demonstrate a recently invented three-ball juggle. ‘Maths is not sums, calculations and formulae. It is pulling things apart to understand how things work.’
Nothing in math is ever wasted.
The Encyclopedia has a function that allows you to listen to any sequence as musical notes. Imagine a piano keyboard with 88 keys, which comprise a spread of just under eight octaves. The number 1 makes the piano play its lowest note, the number 2 makes it play the second-lowest note, and so on all the way up to 88, which commands the highest note. When the notes run out, you start at the bottom again, so 89 is back to the first key. The natural numbers 1, 2, 3, 4, 5…sound like a rising scale set on an endless loop. The music created by the Recamán sequence, however, is chilling. It sounds like the soundtrack of a horror movie. It is dissonant, but it does not sound random. You can hear noticeable patterns, as if there is a human hand mysteriously present behind the cacophony.
Get the sound file here – Recamán sequence as a symphony at the On-Line Encyclopedia of Integer Sequences®. (before playing or downloading, put A005132 in the box for ‘sequence’ which is empty or you’ll get an error)
The decimal system throws up an excellent example of a Zeno-inspired paradox. What is the largest number less than 1? It is not 0.9, since 0.99 is larger and still less than 1. It is not 0.99 since 0.999 is larger still and also less than 1. The only possible candidate is the recurring decimal 0.9999…where the ‘…’ means that the nines go on for ever. Yet this is where we come to the paradox. It cannot be 0.9999…since the number 0.9999…is identical to 1! Think of it this way. If 0.9999…is a different number from 1, then there must be space between them on the number line. So it must be possible to squeeze a number in the gap that is larger than 0.9999…and smaller than 1. Yet what number could this be? You cannot get closer to 1 than 0.9999…. So, if 0.9999…and 1 cannot be different, they must be the same. Counter-intuitive though it is, 0.9999…= 1.
In Rome, for example, coins were flipped as a way of settling disputes. If the side with the head of Julius Caesar came up, it meant that he agreed with the decision. Randomness was not seen as random, but as an expression of divine will. Throughout history, humans have been remarkably imaginative in finding ways to interpret random events. Superstition presented a powerful block against a scientific approach to probability, but after millennia of dice-throwing, mysticism was overcome by perhaps a stronger human urge – the desire for financial profit. Girolamo Cardano was the first man to take Fortune hostage. It could be argued, in fact, that the invention of probability was the root cause of the decline, over the last few centuries, of superstition and religion. If unpredictable events obey mathematical laws, there is no need to have them explained by deities. The secularization of the world is usually associated with thinkers such as Charles Darwin and Friedrich Nietzsche, yet quite possibly the man who set the ball rolling was Girolamo Cardano.
How probability killed god.
The human need to be in control is a deep-rooted survival instinct. In the 1970s a fascinating (if brutal) experiment examined how important a sense of control was for elderly patients in a nursing home. Some patients were allowed to choose how their rooms were arranged and allowed to choose a plant to look after. The others were told how their rooms would be and had a plant chosen and tended for them. The result after 18 months was striking. The patients who had control over their rooms had a 15 percent death rate, but for those who had no control the rate was 30 percent. Feeling in control can keep us alive.
If society was like a machine that produced a regular number of murderers, didn’t this indicate that murder was the fault of society and not the individual?
This and the previous quote are just a couple of many side-explorations of philosophy, because mathematicians were often philosophers too, trying hard to understand the underlying mysticism of the abstract world of numbers.
Cantor led us beyond the imaginable. It is a rather wonderful place and one that is amusingly opposite to the situation of the Amazonian tribe I mentioned at the beginning of this book. The Munduruku have many things, but not enough numbers to count them. Cantor has provided us with as many numbers as we like, but there are no longer enough things to count.
The books ends with this gem. Such a wonderful adventure it was!