Knowing Numbers
I can all but guarantee that the first overtly mathematical thing you ever did as a very young child was learn to count. Probably through songs and with some help from your fingers and toes you began your journey into the world of number. Since then, you will have learnt that those are not the only numbers, the ones you can make with your fingers and toes, but that there are more numbers, different numbers, numbers which might require a bit more imagination like negative numbers and fractions. Delightfully, you have many more 'different numbers' still yet to discover and delicious questions about why they exist to wonder about.
Being a mathematician is a mysterious mix of being an explorer and an inventor. As explorers we learn new skills so that we can solve new problems until we find a problem we can't solve... yet. Now we must become inventors and imagine what new things must exist in order for that problem to be solvable. We propose ideas, we argue, we fail, we propose better ideas, we argue, we prove and eventually we know we have invented correctly, or correctly imagined the truth as it already was, depending on how you look at it. Now, with our new truths discovered and new skills we can continue to explore new problems and the cycle begins again. The joyous part is that there is no retirement from being a mathematician, there is no moment when we are forced to walk away and say that bit of mathematics will never work, there are always new things to explore and invent. A good illustration of this is the story of the need for new numbers.
Everything is Number
You've heard of Pythagoras on account of that famous theorem about right angle triangles, but in truth he was not one man obsessed by three sided polygons, but rather an entire community of people who were pretty much obsessed with mathematics. Pythagoras (born around 569 BC) founded a school in Crotona, a Greek colony in southern Italy, where people studied mathematics, science & philosophy. The Pythagoreans consisted of men and women (progressive for the time) and the inner circle, called the mathematikoi, all lived together and followed the same vegetarian and bean-free diet. They apparently believed that eating beans was sinful so even these great thinkers were not right about everything. Above the entrance of their school the Pythagoreans carved their motto, which united them in belief and fascination and which ruled all their contemplation of god and the natural world: Everything is number.
Natural Numbers
The numbers the Pythagoreans were primarily interested in were the whole numbers, the numbers we now call the 'natural numbers' or the counting numbers 1, 2, 3, 4, ... . Still today there is some debate as to whether 0 is included (can you really count zero of something?) In modern mathematics we denote the set of all of these numbers by N.
In fairness to the Pythagoreans these numbers are a decent thing to be fascinated by. For most of us it is the first set we meet, when we learn to count, and it's also the first infinity we meet (yes, there are more) when we learn to argue "You're silly times infinity" "Well you're silly times infinity and one" with our siblings. This is the natural feeling for infinity: that it is the 'number' we would 'get to' if we continued to count upwards in the natural numbers forever. The only small caveats are that it is not a number and we can never 'get' there.
The natural numbers are the mathematical embodiment of 'next'-ness. Something comes first, the next thing comes next and each next thing is counted. Simple, elegant, powerful.
All the Integers
By now we have got very friendly with the natural numbers, our most familiar, fingers and toes to infinity numbers. But we haven't thought too much about what we can do with the natural numbers without coming out of their world. For example, one of the games we can play is addition, take any two natural numbers and add them together you will get another natural number. In fact the game is not so strict, you can take any natural number of natural numbers and add them together and you will get another natural number. It is only when you try to add infinitely many natural numbers that you might run into trouble. So addition is a safe game that we can play in the world of natural numbers. Multiplication is another one. Take any natural number of natural numbers and multiply them together and you will get another, probably rather large, natural number. So multiplication is also a safe game for the natural numbers to play.
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We need not look much further though to hit our first bump in the road. If there is addition then there must be an 'undo' for addition, an inverse: subtraction. At first it might seem that subtraction is also a safe game to play in the natural numbers: 10-8=2, 9095-9091=4. But, as soon as we ask a question as seemingly simple as 2-13=? Well, if all we have is the natural numbers that question is unanswerable, there is no solution in our world of the counting numbers, the game is broken, we need new numbers. The solution of course is to introduce the negative of each natural number extending in an infinite queue downwards from zero in the same way as the naturals queue infinitely upwards. This might seem to be a rather undramatic and obvious move because we were all introduced to the negatives in primary school so they hardly seem ground-breaking, but to take the world of the naturals and imagine a symmetrical world on the other side of nothing is really quite an impressive and significant jump. In modern mathematics, the set of all of the natural numbers with their negative partners and zero is called the integers and is denoted by Z. The word integer is Latin and means 'whole' while the Z is from 'ganze Zahlen' which is German for 'whole number' used on account of the many German mathematicians who have contributed to number theory. We should also note that every natural number is an integer so N is contained in Z. Now that we inhabit the slightly larger world of the integers we can safely play with addition, multiplication and subtraction. Admittedly, there is a little work to be done in order to establish how these games can be played together, especially in making sure the negative integers don't break any of the rules we already know to be true.
Rational Numbers
The next step is to investigate the 'undo' of multiplication, its inverse: division. At first, again, division is also a safe game to play 90÷9=10, 91÷13=7 even the negatives play by the rules and make sense doing so -20÷5=-4. However, before long we can ask a question that spoils the game, say, 7÷3=? The first answer we learn at primary school is '3 into 7 doesn't go', mic drop, walk away, mathematics is broken. If we want to do better than that though we can sort of wriggle out of this difficulty by giving an answer that stays in the integer world 7÷3=2 with a remainder of 1. This is a comfortable enough side step around the problem which allows us to stay in a world we know, hence why we learn to think this way first. The trouble with it though is this: 7÷3=2 with a remainder of 1 and also 4001÷2000=2 with a remainder of 1 so both of those very different questions appear to have identical solutions: 2 with a remainder of 1. But while those answers look the same in truth they should be different, a remainder of 1 when dividing by 3 is an altogether different beast from a remainder of 1 when dividing by 2000. Hence the need for a better solution to the problem of division, we need new numbers.
We need numbers which allow us to express division by numbers that 'don't go', we need fractions to exist. Once you have the integers and the concept of division, you can define a whole new set of numbers which is made up of all the fractions (in simplest form to avoid repeats) that can possibly be made using the integers. In modern mathematics this set is called the rational numbers and is denoted by Q for quotient, a word for division which comes from the Latin 'quotiens' meaning 'how many times'. We should also note that every integer is a rational number because it can be written as a fraction with numerator equal to itself and a denominator of one. This means that the set of integers is contained within the set of rational numbers.
The beauty of fractions is their simplicity, all they are is an expression of division, but crucially they are an accurate one. What is 7÷3? Well it's seven divided by three, seven thirds. What is 4001÷2000? It's four thousand and one divided by two thousand, four thousand and one two thousandths. Now these answers are no longer identical, they are similar in that they are both two and a bit but we can now see that the extra third and the extra two thousandth are very different 'a bit's and not the same old remainder of one. Not only do fractions allow us to divide any of our numbers by any of our numbers, even the ones that 'don't go', once we learn about equivalence they also give us immediate access to infinitely many more divisions that result in the same value. Fractions are brilliant. Love them.
Numbers to Drown For
So by coming up with the new numbers we need in order to solve a problem we only need to consider four simple operations to expand our world from the natural numbers N to the integers Z to the rational numbers Q. So far, each of our worlds sits neatly inside the next: the natural numbers are all integers and all integers are rational. There are no problems yet in terms of moving between these different sets.
Now when the Pythagoreans said 'everything is number' they knew that there was more going on than just the natural numbers, they were also deeply interested in the ratios that could be made between them, what we now recognise as fractions, so they were fully on board with the rational numbers. In fact, the school believed that the entire physical world could be understood through the rational numbers, they applied this to every field including investigating sound: if the ratio between two sound frequencies is a whole number then we like how they sound together, if not then we don't.
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As far as the Pythagoreans were concerned these were all of the numbers, they studied the entire world through this lens and lived in almost religious devotion to the truthful language of this world: the rational numbers. We can imagine then that it would throw a considerable spanner in the works should a number ever appear that didn't fit...
The legend goes that when the Pythagorean philosopher, Hippasus, discovered that numbers exist outside of the rationals, this so shocked the Pythagoreans that Hippasus was drowned at sea as punishment by the Gods. There may not be much truth in the story as there is debate over which Pythagorean made the discovery and whether the drowning punishment was attributed to this mathematical crime rather than some others, but the sentiment seems accurate. The Pythagoreans built their learning on the rational numbers, they lived by that mantra 'everything is number' and so to learn that they had been so wrong about numbers must have been earth-shattering.
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Historians are also fairly certain that the Pythagoreans had a proof for the irrational (not rational) number which is the first irrational most of us meet: the square root of two. To add insult to injury, it is likely that they proved its irrationality using the Pythagorean theorem, brutal.
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This discovery opens up a whole new word of irrational numbers which have quite different characteristics from the rationals we have known so far. If we take the set of rational numbers and add in all of the irrational numbers then what we get is the set called the real numbers, denoted by R. The naming of this set is in equal measures odd and exciting; if these are not all of the numbers, what numbers must come next?