Chapter 0: How nothing gave us something
Count for something
In the beginning, there was nothing. Well, actually, no. In the beginning there was always something. It might have been beans, successful hunts, or victories in battle, but for millennia people were using maths to describe something – counting it, measuring it, dividing it up. A mathematical description of nothing, zero, was still a long way off.
It’s most likely early humans counted on their fingers in just the same way we all first learn to count. (It’s handy having a set of counting sticks at the end of your arm in your pocket, or, rather, in the fold of your animal skin.) One of the first pieces of evidence of our use of numbers are what are believed to be tally marks cut into a 20,000 year old bone, known as the Ishango Bone, found in Zaire, Africa. A tally system is a very sensible way to keep track of accumulating quantities, whether you’re keeping track of a score or you are a prisoner marking your days inside on your cell wall. The way we keep control of a large number of tally marks today is still firmly connected to our early days of counting – we group them into fives, like the five fingers on our hands. The first four are marked individually, the fifth a line crossing the first four, making a complete set. It makes sense that our sense of an easily manageable set is the same as the count of the digits on our hand.
What we called these quantities, whether we even had words for them, is another question. There are still cultures today, including the Pirahã and Mundurukú from the Brazilian Amazon, who have a name for small numbers or quantities, but refer to anything larger just as “many”.
But over the centuries almost all cultures developed names and symbolic representations for numbers, and a way to combine these so as to write any number they could possibly need. Inscriptions found in Egyptian tombs from over 5000 years ago (3000 BC) show that the Egyptians were using beautiful hieroglyphs to represent numbers, such as coils of rope, lotus flowers and frogs, to represent 100, 1,000 and 100,000. These symbols would then be repeated, to build up the number they required, some numbers requiring a large collection of symbols to be represented.
The ancient Greeks built up numbers in a similar way, using letters from their alphabet to write numbers – for example α for 1, β for 2, γ for 3, κ for 20, or τ for 300. The Romans used combinations of symbols such as I for 1, V for 5, X for 10, L for 50, C for 100, D for 500 and M for 1000 to write numbers. Generally numbers were built up essentially by adding together the symbols’ values (though there were some conventions for subtraction too, for example IV meant 5-1=4). It’s a system we still use today for counting royalty and giving dates at the end of movies and TV shows.
But no matter what names or symbols a culture used, they meant the same thing. The number 3 means the same thing whether it is written in tally marks or in Egyptian, Roman or Greek numerals. A number, no matter what language it is written in, is just a name or symbol for the quantity of things that are being counted. The one-ness of a set of one thing, the two-ness of a set of two things, the three-ness of a set of three things is one of the very first mathematical abstractions that we all intuitively make. The number of the things we are counting is independent of what those things actually are, be they kittens, kills or cabbages.
None of the systems we just looked at included a symbol for the set of no things – it just wasn’t necessary. All of the systems are additive - you simply add up the values of the symbols in the number to get its value. There may be a convention for ordering in such systems, for example starting with the largest blocks on the left of the number. But it doesn’t really matter which order you write these symbols in, there isn’t usually any ambiguity, it’s just about adding up the individual pieces. This is good, but it does lead to complications if you are dealing with big numbers or trying to do complicated sums.
Take, for example, the numbers MCMLXXIV and XXXIX written in Roman numerals.
Now add these together (the answer is MMXIV). You’ll find that that’s a far harder task than adding 1974 and 39 (to get 2013) with our modern numerals. This is where tally-based number systems fail. To really get control of large numbers and to simplify mathematical calculations takes a cleverer way of writing numbers. The vital piece to make such a system work is zero.
Something for nothing
A people we now loosely call the Babylonians lived in Mesopotamia between the Tigris and Euphrates rivers (now Iraq). As far back as 3000 BC the indigenous Mesopotamian people, the Sumerians, used soft clay tablets to write their numbers, imprinted with the wedged end of their writing stylus. This way of writing numbers evolved into a system that used two wedge-shaped symbols arranged to represent all the numbers 1 to 59.
But rather than continue in this way, inventing ever new arrangements and symbols, the Babylonians, some 4000 years ago, made a brilliant leap: they invented a place value system very similar to what we use today. The numbers are written side by side in a string, and the value each number represents depends on its place in this string.
We can illustrate this using our own number system. For us the “4” in 4622 no longer represents the value 4. Instead, it tells us that our number contains exactly 4 multiples of 1,000. Similarly, the 6 tells us that there are 6 multiples of 100. And the two 2s in the number represent different values: the left-most 2 means that there are 2 multiples of 10 and the right-most 2 that we have 2 multiples of 1. What do 1000, 100 and 10 have in common? They are all powers of the number 10, that is, numbers you get by multiplying 10 by itself some number of times:
1000 = 10 x 10 x 10 = 103,
100= 10 x 10 = 102
and 10 = 101.
The Babylonian system worked in the same way only that rather than being based on powers of 10 it was based on powers of 60. A digit within a number told you how many multiples of 1, 60, 602=3600, and so on there were in the number, based on where in the string the digit appeared.
The place value system was a great advance. It made it possible to write very large numbers without having to invent new symbols to represent greater and greater orders of magnitude. It also made complicated sums easier: the way we write the numbers does some of the work for us. If one number contained 3 multiples of 60 and another 4, then clearly their sum would contain 7, telling you exactly what to write in the slot allotted to multiples of 60. The only complication, that the multiples of 60 in the sum might give you something bigger than 602, is dealt with by carrying digits to the next slot along.
But still, there was a hitch. What would you write when there isn’t a multiple of 60, or 602 or some other power of 60 in a given number? For example, this happens with 3,601 = 602+1 which doesn’t have a multiple of 60 in it. Originally the Babylonians indicated such a missing power just with a space, leaving lots of scope for ambiguity: is this really a space, or just the result of the writer having a hic-up? The Babylonians seemed able to cope with this ambiguity by an intuitive understanding of the size of the numbers they were dealing with for any particular calculation. But what they really needed was a placeholder to separate the powers of 60.
Around 300 BC such a new symbol began to appear in the shape of two angled wedges. Whenever you came across those you knew that a power of 60 was missing. The new sophisticated number system allowed Babylonian mathematics to flourish. Complex calculations now became possible and spawned extremely accurate astronomical tables.
The place value system was invented at least another couple of times before the origins of our own came along: by the Chinese, from around 300 BC, and by the Mayans, whose culture began as far back as 2000 BC but peaked around 500 AD. Both systems also developed a placeholder symbol; zero had begun its inexorable spread in mathematics.
Nothing really is something
What none of these cultures seem to have recognised, however, was that their placeholder symbol – their zero – was really a number in its own right. That realisation, along with a number system we use today, comes from India. Indians were using that system, which was positional and had a decimal base, as far back as 500 AD. In 499 AD the mathematician and astronomer Aryabhata beautifully captured its essence in his book Aryabhatiya:
“From place to place, each is ten times the preceeding.”
Indians used the Sanskrit word for “void”, śūnya, to refer to zero. Our first record of a little round circle used to describe it is from 870. It later mutated into the symbol for zero we use today.
But most importantly, Indian mathematicians treated zero as a number onto itself, a number you could do calculations with and that might even pop out as the answer to a problem.
In his book Brahmasphutasiddhanta, published in around 628, mathematician and astronomer Brahmagupta laid down rules for arithmetic. In doing so he captured the essence of zero’s nothingness, at least as far as arithmetic was concerned. We can express it as follows:
When zero is added to a number or subtracted from a number, the number remains unchanged.
b+0=0+b=b b-0=b
This fact makes zero unique among numbers: no other number leaves its partner in addition (or subtraction) quite so undisturbed. For suppose that there was another such number and call it u for “unknown”.
Since adding u to any number leaves that number unchanged we have
0 = 0+u.
And since adding 0 to any number also leaves that number unchanged, we also have
0+u = u. Putting the two together gives
0 = 0+u = u. So u was equal to 0 all along!
This, incidentally, is our first example of a mathematical proof -- an argument that shows, beyond any doubt, that something is true. The concept of proof belongs to mathematics like fish belong to water and we will be meeting it plenty more times later on.
Another rule attributed to Brahmagupta concerns the behaviour of zero under multiplication:
Zero multiplied by any number is zero.
This little number, so unobtrusive when it comes to addition, under multiplication becomes all absorbing.
What, then can we say about zero and division? What is 5 divided by 0, or 0 divided by 0? These turn out to be tricky questions that plant the seed for mathematics that was developed centuries later. Brahmagupta hedged his bet on the former, but was categorical about the latter, asserting that 0/0 should be 0. In the modern view he was wrong. It was another Indian mathematician who provided a deeper insight into this hairy problem: Bhaskara II.
To the limit
Bhaskara II, who lived in the twelfth century AD, is regarded by many as the greatest mathematician and astronomer to emerge from medieval India. His most famous contribution to mathematics, however, seems to have been the result of something we today regard as astronomy’s evil twin: astrology. According to legend, Bhaskara consulted his beloved daughter’s horoscope and found, to his horror, that she was to remain childless and unmarried. Not prepared to bow to that fate, Bhaskara determined an auspicious moment at which her wedding should take place. To make absolutely sure the moment wouldn’t be missed, he constructed a water clock. But his daughter, by the beautiful name of Lilavati , could not suppress her curiosity. When looking at the clock from close up, a pearl from her bridal dress fell into it. It blocked the hole through which the water flowed and thus the auspicious moment could never come. The wedding was off! To console Lilavati, a devastated Bhaskara promised to write a book in her name, a book that would exist forever. Luckily for her, the book was a maths book.
The Lilavati is just one part of a greater work, called Siddhānta Shiromani, which translates from the Sanskrit as Crown of Treatises. It covers an eclectic collection of mathematical questions: there is a lot of arithmetic, but also geometry and algebra. Some questions are directly addressed to Lilavati, “whose eyes are like fawn’s”, and many with a poetry our textbooks can only dream of, for example:
The square root of half the number of a swarm of bees is gone to a shrub of jasmine; and so are eight-ninth of the whole swarm: a female is buzzing to one remaining male that is humming within a lotus flower in which he is confined, having been allured to it by its fragrance at night. Say, lovely woman, the number of bees.
If you can’t work out the answer, see the box at the end of this chapter.
In the Lilavati Bhaskara gives rules for calculating with zero, including one that appears to say that for any number a,
(a x 0)/0 = 0 x 0 = a.
This seems to suggest that 0/0 can be anything -- any number a you care to choose -- and we will see an echo of this below. Bhaskara’s great insight, however, came in a lesser known work of his, the Vija-Ganita, where he considers what a/0 should be:
Quotient the fraction 3/0. This fraction of which the denominator is [zero], is termed an infinite quantity. In this quantity [...] there is no alteration, though many be inserted or extracted; as no change takes place in the infinite and immutable God.
So according to Bhaskara, the result of division by zero should be be infinity, a number he equates with an unchanging God as infinity is unchanged by addition or subtraction. Modern mathematicians do not agree with this idea, but it is quite easy to see why Bhaskara came up with it. If you divide a line segment into smaller and smaller pieces, the number of pieces gets larger and larger. As the length of your pieces gets closer to zero, the number of them gets closer to, well, infinity.
Under normal circumstances division behaves in a nice continuous fashion. If I divide 1 by a sequence of numbers that get closer and closer to 2, then the result will get closer and closer to ½ = 0.5:
1/1.9 approx 0.5263
1/1.99 = 0.5025
1/1.999 = 0.5003
and so on. If we require that the same should go for dividing a number by numbers that get closer and closer to 0, then this suggests that any number divided by 0 gives infinity.
But unfortunately things are not quite as simple as this. Imagine the number 0 as it appears on your thermometer, with positive temperatures above it and negative ones below it. If we mark the sequence of numbers we divided by in the argument above -- the lengths of the smaller and smaller line pieces -- on the thermometer we would get a sequence of numbers creeping up on 0 from the positive side of the thermometer. But we could equally have snuck up on it from the negative side, dividing by negative numbers that get closer and closer to zero, for example -0.001, -0.0001, -0.00001, -0.000001 and so on. Dividing a positive number, such as 1, by a negative one gives you a negative answer, so the results are now
-1/0.001 = -1000
-1/0.0001 = -10,000
-1/0.00001 = -100,000
-1/0.000001 = -1,000,000.
This sequence seems to tend to something infinite as well, but it seems to be infinite in the opposite direction! Rather than climbing higher and higher on the thermometer we are dropping lower and lower. Is there such a thing as minus infinity? And if yes, is it different from plus infinity? These are difficult questions which we will explore later on. Suffice to say that modern mathematicians refuse to commit when it comes to dividing a number by 0: they simply state that the result of such a division is undefined.
What, then, about dividing 0 by 0? Nothing divided by something is still nothing, that’s an uncontentious issue that was already decided by Brahmagupta. So if we divide zero successively by a sequence of numbers that get closer and closer to it we always get 0
0/0.1 = 0
0/0.001 = 0
0/0.0001 = 0
The same is true if we divide by the negatives of the numbers in that sequence:
0/-0.1 = 0
0/-0.01=0
0/-0.001 = 0
and so on. So in accord with Brahmagupta, we could be tempted to decide that 0/0 = 0.
But again there is a hitch. What if I take two sequences of numbers, both creeping up on zero, say
0.01, 0.001, 0.0001, ...
and
0.02, 0.002, 0.0002, ...
and divide the corresponding terms by each other? We get
0.01/0.02 = 2
0.001/0.002 = 2
0.0001/0.0002 = 2
0.00001/0.00002 = 2
and so on. Since both sequences, the one that gives us the top of the fractions and the one that gives us the bottom, creep up on 0, this might suggest that 0/0 should be equal to 2. Equally, if I had turned the division around and divided the numbers in the second sequence by those in the first, the same reasoning would suggest that the result of 0/0 should be ½! As it turns out, by choosing the two sequences just right you can make a case for any number being the result of 0/0. Which is why mathematicians have opted out of this one too. The answer of 0/0 is officially undefined. Nothing divided by nothing is no thing!
Despite these difficulties, and thanks to the initial efforts of Bhaskara and his contemporaries, we are today happy to use zero both as a placeholder in our number system and as a number unto itself. And zero has become even more valuable in today’s digital age. But to unlock the secret of information we need to combine the power of zero with the number 1.
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Answer to Lilavati bee question:
Let x be the number of bees. Then \sqrt{1/2x} + 8/9 x +2 = x, get x =72.
Further reading
The Book of Nothing, John D. Barrow, Pantheon