By Brian Clegg
An infinite series is a familiar mathematical concept, where ‘…’ effectively indicates ‘don’t ever stop’ – for example 1 + ½ + ¼ + ⅛… an infinite series totalling 2. This, though is a short series of posts about infinity.
Strictly, this one is just about big numbers – but it’s on the way to the real thing. There’s something special about big numbers. It’s almost as if by being able to give a big number a name we demonstrate our power over it – and, of course, the bigger the number is, the more power we have. A classic example of this is in the reported early life of Gautama Buddha. As part of his testing as a young man in an attempt to win the hand of his beloved Gopa, Gautama was required to name numbers up to some huge, totally worthless value – and managed to go even further to show how clever he was.
But you don’t need to go back in history to examine this fascination. Anyone with children will have heard them counting, running away with sequences of numbers as if they’re trying to find the end. It’s not just for fun, of course. These counting sequences are hammered into us at school to help build familiarity with the numbers. Most of you will also be familiar with rhymes, used for the same purpose. For example, One two three four five, once I caught a fish alive, and so on. But there is a slight danger that arises from the use of these sequences – our tendency to remember in strings of information can become a hindrance to flexibility. Try counting 1 to 10 then 10 to 1 in a language you aren’t totally fluent in – you will find 10 to 1 much harder.
It’s fine giving names to numbers we need to use (or useless ones to show off), but realistically how many of us ever use this number:
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
As it happens it does have a name, one that proved something of a problem for the unfortunate
Major Charles Ingram when it was his million pound question on
Who Wants to be a Millionaire. He was asked if the number – 1 with 100 noughts after it – was a googol, a megatron, a gigabit or a nanomol. Major Ingram favoured the last of these until a cough from the audience came along to push him in he right direction, and to be honest, who can blame him? A googol frankly sounds childish. And there’s a good reason for that. It is. Literally.
In 1938, according to maths legend, mathematician Ed Kasner was working on some numbers at home and his nephew, 9-year-old Milton Sirotta, spotted this and said something to the effect of ‘that looks like a google’. I’m not convinced – I just can’t see why Kasner would bother to write that on a blackboard, and think it’s more likely he just asked his nephew, what you call a really, really, REALLY big number (say 1 with 100 noughts after it)’ and up pipes Milton ‘a google.’
But this urge to give names to large numbers dates back even further than Buddha. Right back, in fact, to Archimedes, who was born some time around 287 BC. In one rather strange little book, Archimedes made an attempt to count the number of grains of sand that would fill the universe. This clearly wasn’t a practical proposition. But the Greek number system was extremely clumsy, and what Archimedes seems to have been doing in his book, The Sand Reckoner, is to demonstrate that it’s perfectly possible to extend the system as far as you like.
Archimedes starts the book, addressed to King Gelon of Syracuse, by pointing out that some people reckon that there are an infinite number of grains of sand in the world. What he meant by this was not literally infinite, but an uncountable and incalculable number. And then, perversely, Archimedes set out to show this wasn’t true – and just to rub it in, calculated how many grains it would take not only to cover the Earth, but to fill the whole universe. It’s an impressive feat, though we need to be a little careful about what we mean by universe. Archimedes had in mind a picture of everything that had the Earth at the centre of a number of spheres. Around us went the moon, the sun, the planets and finally the stars. So the ‘universe’ he would fill with sand was more like our idea of the solar system – even so, quite a size.
Tantalisingly, Archimedes goes a little further. He imagines an even bigger universe, suggested by an off-the-wall theory around at the time that instead of the sun moving around the Earth as it obviously does, the Earth moved around the sun. It’s tantalising because Archimedes’ passing mention of this theory of Aristarchus is the only surviving reference to the earliest known person to spot that we move around the sun rather than the other way around.
After making a few assumptions like ‘the diameter of the earth is greater than the moon, and the diameter of the sun is greater than the earth’ (many close to the truth) and some elegant geometry, Archimedes comes to the conclusion that the universe is no more than 10 billion stades across. That’s a measurement based on a stadium, just as we often estimate in football pitches, at around 180 metres, so it probably makes his universe 1,800 million kilometres, which at just outside the orbit of Saturn really isn’t bad. I say probably because stades were a local measure – a stadion was the distance around your local stadium and didn’t have a universally accepted value. But it’s order of magnitude correct.
Archmedes then begins to work up from a grain of sand, through a poppy seed to greater and greater size. He had a bit of a problem, though, as the biggest number in the Greek system was a myriad – 10,000. Archimedes set up a whole supersystem. Everything up to a myriad myriads was the first order. Multiply that by itself and you got the second order. And so on. Do that a myriad myriad times and you’d reached the end of the first period. And so on one again. With a few rules to make this ‘new maths’ work, he was able to put the number of grains of sand in the universe as less than 1,000 units of the seventh order (1051), or in Aristarchus’s bigger universe, less than 10 million of the eighth order (1063). Even by today’s standards these are pretty big numbers.
Now Appearing is the blog of science writer Brian Clegg (www.brianclegg.net), author of Inflight Science, Before the Big Bang and The God Effect.
Source:
http://brianclegg.blogspot.com/2025/02/a-short-infinite-series-2-big-numbers.html
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