Is there really a way of understanding big numbers?
In this shortened extract from Is That a Big Number? by Andrew Elliott, examines how to create big numbers.
What is a number anyway?
In the beginning were the counting numbers: 1, 2, 3 . . . From that point, people have extended the concept of ‘number’ in more and more ‘unnatural’ ways. Fractions simply inverted the concept of number. If there was one loaf for four hunter-gatherers, well, the whole had to be broken (‘fractured’) into fragments or fractions. The concept of zero was a necessary adjunct to the ‘Arabic’ (really Indian) system of numerals that allowed an open-ended notation for numbers and it may have been used as early as the seventh century CE.
Negative numbers were plainly an absurdity. How could there be a number that was less than nothing, and that, when added to five, gave a result of three? But, in the world of trading and credits and debits, you could see how this backwards arithmetic could work, and work consistently, and be useful. And so negative numbers were admitted into the club…
Those qualities, namely consistency and utility, also lie behind the adoption of imaginary numbers and complex numbers. Many people still struggle with these (the name imaginary doesn’t help — these numbers are no more imaginary than zero is imaginary — like all numbers, they are mathematical constructs). But being consistent and astonishingly useful makes them highly valued newcomers, becoming widely accepted in the eighteenth century.
What about googol and googolplex?
When Edward Kasner was writing his book The Mathematical Imagination (1940), he asked his nine-year-old nephew Milton Sirotta for a name for 10100, a number he was using as an example to show the difference between a very large number and infinity. ‘Googol’ it was dubbed, and although the number has no specific significance, other than its memorability, it has established itself as a truly landmark number. In physical terms, it is more than the ratio of the mass of the visible universe to the mass of an electron.
Once you have a recipe for creating big numbers, you can make as many as you like, bigger and bigger still.
So, the googolplex, coined also by Kasner and his nephew Sirotta, is 10googol. This number could never even be written down in full: the universe doesn’t have space to hold it, or materials to do it.
What about Graham’s number?
The 1980 edition of the Guinness Book of World Records first cited Graham’s number as ‘the most massive finite number ever used in a serious mathematical proof ’. It no longer holds that crown, but it is representative of a class of numbers where even saying what the number is, is problematic. Unsurprisingly, Graham’s number arises from a problem in combinatorics.
As we’ve tackled bigger and bigger numbers, from the small counting numbers, past the comfort zone of a thousand, into the astronomical numbers, we’ve needed to use different strategies to mentally accommodate them. We’ve rearranged the way we think about the numbers, to shift and rebalance the conceptual burden. This has sometimes meant a move from a direct understanding of a number to an understanding of the algorithms needed for creating the number.
For example, when we decide that ‘quintillion’ is no longer very much clearer than 1018, and we switch to using the scientific notation, then we are adopting an algorithm (in this case calculating a power of 10) as our primary way of understanding the number. Graham’s number is an extreme version of this shift. Any explanation of it is almost entirely an explanation of the algorithm that would be used to construct it. It’s a recipe for taking powers of numbers unimaginably many times. Grasping the algorithm is hard enough. The number itself is beyond conceptualising. No one even knows what the first digit is.
Even though the method of constructing the number is well defined, no-one has ever been through the steps — no-one ever could.
At this point, you might ask: is there any way of understanding how big these numbers are? And the answer is no, not really, not directly. The only way is by understanding the process by which these numbers are reached, and expressing the steps in these processes using small numbers from our comfort zone. You might find yourself asking: Is Graham’s number even a number at all? That’s a very good question. All we know is a method to construct it, but it’s a method that could never be followed, as it would require more space and time than the universe provides. But when you think of it, every number we write down, even the £2.50 price of my coffee, is not the number itself, but a method of constructing the number, an algorithm. The difference is just one of the degree of familiarity with the process.
Andrew Elliott grew up in the Eastern Cape region of South Africa, and studied statistics and actuarial science at the University of Cape Town. He emigrated to the United Kingdom in the late 1980s to work on computer systems, eventually becoming a self-employed management consultant. He has co-founded several companies, Open Square in 2003, and Revolutionary Systems in 2014 to explore systems based on flexible models.
Frustrated by how quantitative information is presented in the media and public discussion, in 2016 Andrew started Is That A Big Number?, a project to promote numeracy and the development of intuitive number sense.
