Big numbers

When you move beyond trillions, there are some extremely mind-bending numbers, says Richard Fisher. Some of them are too large to fit in the mind – or even within the known Universe. What's the biggest number you can think of? When I was a child, it's the kind of question we'd ask each other in the school playground. Someone would say something hopelessly naïve like "a billion billion billion", only to be outstripped by a peer who knew about trillions, squillions or kajillions (it didn't matter if only one of those is real).

Eventually, someone would remember that they knew the winning answer: "infinity!" But the smugness was short-lived. Another kid – with a mathematical mic drop – soon pointed out that they could beat it, with "infinity… plus one".

Trying to imagine and understand very big numbers, however, is more than just playground game. It's a task that mathematicians have thought about for centuries. They've proposed the existence of numbers that are so enormous that no human being has ever successfully brought them to mind in full, let alone written them down. And as for infinity, it turns out there is more than one of those – and, counterintuitively, some infinities are bigger than others.

Let's start with an obvious point that was lost on my 10-year-old self. There is no specific number you could describe as the biggest, since natural numbers are infinite. You can't win the playground game. However, that doesn't mean that all the big numbers have been thought of, expressed, written down… or even represented by computers.

First let's climb up the ladder of numbers directly beyond those used in day-to-day life. In news headlines, the biggest numbers – of national debt, for instance – tend to be expressed in the trillions. But there's a hiererchy of ever-bigger numbers that come afterwards, the names of which rarely get mentioned. It starts with quadrillions, quintillions, sextillions and so on. A quadrillion (the US version) has 15 zeroes, a quintillion has 18, and a sextillion has 21.

These numbers are enormous. The human body has around 30 trillion cells – so to get a quadrillion cells in a room, you'd need 34 people. And quintillions only really come into play if you want to talk about, say, how many insects there are on Earth (around 10 quintillion). The number sextillion, meanwhile, is so big that a tower of sextillion people would be 180,000 light years tall – bigger than the diameter of the Milky Way.

You can keep going up to a centillion, which has 303 zeroes in the US version (and beyond, with duocentillion, trecentillion, but these are less standardised). Realistically, only physicists and mathematicians would have much use for a centillion, and even then, only in specialist fields like string theory. If Elon Musk wanted to become a centillionaire, he would have to earn his current wealth every millisecond for the next 1.7 x 10^282 years – a number 283 digits long. 

Googols and googol plexes

Another big number, which is not as big as a US centillion, but perhaps better-known, is a googol. This a one followed by 100 zeroes – 10^100, and also happens to have provided the inspiration for a well-known search engine. Google's founders were drawn to it because it gave a nod to the vast amount of information found online. However, so far the internet isn't anywhere near that big: to date, the Internet Archive's Wayback Machine has indexed only 801 billion web pages since the 1990s.

It's possible to supercharge a googol by making it into a googol plex (the name of Google's California headquarters.)  This number is 10 to the power of a googol – or 10 to the power of 10 to the power of 100.

To get my head around just how big this is, I spoke with the mathematician Joel David Hamkins of the University of Notre Dame in the US, who writes a newsletter about enormous numbers and infinity called Infinitely More.

A googol plex, he explains, is a one followed by a googol number of zeroes. How long would it take you to write that down? Well, you certainly couldn't do it in your own lifetime, even if you started when you first picked up a pencil as a child.

To get a handle on just how many digits we're talking about, Hamkins proposes the following thought experiment:

"Suppose that I gave you this printing device: a super-fast printer that would print numbers, and let's suppose, for example, it could print a million digits every second," he says. Now imagine it began printing at the beginning of the Universe, 13.8 billion years ago – or 10^18 seconds. "Even if you're printing a million digits every second, if you let this thing go from the beginning of time, from the Big Bang, you wouldn't even be close, you would have just the tiniest fraction of a googol plex."

Hamkins also points out something intriguing – there are large numbers smaller than a googol plex that cannot be reduced to a simpler notation or a single word, and therefore are "fundamentally beyond our comprehension". They've never been imagined or expressed.

"The only way to say what those numbers are is to say their digits. But even if you printed a million digits every second, since the beginning of time, you wouldn't be able to say those numbers," he says. "So this is an interesting situation, because it means that we have simple descriptions of enormous numbers, but lots of numbers in between are extremely difficult to describe. There are milestone numbers that are simple in terms of their descriptive complexity, but there are these oceans of complexity between them."

Even if you printed a million digits every second, since the beginning of time, you wouldn't be able to say those numbers

However, mathematicians have described numbers even bigger than a googol plex. The most famous is Graham's number.

Conceived of in the 1970s, the mathematician Ronald Graham used it as part of a mathematical proof. He proposed it to solve a problem in a branch of mathematics called Ramsey theory, which deals with how to find order in chaos.

Understanding the maths behind it is a little involved, but the main thing to know is that creating it involves exponentiation to a truly brain-shattering degree. Graham himself explains why in this video for the mathematics YouTube channel Numberphile.

Oh, and you should also know that even if you did try to write it down on paper, there wouldn't be enough room in the visible Universe to fit it in.

What about infinity though? For the average person, infinity seems a straightforward concept – it's not a number, rather something that goes on forever. Whether the human mind is capable of truly understanding it, however, is another question.

In the 1700s, the writer and philosopher Edmund Burke wrote that "infinity has a tendency to fill the mind with that sort of delightful horror which is the most genuine effect and truest test of the sublime". For Burke, the concept evoked a mixture of astonishment and fear; pleasure and pain, both at the same time. And there were few times that people would ever encounter it in the world, apart from in the imagination, and even then they could not truly know it.

However, the following century, the logician Georg Cantor took the concept of infinity and made it even more mind-bending. Some infinities, he showed, are bigger than others.

How so? To understand why, consider the numbers as 'sets'. If you were to compare all natural numbers (1, 2, 3, 4, and so on) in one set, and all the even numbers in another set, then every natural number could in principle be paired with a corresponding even number. This pairing suggests the two sets – both infinite – are the same size. They are 'countably infinite'.

However, Cantor showed that you can't do the same with the natural numbers and the 'real' numbers – the continuum of numbers with decimal places between 1, 2, 3, 4 (0.123, 0.1234, 0.12345 and so on.)

If you attempted to pair up the numbers within each set, you could always find a real number that does not match up with a natural number. Real numbers are uncountably infinite. Therefore, there must be multiple sizes of infinity.

This is hard to accept, let alone picture, but that's what happens to the mind when it tries to grapple with mathematical enormity. Such enormous numbers are a great deal more difficult to understand than a 10-year-old me could ever have imagined.

Richard Fisher