What’s the biggest number you can think of? A googol? TREE(3)? Fourteen?
How about an infinite number of nines? No, we’re not being facetious – in higher math, that’s not just a legit possible answer, but it’s also a really bad one. Why? Because an infinite number of nines is actually equal to negative one. It’s not very big at all.
Okay, let us explain.
A crash course on counting
Unless you’re reading this from ancient Babylonia or the northwest of Papua New Guinea, your go-to number system is practically guaranteed to be base ten, or decimal.
What that means in practice is – well, it’s something so basic that you may not have really thought about it since you were first learning your numbers back in pre-school. We count using ten individual symbols – 0,1, 2, 3, 4, 5, 6, 7, 8, and 9 – and when those run out, we just go back to the start again, this time with the prefix “1” to remind ourselves that we’ve already finished the whole set once.
If we continue counting like this, we may find ourselves having to change that prefix to a “2”, a “3”, and eventually, a “9”. We may even reach 99 – meaning we’ve counted all ten digits ten times – and at that point, we just level up again, adding a second “1” prefix and starting once more at 100.
Now, that’s the first principles explanation, but it’s not really the most useful one – so, instead of thinking about our numbers in terms of adding, let’s move to multiplication. How should we understand a base ten system now?
In fact, it’s rather beautifully straightforward. A number like, say, 12,345, can be thought of as the sum of five numbers: 1 × 104, 2 × 103, 3 × 102, 4 × 101, and 5 × 100 – those indices, or powers, above the 10s referring to how many times we multiply 10 by itself. It’s both a lot easier than plussing one twelve thousand three hundred forty-four times, and also lets us do something pretty cool – it allows us to enter the world of decimal fractions.
More than zero, less than one
Ever since two cavepeople first looked at a single hunk of delicious mammoth meat and decided there were better solutions than warfare to solve their problems, we’ve needed a concept of fractional quantities. And for rather a lot of Western history, these fractions – numbers like 1/2 or 1/3, formed by setting the number of things you want to divide on top of the number of things you want to divide by – was the only way we had to express them.
Other cultures, though, were smarter about it. In China, the Islamic world, and even ancient Mesopotamia, mathematicians had figured out a way to express fractions as decimal expansions – and now that we’ve seen how to create decimal numbers, we can too.
So: rather than confining our indices to non-negative numbers, what if we extend them to include numbers less than zero? In other words, what is 10-1?
In fractions, this is simple once you know the rule: when you see a minus sign in the index, it just means “one divided by this number”. 10-1 is equal to 1/10, for example; 10-2 is equal to 1/102; 10-163 is equal to 1/10163.
But what if we chose instead to take our way of writing base ten numbers and extend it in the opposite direction? Basically, if 12,345 means 1 × 104 + 2 × 103 + 3 × 102 + 4 × 101 + 5 × 100, then we can stick in a decimal point after the 100 term and continue for as long as necessary: 12,345.6789, for example, would then be equal to 1 × 104 + 2 × 103 + 3 × 102 + 4 × 101 + 5 × 100 + 6 × 10-1 + 7 × 10-2 + 8 × 10-3 + 9 × 10-4.
And the cool thing about these numbers is that they can go on literally forever – with some unexpected results. One of the best known counterintuitive facts is that 0.999… – that’s math-speak for “an infinite number of 9s, but we don’t have time to write them all out” – is equal to 1.
No, not approximately equal. Literally, exactly, equal.
What are 10-adic numbers?
Now, you might have seen “0.999… = 1” and thought, “no, that’s weird enough for me” – and you know what? That’s fair. But if you, like number theorists, hunger for more, then allow us to introduce you to the world of 10-adic numbers.
The basic idea behind 10-adic numbers is this: what if, instead of extending decimal numbers indefinitely to the right like this, we extended them to the left? Is there some way in which a number like, say, …999 – that is, 9 + 90 + 900 + 9000 + 90,000 and so on ad infinitum – even makes any sense?
Well, let’s take our cues from repeating decimals. Remember how 0.999… = 1? We can prove that using the following neat little argument: first, note that
0.999… × 10 = 9.999….
Then, subtract one from the other to see that
0.999… × 10 – 0.999… × 1 = 9.999… – 0.999…
= 9.
In other words,
0.999… × (10 – 1) = 0.999… × 9 = 9,
which means that 0.999… = 1.
Got that? Great! Now let’s apply the same argument to 10-adic numbers and see what happens. We have:
…999 × 10 = …9990.
In this case, subtracting one from the other gives us
…999 × 10 – …999 × 1 = …9990 – …999
= -9
But that means
…999 × (10 – 1) = …999 × 9 = -9,
which means that – hang on, that can’t be right:
…999 = -1.
Weird math
Okay, at first glance, it seems extremely obvious that a never-ending row of nines simply cannot be equal to negative one. But we dare you to prove us wrong – look:
.…99999
0000001 +
Try adding the two together, and you’ll be locked into an infinite row of repeating zeroes – in other words, zero itself. If …999 + 1 = 0, then …999 = -1.
Now, it’s tempting to stop here and say “well, if that’s what math says, then obviously math is nonsense.” But is there a way we can make it make sense? In fact there is – and, just like when we leveled up to discover decimal expansion, it takes some lateral thinking to get there.
So: let’s think about what we mean when we say “close to zero”. Normally, we’re thinking additively, right? The number 5 is the size it is because it is five unit-lengths away from zero; the number 13 is thirteen unit-lengths away from zero; the number 18.75 is eighteen and three-quarter unit-lengths away from zero.
It’s a very natural way of thinking about distance, and we’re going to completely forget about it from now on. Instead, let’s think multiplicatively, and consider numbers in terms of their factorizations by 10.
Let’s start with zero, since that’s the number we measure from. In the world of integers, zero is the most divisible number of all – you can literally divide any number into it, and it will work fine (just don’t try it the other way around).
Now let’s consider a couple of examples: let’s take, say, 30, and 1,000,000. We can divide 30 by 10 just once before we leave the integers – 1,000,000, on the other hand, can be divided by 10 six times. So, in a sense, we can say that 1,000,000 is “closer to zero” than 30 – it is more zero-like in terms of divisibility.
It may not seem intuitive, but you can – and people have – prove that it is a perfectly sensible mathematical metric. All we have to do is set up the following rule:
If a can be divided by ten a maximum of k times and remain an integer, then set |a|ten = 1/10k.
Applying this to …999, we can now see that the number really is getting closer and closer to negative one. After all, we have:
|9| = |10 -1| = 0.1 – 1 = -0.9
|99| = |100 -1| = 0.01 – 1 = -0.99
|999| = |1,000 -1| = 0.001 – 1 = -0.999
|9,999| = |10,000 -1| = 0.0001 – 1 = -0.9999
And so on. We can extend the idea to fractions, too: for example, we can get 1/3 by asking “what 10-adic number, when multiplied by 3, results in 1?” We’ll spare you the working, but you can check for yourself that the answer is …6667 – that is, an infinite number of sixes followed by one final seven.
Everything is looking great, right?
A problem
What would you say is the “first” fraction? Like, the most basic one? A half?
What’s one-half as a 10-adic number?
By the same logic as above, we’re looking for a 10-adic number which, when you multiply it by two, gives us one. And here’s the problem: no such number exists.
The same problem turns up if you try to divide by five – and you may now be getting an idea of what the reason is. Basically, 10-adic numbers suffer from the problem that 10 is non-prime: it’s equal to the product of 2 and 5, which leads to a weird situation where you can multiply two non-zero numbers in the 10-adic system and get a result of zero.
The solution is, in true number theory style, to simply only consider prime numbers as the basis for your p-adic numbers – p here stands for “prime”. By doing that, you avoid these issues and can use these infinitely long numbers to your heart’s content.
But why would you? Well, p-adic numbers may be weird, but they’ve been used to solve some of the most famous problems in mathematical history – namely, Fermat’s Last Theorem. They’re useful for both ancient problems, like Diophantine analysis, and modern ones like quantum mechanics; mostly, they’re just a neat way to look at problems that would be much more difficult using our normal, boring number system.
Or, to put it more poetically by quoting the Japanese mathematician Kazuya Kato: “Real numbers are like the sun, and the p-adics are like the stars. The sun blocks out the stars during the day, and humans are asleep at night and don’t see the stars, even though they are just as important.”
