Ever wondered why the people at Google spend so much time trying to calculate eleventy-billion digits of pi? Why a number like one third can be so simple to write as a fraction, but impossible to jot down in decimal form? Or why there’s a number that’s literally just called “e”?
Well, we have the answers – but be careful: things are going to get a little bit irrational.
What are irrational numbers?
Whether or not you’ve heard the term “irrational numbers”, you’ll almost certainly know some examples already. Pi, , is probably the most famous of all: the ratio between a circle’s diameter and its circumference. Then there’s e – not the letter, but , seen pretty much anywhere where people are measuring growth or decay, writing down the , or (appropriately enough) .
The square root of two, aka √2, is irrational too, as is its sort-of (we’ll get to that) neighbor √3. The square root of four, however, definitely isn’t. An infinite decimal expansion of any number – 0.111…, 0.222…, 0.333…, etc – none of those are irrational. But 0.123456789101112…? So irrational.
How do we know? Well, irrational numbers – and their opposites, the rational numbers – are distinguished by just one property: can they be written as a ratio of two whole numbers?
“The most simple numbers we know are just the whole numbers: one, two, three, four, and so on,” explains number theorist (and Fields medal winner!) James Maynard in a 2019 video for .
“Slightly more complicated [are the] rational numbers, which are just ratios or fractions of whole numbers,” he continues. “So, one half, one third, two thirds…”
Given this explanation, you may not be surprised to learn that an irrational number, then, is one that can not be written as a fraction of two whole numbers. Take pi, for example: there’s a chance you may have seen it expressed as the number 3.14, or if you’re feeling particularly , the fraction 22/7. Neither of those are accurate, though – in fact, there’s only one way to write the value of pi exactly, and it’s like this: π.
“We’ve that you can’t write it as just a ratio of two whole numbers,” Maynard says. “And in fact[…] in many ways most numbers can’t be written as just a ratio of two whole numbers.”
What do irrational numbers look like?
So now we know what irrational numbers are – and what they’re not – how do we go about spotting them in the wild?
Well, a good clue is when – like pi, e, phi, tau, and so many others – they aren’t written using numbers, but letters (albeit quite often Greek ones). That’s not always the case though: physical constants are , and they irrational, since they come from nature, rather than logic.
Another tell-tale sign that a number might be irrational is when its decimal expansion is followed by a dot-dot-dot. “What [an] expression [like 3.14159…] is saying is: I have lots of different approximations to the number pi, which get more and more accurate,” Maynard says.
“So maybe a first approximation is that pi is roughly equal to three, and then a second approximation is pi is approximately equal to 3.1, which is 31/10, and then a better approximation is that pi is approximately equal to 314/100, 3.14, and so on,” he explains. “This is really what we mean by an irrational number; we can’t just uniquely define it in a very easy way typically, but we can define it by the sequence of simple approximations.”
But again, this is far from foolproof. Any decimal expansion with a repeating pattern, for example, is rational: 0.333… is equal to 1/3, so it’s rational; 0.857142857142857142… is equal to 6/7, so it’s rational; 1.982456140350877192982456140350877192982456140350877192 – it repeats every 18 digits, if you can’t find it – is equal to 113/57, so again, it’s rational.
Square roots are often irrational – in fact the square root of any prime number is proven to be so – as are most general roots. Obviously, again, this is not always true: the square root of any square number is rational, for example, as is the nth root of any nth power.
And then there’s the handful of seemingly random examples that mathematicians have picked up along the way – many of which have very specific notation that comes directly from how they were designed in the first place. But the vast, vast majority of irrational numbers don’t have any particular notation to distinguish them – so the only way to know for sure whether they’re irrational or not is to prove it with your bare hands. That’s pretty difficult, which is why some numbers out there that you’d really think must be irrational are technically .
But why wouldn’t we bother giving them all handy names, like pi? Well…
How many irrational numbers are there?
All infinities are infinite – but . And unfortunately for anybody setting out to list all the irrational numbers, there are… well, there are literally too many of them to count.
“Everyone understands – or pretends to understand – the distinction between being finite and being infinite,” says Justin Moore, a math professor at Cornell, in Quanta Magazine’s podcast. “[But] for an infinite set, you can make further distinctions. You can distinguish between it being what’s called countable and then what’s called being uncountable.”
For those who haven’t studied math past high school, the name might be misleading – after all, how can something infinite be “countable”? But the term “just means that you can assign a natural number to each element of the set so that no natural number gets used twice,” Moore explains. “So the natural numbers [that is, whole positive numbers] are obviously countable because they count themselves.”
But what about the rational and irrational numbers? At first glance, it seems obvious that there are more rational numbers than natural numbers – after all, you have an infinite number of potential numerators, and for each one of those, an infinite number of denominators too! But here’s the surprising thing: the two sets – natural numbers and rational numbers – are actually the same size.
“That’s actually pretty easy to see when you, when you think about it,” Moore says, “because you can just list all fractions with denominator 1 – or numerator and denominator absolute value at most 1. And then, at most 2, at most 3, at most 4. And at each stage, there are only finitely many fractions where the numerator and denominator are at least in magnitude at most n. And then you can exhaust all of the rationals that way.”
But irrational numbers? Well, that’s a different story: “the real numbers, the set of decimal numbers, are uncountable,” Moore explains. “If you hand to me a list, a purported list of all the elements on [the number line], there’s a procedure called the diagonal argument, which allows you to produce a new point that’s on the line, but not on your list.”
What does this mean, in practice? Basically, when you’re looking at the number line and wondering “what percentage of that is irrational numbers?” the answer is, for all intents and purposes, 100 percent. The fact that literally infinitely many points along it are rational doesn’t really matter – there are still so many more irrational numbers there that, in the grand scheme of things, the rationals may as well not be there at all.
Why should I care about irrational numbers?
If this is all sounding a bit abstract and pointless, then – well, . But just because we’ll never be able to write down the exact value of some irrational number using digits, that doesn’t mean they’re not important: “pi for example was historically very important for engineers to do certain bits of architecture,” Maynard points out, “and they needed to know it to a certain level of precision.”
So, what do we do? “To a given number position you can just approximate it by some rational number,” he explains. And you’d be surprised at just how few digits you need to know: NASA never goes above 15 decimal places for pi, for example, even when they’re calculating the intricacies of interplanetary navigation. In fact, even to measure the size of the entire known universe at an accuracy equal to the diameter of a hydrogen atom, of the constant.
Why, then, you might ask, do we in fact know the value of pi to some ? Honestly, the answer is… because we can.
“It’s a computational challenge,” David Harvey, an associate professor at the University of New South Wales, told . “It is a really seriously difficult thing to do, and it involves lots of mathematics and these days computer science.”
“You do pi because everyone else has been doing pi,” he added. But as we’ve seen, the dessert-sounding constant is just one wave in an infinite sea of irrationality – maybe it’s time for those computer scientists to turn their attention to some of the less loved numbers out there.
“There’s plenty of other interesting constants in mathematics,” Harvey pointed out. “If you’re into chaos theory there’s Feigenbaum constants, if you’re into analytic number theory there’s Euler’s gamma constant… [there’s] e, the natural logarithm base, you could calculate the square root of 2. Why… do pi?”
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