Irrational Numbers: The Unobservable, Indescribable Foundations of Physics
Real numbers are not real, but irrational numbers are not irrational.
Hippasus of Metapontum was a Greek man who lived in the fifth century BC. In a world convulsed by war and violence, he found a profession that must have seemed blissfully safe — mathematics. While the three hundred Spartans made their last stand at Thermopylae, we can imagine Hippasus sitting comfortably at his desk studying his triangles and dodecahedra. This is the life, he must have thought. Only too late would he discover a surprising truth — mathematics is not for the faint-hearted.
For tradition says that Hippasus’ mathematics cost him his life.1 As a follower of Pythagoras, Hippasus was dutifully studying the side lengths of right-angled triangles. According to Pythagoras’ famous theorem, if you draw such a triangle with two sides that are each one unit in length, the third side will have a length equal to the square root of two. Hippasus’ fateful discovery was that the square root of two is a peculiar number as it cannot be expressed as a fraction. In ancient Greek parlance, it was alogos — inexpressible or indescribable. In modern parlance, it is irrational. The existence of irrational numbers was such an affront to Pythagorean orthodoxy that it is said that when Hippasus revealed his work to his comrades, they swiftly had him drowned for impiety.
What should we make of this response? Today, it seems not just extreme but also a little odd. Irrational numbers are a routine part of modern life, found everywhere from school classrooms to construction sites to GPS systems. Far from their unholy power bringing the world to chaotic ruin, they have proved vital to the scientific developments that have built the modern world. Yet this should not blind us to the truth that was so evident to the ancient Greeks — there is something very strange about irrational numbers. The more closely we examine them, the more we may find ourselves wondering if believing in them is indeed as irrational as their name suggests.
From Counting to Rationality
Long before there were mathematicians and theorems, people used their fingers to count. The numbers that arise during counting — 1, 2, 3, 4, … — are called the counting numbers. For countless millennia, these numbers sufficed to describe the natural world of plants, animals and the passing of days. But civilisation brought new challenges.
The progression of arithmetic through history can be understood as a process of introducing new numbers to meet these new challenges. Money and accounting naturally led to the idea of negative numbers to describe debt. If a person has $5 and spends $2, we can easily calculate that their remaining wealth is $3 because 5-2=3. But what about a person who has $2 but spends $5? Their wealth is 2-5, which is not equal to any counting number. How can we deal with this? We simply invent a new type of number — negative numbers! Since 2-5 is the opposite of 5-2, we call its solution “-3”. Indeed, if we insist that every subtraction of counting numbers must have an answer, then we end up with the integers — the positive and negative counting numbers together with zero.
Next comes fractions, which address the need to fairly distribute limited resources. If six fish are to be shared equally between three people, we can divide six by three to determine that each person should get two fish. But if only five fish are to be shared between three, then we must divide five by three which cannot be solved by any integer. How can we deal with this? Invent new numbers of course! We declare that there is a number that equals “5÷3” and call this number “5/3”. If we assert that there must be a number that is equal to the division of any two integers, then we create the rational numbers. These are all of the numbers that can be written as fractions or, equivalently, as an exact decimal (either terminating or recurring).
This is a lot of new numbers to have invented! We may wonder whether a prehistoric hunter-gatherer would reasonably have been as just as affronted by the unwieldy smorgasbord of fractions as the Pythagoreans were by the heresy of irrational numbers. Yet, rational numbers have important properties that make them orderly. Every rational number can be interpreted as a representation of an imaginable real-world scenario; 4/3 describes four pies shared between three people, -1.45 encapsulates the finances of a person who owes a lender $1.45, and so on. Moreover, every rational number can be written down on paper, entered into a calculator or described in words; they are precise concepts with which we can calculate.
It is a long way from counting numbers to rational numbers, but the progression is fundamentally one of natural extension. There is nothing mysterious about rational numbers. They are simply an elegant manifestation of human’s capacity for abstract thought. The same cannot be said for the irrational numbers.
Irrational Numbers: Unknowable and Uncalculable
Like negative numbers and fractions, irrational numbers were invented to answer otherwise unanswerable questions. Hippasus’ example of right-angled triangles with a side length of √2 is one example. Another is the circumference of a circle. If you draw a circle with a diameter of one unit, there is no rational number that measures the length of the path around the circle. How can we fix this? We can invent a new number, called π. π is famous for the fact that its decimal representation continues forever without pattern, but there are infinitely many other numbers with the same properties. Collectively, these are the irrational numbers. The combination of these numbers with the rational numbers is called the real numbers.
Yet real is a strange name, since irrational numbers fail to meet basic criteria for reality. Indeed, they can never be observed in the real world. Whenever we make a measurement, we always use a piece of apparatus that can only give a reading from a discrete set of possibilities. Typically, this is expressed as a decimal. For example, a typical tape measure with millimetre precision can give us readings of 1.414 metres or 1.415 metres, but cannot be used to measure a length of 1.41421356 metres. A more accurate piece of equipment may give a more precise measurement (e.g., 1.41421 metres), but no apparatus could ever measure the infinitely long decimal string that represents an irrational number like √2. This means that even if we construct the right-angled triangle studied by Hippasus and measure its sides, we will never observe the irrational length he inferred it should have.
Of course, we may consider this to reflect our limitations. While we can only measure rational numbers, we may reason that these are merely approximations to the true irrational values. Yet this begs the question of what it is we are approximating. In the real world, there is no such thing as a perfect circle or a true right-angled triangle. If you examine a circle under a microscope, you will always find flaws where the pen slipped or a speck of dust got in the way. If you look closely enough, you will ultimately see that there is not even truly a continuous curve on the page; the ink that forms the circle really consists of a discrete collection of atoms largely consisting of empty space. It is not our measurements which approximate the true geometric perfection of the world. It is the perfect circles and polygons of our imagination that are approximations of the messy reality of the physical world. It is only in this idealised world of our thought that we can see irrational numbers.
Indeed, even in the world of ideas, irrational numbers’ reality is elusive. Even as concepts, they cannot be written or described exactly. A symbol like π can represent an irrational number, but it is useless for calculation since it does not express how the number is related to other numbers. Moreover, the irrational numbers are so numerous that it is not even theoretically possible to devise a system of symbols that could represent each one.2 With this in mind, the best we can do is to approximate irrational numbers with rational numbers that are similar in value, such as 3.14 for π. This is what a calculator does when asked to perform arithmetic with irrational numbers. Yet such calculations can never give exactly correct answers, but instead themselves yield only approximations.
Irrational numbers therefore cannot be seen in the physical world, cannot be written down or spoken of and cannot even be used correctly in calculations. Whatever the nomenclature of real numbers may suggest, we surely cannot avoid the conclusion that irrational numbers are not real in any meaningful sense.
An Indispensible Fiction
So were the Pythagoreans correct to try to suppress irrational numbers? Are they a pernicious fantasy that distracts mathematicians and scientists from the real world? On the contrary, irrational numbers are an indispensible tool without which modern science would not be possible! This is because, while irrational numbers themselves are an unwieldy nusiance, they usefully fill up all of the gaps between rational numbers. This means that accepting irrational numbers transforms the discrete patchwork of points described by rational numbers into the continuous number line of the real numbers.
This is particularly vital to studying motion. Even as Hippasus was uncovering irrational numbers, Zeno of Elea was posing a series of paradoxes that purported to undermine the notion that anything can truly move. They were based on the principle that an object could not pass through the infinite number of points between two places in a finite period of time. The continuous number line solves this problem by allowing for time to be divided into infinitesimally short instants of which an infinity can indeed fit into a finite time. This reasoning about motion ultimately developed into the mathematics of calculus which, since the time of Isaac Newton, has formed the foundation of physics.
In 1960, Eugene Wigner wrote of the “unreasonable effectiveness of mathematics in the natural sciences.” From his perspective, that mathematics has proved such an effective tool for studying the physical world seemed an inexplicable miracle. Yet perhaps it should not be such a mystery. Mathematics is a tool for approximating the natural world by simpler forms, such as basic geometric shapes. These are the same approximations that physicists use to develop models that approximate the behaviour of the natural world. The cost of this approach is that mathematicians were compelled to invent irrational numbers as artificial tools designed to facilitate their abstraction of the world. But the reward was the immense wealth of possibilities of modern physics. All things considered, we must conclude that: real numbers are not real, but irrational numbers are not irrational.
We must say “according to tradition” because the historical basis for this story is quite flimsy. Only some ancient sources report Hippasus’ drowning, and even they are somewhat unclear about whether the discovery of irrational numbers was the reason. This is unfortunate for those of us wishing to tell dramatic stories from the history of science, but it may have been very fortunate for Hippasus who may well have lived happily to a ripe old age after all!
This is because the irrational numbers are uncountable, meaning they cannot be organised into a list, even one which continues indefinitely. This is proven using the diagonalisation argument of Cantor, described here.
My favourite subset of irrational numbers are the uncomputable numbers. Those are the numbers for which no set of instructions can be written down as to how to compute them. By contrast, computable irrational numbers like pi seem very concrete (at least they have formulas!). Most real numbers are uncomputable. This remarkable fact can be seen simply by noting that the entire set of formulas you could imagine writing down is countable. Therefore, there must be uncountably many uncomputable numbers! Some of them can be defined, however (though not computed). I seem to recall from a book of Scott Aaronson's that one example is 'the probability that a turing machine will stop given its own code as input'. From memory, if that probability could be written down it would solve the halting problem, therefore it must be uncomputable!
Very nice post! One minor nitpick for footnote 2: I think calling it 'uncountable' does not explain so much. It is maybe worth pointing to how one would prove that they are uncountable, that is, every possible counting scheme for these numbers can be proven to be incomplete. Therefore, there is no counting scheme for them.