On Irrational Numbers and Deep Thinking

Today is March 14th, which to us math nerds is also known as Pi Day, in reference to the first three significant digits of the mathematical constant pi (3.14).

Part of what makes pi interesting is that it’s one of the most famous irrational numbers, meaning that it cannot be expressed as the fraction of two whole values (though 22/7 comes pretty close). In honor of Pi Day, I thought I would learn more about the history of irrational numbers, so I turned to one of my bookshelf favorites, The World of Mathematics: a four-volume history of math, edited by James R. Newman, and published in a handsome faux-leather box set in 1956. (I picked up my copy at a used book sale five years ago.)

The first volume contains an extended essay (originally a short book) titled “The Great Mathematicians,” written by Herbert Western Turnbull, the late Scottish algebraist. Turnbull dedicates much of the essay to great innovations from ancient Greek mathematics. It was Pythagoras (570 – 495 BC), he notes, who is most often credited for discovering that irrational numbers exist.

We know something of the proof that Pythagoras used due to a later account by Aristotle. The proof is elementary by modern standards (indeed, it’s a common example in undergraduate-level discrete mathematics courses). It goes something like this…

Consider a square with sides of length one. Applying the Pythagorean Theorem (which, of course, was brand new in Pythagoras’ time), it follows the length of the diagonal of this square is the square root of 2. It is this common value that was shown to be irrational.

To make this claim, let us assume, for now, that the square root of 2 is a rational number. We can show this will lead to a logical contradiction. This march towards a contradiction unfolds as a rapid-fire sequence of basic number theory and algebraic claims:

  • If the square root of 2 is rational, then it can be written as x/y, for two whole numbers x and y that share no factors in common, which implies…
  • that if we square both sides, then 2 = x^2 / y^2, and therefore, x^2 = 2 * y^2, which implies…
  • x^2 is even (since it is expressed as 2 times another whole number), which tells us that x must also be even, because if you square an odd number you would get an odd number, which implies…
  • x^2 = (2k)^2 = 4k^2, for some whole number k, which, after doing some algebra, tells us that y^2 = 2k^2, which implies…
  • that y^2 is even, and therefore y is even (by the same arguement that we applied to x^2).

We have now shown that both x and y are even. But earlier we assumed they had no factors in common. If they are even, they would have a factor in common (namely, 2). This is a contradiction! Math cannot have contradictions, so our original assumption that the square root of 2 is rational must have been wrong.

We might think of this proof as easy, but as Turnbull argues, the result should not be dismissed:

“That will ever rank as a piece of essentially advanced mathematics. As it upset many of the accepted geometrical proofs it came as a ‘veritable logical scandal.'”

The discovery of irrational numbers turned out to be more than just a logical scandal, it also caused theological issues. Pythagoras’s cult had been convinced that everything in the universe could be reduced down to whole numbers. The idea that values existed that could not be expressed with whole numbers alone destabilized this understanding of existence —  spawning a number-theoretic crisis of faith.

It’s worth briefly visiting this history of irrational numbers because it’s interesting, and because today we celebrate one such value. But for anyone who shares my interest in cultivating a deep life, this story holds a more compelling layer. It reminds us that there was a time, almost 3000 years ago, when a select group of lucky ancient philosophers could build an entire system of meaning out of simply thinking hard and then marveling at what they discovered.

The mind, when properly cultivated, really does contain multitudes.

20 thoughts on “On Irrational Numbers and Deep Thinking”

  1. I just spent two hours today working on proof by contradiction problems while preparing for my discrete math final and then you posted this. It feels really weird being able to semi-understand the square root of two is irrational proof now…

  2. Happy ? day!
    Can you give an update as to how you’re committed to a deep life nowadays and what actions you could take to go a step ahead? Do you feel like you can go a step ahead? Do you need to?

      • To rule out more shallow work and limit your time and energy even more to the most important and challenging aspects of your work and to train your ability to concentrate even more… basically just doing more deep work which is deeper (more intense and focused) in the hope of getting greater outcomes in the form of success in one’s career and wellbeing from a deep lifestyle.

  3. I have a somewhat academic question for you Cal: Since you professionally prove theorems for a living, what advice would you give to people trying to get better at writing proofs?

    My only thinking on this is that much of proof writing is being able to see patterns, and the only way to get better at that is to do lots of proofs. Perhaps you have more specific advice?

    • By far the most important activity for improving your ability to prove theorems on a given topic is to study and learn other peoples’ theorems on the same topic (where, by “learn” I mean understand to the point where you could teach them). This is to mathematicians what time in the batting cage is to professional baseball players.

      • That’s for discovering proofs. For writing up proofs (to submit to journals), you also need to be able to write, period.

        In graduate school, one of the guys I knew gave his defense, and he was told afterwards to rewrite his thesis from scratch, it was so bad.

  4. Brilliant. Can you help to uncover the reason why humanity living thousands of years ago were able to go beyond reasoning and marvel when today we have to spend our time wondering where we are going to obtain our next 12-pack of toilet paper?

    • Well, everyone living in today’s world should be able to prove that the numbers of toilet paper rolls desired is not a rational number.

  5. Funny how at one time the mere discovery of irrationals caused a number-theoretic crisis. It makes me wonder which layers of assumed knowledge we’re currently sitting on 🙂

  6. I bought that exact same 4 book edition at a used book store 15 years ago for about $10. One of my most prized possessions. There are so many real hidden gem articles written by some of the greatest minds of all time.

    • Yes, but you also need to remember that it’s an old book (series). About 15 years ago, a colleague came up to me and asked, “Did they ever solve the Four Color Theorem?”

      I wasn’t sure whether he was joking. When he said he was reading these books, I decided he wasn’t.

  7. The Pythagoreans were upset when they found out that the square root of 2 was irrational. They never seem to have found out that their favorite non-integer (the golden section, 0.618…) wasn’t rational, either.

  8. Here’s a proof that the square root of 10 is irrational.
    Suppose to the contrary that it is rational, namely m/n.
    Then m^2 = 10*n^2. Now every square ends in an even number of zeros, including n^2. But then 10*n^2 must end in an odd number of zeros, contradicting that m^2 ends in an even number of zeros.

    This argument works for decimal, but it also works in binary, proving that the square root of 2 (which is 10 in binary) is irrational.

    Wait, wait. Does it work in base 4? Clearly not since the square root of 4 is 2.

    What’s wrong with this argument?

    Here’s the problem. In base 4 the square of 2 in base 4 is 10, which ends in an odd number of zeros.

    So only some bases have the property that every square ends in an even number of zeros.

    Problem: prove that both 2 and 5*2 have that property.

  9. Is someone so kind to help me verify some ideas about irrational, transcendental numbers? I tried to find an example irrational number that has this property: on any possible approximation (like series), to calculate the first N decimals you need an arbitrary high number of decimals, that you cannot predict from N. I suspect all transcendental non-Liouville numbers have this property, however my math is kind of weak.

    For example, you can predict how many decimals you need to approximate first N decimals of the square root of 2: I think 2*N decimals should be enough. For some other numbers, maybe even PI, it’s harder to predict how many additional decimals you need to calculate the first N decimals. I would like to find a case where it is proved that you cannot predict an upper bound of additional decimals for calculating the first N ones.

    Do you know more about this? I need this info for this:


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