I had not heard of the term incommensurable prior to reading it in the History of Mathematics (Merzbach and Boyer). According to the online dictionary (free dictionary), it means these numbers are “impossible to measure,” or in mathematics specifically, “having an irrational ratio.” This term illustrates the practicality of early mathematics. Merzbach and Boyer assert that early mathematics were used for practical things like measuring. So, instead of saying a number is irrational, early mathematicians more accurately described these numbers as un-measurable.

What fascinated me about the term incommensurable, and the ratios or numbers that fit the definition, is how mysterious and troubling incommensurables must have been to these earlier cultures. I imagine early mensurers, or as we call them today, surveyors, were befuddled that nearly every object had a length or a side that was not measurable or re-creatable. To paraphrase Alex Bellos (Numberphile, Pi), pi “is the simplest possible ratio (circumference to diameter) of the simplest possible shape (a circle) and yet it is this ugly, complicated number,” and it is everywhere. Sure, the length of an incommensurable is exactly equal to a ratio, but that doesn’t help someone who is actually measuring or creating an object.

I imagine one of the King’s workers saying, “Yes, that circular wall on the side of your castle will be about 3 units. That’s as close as I can estimate.” Now obviously, estimating pi to about 3 worked, and worked very well, but it does seem funny none-the-less. Early cultures came up with close approximations like “about 3” or square root of 10, and these were surely close enough for most tasks required of them. Archimedes’ approximation of Pi as 3 + 10/71 was used for more than 1000 years (Numberphile – PI).

The square root of 2 as the hypotenuse of a right triangle. Image: public domain, via Wikimedia Commons.

The square root of 2 as the hypotenuse of a right triangle. Image: public domain, via Wikimedia Commons.

That some early mathematicians avoided incommensurables is no surprise to me. Apparently, Pythagoras was afraid of the incommensurables. Merzbach and Boyer describe the Pythogorean belief as possessing the tenet, “that the essence of all things, in geometry as well as in the practical and theoretical affairs of man, is explainable in terms of arithmos, or intrinsic properties of whole numbers and their ratios.” We now know that whole numbers cannot even describe something as simple as the diagonal of square. I imagine the Pythagoreans believed they simply had not discovered or understood something about numbers, shapes, and math. To the Pythagoreans, incommensurables probably made the universe seem poorly planned or unorganized. However, I think the opposite view could also be taken. The universe was so delicately planned or created or happened in such a way that every object is made up of an unmeasurable number. How planned and organized and amazing is that? That is way more amazing than a universe based on whole numbers.

I am awestruck by the fact that we live in a universe where nearly every object has some incommensurable aspect to it? We can exactly represent the incommensurables or irrationals in numbers, thanks to Math, but we cannot measure them. This may be a leap, but to me, that means the universe is unmeasurable. There is something satisfying about living in an unmeasurable universe. Of course, if I were a god or the master of all creation, I think I would do the same thing. I would create objects that could never be recreated exactly the way I created them, and every object would have this small mystery to it. I love a universe that is unknowable. A Pythagoreans universe would be boring and unpleasant.

In the end, however the universe came into being, the existence of incommensurables is one convincing reason to believe that humans did not create math. If humans created math, I doubt they could have ever thought to have such numbers or for them to be so pervasive. Score one point, for math existing, not being created.

Also, on your next date, remember to tell the person you are with that they are incommensurable. It is not synonymous with irrational, which would be an insult. Being incommensurable means they are mysterious, unmeasurable, and a perfect ratio that is undefinable any other way.


History of Mathematics, 3 ed., Uta C. Merzbach and Carl B. Boyer
incommensurable. (n.d.) The American Heritage® Dictionary of the English Language, Fourth Edition. (2003). Retrieved September 14 2014 from

Pi – Numberphile


Bellos, A. Here’s Looking At Euclid.

tags: incommensurable, irrational, transcendental, pi, sqrt(2)

5 thoughts on “Incommensurability

  1. Pat Ballew

    You have a really “friendly” writing style and I think it would be interesting and entertaining to students; so well done. In statements like ” Archimedes’ approximation of Pi as 3 + 10/71 was used for more than 1000 years” you may give a false impression, and perhaps not emphasize a really BIG idea. Archimedes approximation that pi was between Archimedes’ approximation of π as 3 + 10/71 was used for more than 1000 years 223⁄71 < π < 22⁄7, was used well into the 20th Century in middle school textbooks, but it was not the most accurate known approximation, it was just handy. Ptolemy,figured out 377/120, which is accurate to the first three decimal places; and if you truncate, you get a really good approximation 3.1416. What was really nice, the big idea to me, about Archimedes approximation is that for a really long time, anyone who approximated pi used HIS approach, inscribing and circumscribing a circle with ever larger regular polygons. When Ludolph van Ceulen (1540-1610 A.D.) approximated pi, he used the Archimedean method for a polygon with 1560 sides to calculate π to 35 decimal places. Ludolph’s 36-digit so impressive that π was called (and is still sometimes referred to ) in Germany as the Ludolphine Number. He was so proud of his accomplishment that he had it inscribed on his tomb. A new memorial in Leiden, I think the old one was destroyed, still has his estimate on it.

    As an afterthought, I would add that if you attempt to replicate the 96-gons that Archimedes, you will see that his approximations were actually formed on other approximations of irrationals, specifically, the square roots of 2 and 3 which come us as you double the number of sides. Viete's infinite approximations of pi with nested radicals is obtained by the same process.


    1. ineedmath Post author

      Thank you for the additional history, clarification, and information. I will definitely investigate your suggestions.


  2. benbraun

    Here is an interesting question to ponder: do the real numbers actually exist? For that matter, are there ACTUALLY infinitely many integers? What if what we visualize as the number line of integers is actually a giant number circle of integers, say Z/pZ (this is the set of integers “mod p”) for some unimaginably large prime p? What if what we perceive as a continuum is actually just a huge number of small discrete steps, so small that we can’t perceive them?

    This is more than just idle philosophizing. There are lots of very interesting applications in mathematics that rely on the idea that the reals are not always the best model for the physical world. For example, at the beginning of Audrey Terras’s lovely book “Fourier Analysis on Finite Groups and Applications,” she has the following quote:

    “It is unfortunate that so many scientists have been conditioned to believe that, say, 10^30 particles can *always* be well approximated by an infinite number of points” — D. Greenspan, 1973

    You might also find it interesting to read Doron Zeilberger’s opinion on this matter (he’s a professor at Rutgers): see Opinion #108 here


    1. ineedmath Post author

      Yes, I will investigate those further. I am beginning a project on Cantor and enjoying how slippery or mysterious both infinity and the number line appear to be.



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