Proof Of Existence Of Square Root Of Two For Ten Year Old (Take That, Euclid!)


Yesterday my ten year old daughter cried like I have not seen her cry since her grandmother died. Tears were running down her cheeks:”Cela n’a aucun sens!” It makes no sense! It was a telling choice of words: the Greek mathematicians made the exact same observation, they called such numbers “irrational”. I reassured her: she was brutally confronted by what drove ancient Greek mathematicians crazy: the square root of two.

How to define square roots algebraically is simple: x is the square root of a if and only if: xx = a… No problem if a is one, four, nine… But put a = 2, and a mystery arises: one can write down numbers ever closer to x = square root of two, but then what? Their squares are never quite two… Worse: ancient Greek mathematicians knew (positive) integers, and their ratios, say m/n, where both m and n are integers. They could demonstrate, in a few lines, that square root of two was not a fraction. 

Mathematics is the way of thinking that is most powerful in some situations. For example, as I wrote: xx = 2, I used no less than three mathematical notions that the ancient Greeks mathematicians did not have: the notion of equation, the equal sign, and the notion of unknown x.  

When the Greeks were confronted to this, they realized that some things existed that should not exist, according to their (mathematical) system of thought. Obviously, their mathematics came short. They discovered “irrational” numbers, the hard way. Further reasonings were halted by the rise of fascism in the Hellenistic and then Roman world. Do we have similar situations nowadays? Obviously yes. Nonlocality in Quantum Physics is an obvious example.

There is a mysterious relationship between numbers and geometry. Draw a line: that’s the x axis. Some distance mark one (1); it is the unit of distance. It does more: it represents the number one. Draw the perpendicular axis, traditionally called the y axis. There too, mark a one. Use the proverbial compass to make the units on the x and y axis the same length. Now one is facing a two dimensional plane. It naturally defines a unit of area, the square of sides equal to one. Cut that square in two to get the area of surface ½, etc… A bit of playing around shows that any square of side s has area ss…

Consider the rectangular triangle from the origin with sides one. The longest side can be used as the base of a square. A quick look shows that this square has area two (2). Thus its side is square root of two. 

This demonstrates that the square root of two exists. 

Ancient Greek mathematicians went that far. And that drove them nuts, as, for them, any “number” had to be of the form m/n, with m and n integers… And the square root of two was not such. 

The reasoning above is mine: it was designed to be understood by a very curious ten year old who is not fanatical about mathematics. So the Pythagorean theorem is demonstrated, in a particular case (the general proof is a somewhat confusing generalization; similar ad hoc geometrical proofs work for other numbers, for example square root of 5). 

Patrice Ayme

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Why the Greeks knew square root of two was no fraction (= “rational” number):

If (m/n)^2 = 2, with xx = x^2 and m and n having no common factors… Then:

mm = 2 nn… Thus m is even (if m is odd, mm is always odd). So m = 2u. This forces n to be odd, because by hypothesis, m and n don’t have common factors, so can’t both be divided by 2…

Hence, n = 2v + 1, for some integer v… Plugging back in the initial equation mm = 2 nn we get: 2u2u = 2 (2v+ 1) (2v + 1). Dividing by 2, we get:

2uu = (2v + 1) (2v +1) = 4vv + 4v + 1…

Now that latter equation is impossible: the left side is even, and the right side is odd… Thus the initial hypothesis, (m/n) (m/n) = 2 is impossible…

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17 Responses to “Proof Of Existence Of Square Root Of Two For Ten Year Old (Take That, Euclid!)”

  1. johnscorner Says:

    You have a very lucky daughter!

    Like

    • Patrice Ayme Says:

      Thanks. She didn’t think so when she was initially confronted to the definition of a number through a geometric trick… She got real upset. Hours later she was very cool and self-assured about it, though…

      Liked by 1 person

      • johnscorner Says:

        I have no doubt, on all points. This kind of stuff IS mind-blowing. HOWEVER. Once you’ve had your mind blown–as hers was, and once you calm down about it (as she obviously did), and once you’ve let your mind kind of “soft-focus” into the general CONCEPT and the MANNER OF REASONING . . . yeah. You’re into a completely new realm. You are–or, in this case, she is–able to THINK DIFFERENTLY.

        And what a joy it must be to be the person who gets to introduce your own daughter to this kind of “different-thinking”! It’s certainly part of what I loved so much about homeschooling our kids (now adults in their 30s and early 40s)!

        Like

  2. pshakkottai Says:

    The place value notation arises from a polynomial such as ax**4 + b**3 +c x**2 + d x**1 + e x**0, with x=10 and a, b,c,d,e being numbers with values between 0 and 9. For example 45082 in base 10. This was how India represented that number.

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    • Patrice Ayme Says:

      Hi Partha! Do you have precise information when and how Indian numerals were derived? Looking at ancient Greek business math, one can see they were part way there…. But that didn’t make it to Euclid (who didn’t have the most advanced Greek math… !!!!!!!!!!!!!!!!!)

      Like

      • pshakkottai Says:

        Hi Patrice, Before SURYASIDDHANTA because of complexity of astronomy which came at the end of Rg Veda. Surya Siddhanta talks about both pole stars (north and south) visible from India several times in the past and which were the pole stars. It records Earth’s wobble and knew the nutation period of about 26000 years. This would be before 8500BP, an unbelievably ancient date. Nilesh Oak has several YouTube videos on this topic while discussing dates of epics Ramayana and Mahabharata.
        Astronomical observations are just reading a giant clock with many hands, all planets, Sun, Moon, and many Stars in KALACHAKRA, the Indian Zodiac.

        Like

        • Gmax Says:

          I thought the Indian religion and writing was *just* 3,500 years old? And Patrice asked when the numbers were invented, you know the decimal system, not whether Indians invented all this. People say it’s the Arabs but Patrice pointed out it was the Indians

          Like

  3. D'Ambiallet Says:

    What is perplexing is that the geometry proves the number. Would you please elaborate?

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    • Patrice Ayme Says:

      I don’t understand it myself…
      A reasoning could be that let A(x) = xx be the area associated to the square of side x. That’s the famous parabola. It’s a continuous function, etc So xx = 2 has a solution: just look at the graph. However I am afraid this contains implicitly the advanced dedekind Cuts theory, implicitly… Stay tuned…

      Like

  4. johnscorner Says:

    I should say something–or some THINGS–else.

    I have come back and looked, once more, at your diagram/illustration. No. Not just once. Two or three times more. And, suddenly, I realized you have done with that one illustration much, much more than I recall having ever seen done–i.e., illustrated–by or with such an illustration before. And it is a thing of (mathematical–let alone graphical) beauty.

    It strikes me that you have approached your daughter the way my dad approached his children in these matters. You did not assume or begin with the thought, “Oh. This is Algebra 2-level material. She can’t understand this! She will get to it in [whatever the French equivalent to US 9th or 10th grade is].” No. You assumed she COULD understand it . . . as long as you explained it properly and illustrated your explanation properly.

    I think that approach is a good one. Or, put another way: BON TRAVAIL, MAMAN!

    Like

  5. pshakkottai Says:

    India used a similar diagram with unequal sides of sides X and Y to prove Pythogoras theorem using
    Z**2 + 4*(XY/2) = ( X+Y)**2, Z being the hypotenuse. 2 XY cancels out to show the familiar formula. India’s favorite was algebraic geometry, not pure geometry.

    Like

  6. Gmax Says:

    Amazingly smart. Even I understood it#! Deep… so you think it helps in general thinking to know stuff like that?

    Liked by 1 person

  7. pshakkottai Says:

    Why was Pythagoras theorem needed by India? To design Vedic Altars of many geometries, a model of Solar Sysem with bricks arranged to model the numbers associated with orbital data of various planets.
    Correspondingly, Rg Veda itself models the numbers from Mandalas, Hymns, Groups, Anuvakas listed by Subash Kak in fís book “The Astronomical Code of the Rg Veda” Aditya Prakasan, New Delhi 2016.
    Here is an altar of books of Rg Veda.
    Book10. Book9
    Book 7. Book8
    Book5. Book6
    Book 3. Book 4
    Book 2. Book1. The corresponding hymns are
    191. 114
    104. 92
    87. 75
    62. 58
    43. 191 From this he derives sidereal periods of
    Mercury, Venus, Mars, Jupiter,Saturn as
    87*1; 58+75+92=75*3; 687=191*3+114=43*16; 4332=62*70= 58*75;
    10760= 104*104= 92*117. Another combination of books gives Synodic Periods 120; 583; 779; 398; 377. He discusses why this list is correct.
    India had a passion for Astronomy! It was made a part of Rg Veda which deals with all topics of interest to humanity Including Grammar and prosody and nothing religious dogma. Of course there were gods and goddesses for all natural phenomena, some in common with Europe, the earliest being Earth-mother = Bhu Devi = Gaia = Cybele = Prehistoric Venus.
    Indian geometry box was not only ruler and compass. It was ruler and string to measure curves.

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  8. Gmax Says:

    Psakkottai,
    Why is the Arian invasion theory a fiction?
    That’s what is in the books, like 4,000 years ago… Do you have a link?

    Like

  9. pshakkottai Says:


    Based on SURYASIDDHANTA records

    Like

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