A brood comb

….philosophical and other notes….

Is Square Root of 2 Infinite?

Posted by Tanas Gjorgoski on September 16, 2007

Vallicella at Maverick Philosopher has little discussion of sqrt(2) [square root of 2] in this post. It is related to his previous posts on the issue of actual vs. potential infinites. In the post Vallicella says:

Suppose you have a right triangle. If two of the sides are one unit in length, then, by the theorem of Pythagoras, the length of the hypotenuse is the sqr rt of 2 = 1.14159. . . . And yet the length of the hypotenuse is perfectly definite, perfectly determinate. If the points in the line segment that constitutes the hypotenuse did not form an infinite set, then how could the length of the hypotenuse be perfectly definite?

What I want to point is that the hypotenuse doesn’t have any particular number related to it. It is square root of 2, only in relation with the sides which *are taken to be* 1. The nature of the number to be a ratio is shown here… the number which we assign to the hypotenuse is about the ratio between it and the other sides. However, we could instead take the hypotenuse to be 1, and the sides would be then sqrt(1/2). Or we could take the sides to be 3, and the hypotenuse would be then 3sqrt(2), etc… We see that what doesn’t change is the ratio, and what is perfectly definite here is the ratio.

Now, is there anything infinite here? I don’t think so… the hypotenuse and the other sides of the right triangle have a certain and limited length. Also, such as they are, we can speak of the ratio they form, and hence speak of numbers in relation to them. But numbers are not merely sums – we might *try* to express the ratio of the hypotenuse and the side of the right triangle through a sum, but in the given case we will fail. And this failing, this impossibility to finish with the “translation” of this ratio from its “definiteness” into our numerical notation, IS what we have as infinity here. We just can’t succeed into finishing this translation *because it can’t be done*! Some things can’t be translated into aggregates or sums. That is because there is more to this world than aggregates (or sums).

This ratio also has relations to other ratios which can be expressed as ratio of whole numbers. It can be bigger than some of those ratios and smaller than others. And just because of it, there is possibility of getting as close as needed to this ratio by using ratios of whole numbers. (Of course the ratio of rational numbers also falls under ratio of whole numbers as a/b:c/d is nothing other but the a*d/b*c ratio of whole numbers).

Of course, we can continue this translation as much as we can, and only because it will never succeed, we might be inclined to say that the product of this translation is infinite. But if we search for an ‘actual infinity’, it will not show itself in the results of some repeating iteration, but what is just “imperfectly” expressed by this sum, is as actual, definite and finished in the form of the term sqrt(2) (*and all of different cases which share the ratio with the ratio between the hypotenuse of the right triangle, and it’s side*, or which can be showed as identical).

Seems to me, that if we want to talk about actual infinity, we can just talk about being infinite in relation to something else – i.e. that that former can’t be reduced to the later.

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7 Responses to “Is Square Root of 2 Infinite?”

  1. ornette said

    This leads to an interesting puzzle for anyone who thinks of space as divided up into chunks (rather than a continuum); this is one way to solve Zeno’s paradoxes.

    Imagine you have a right-angled triangle whose sides have lengths 3u, 4u and 5u (the last being the hypotenuse). Then the universe must be divided up such that these three lengths can exist; that is, the chunks must be at most 1u in all directions (never mind what shape that could possibly suggest for them), and can be 1/n of a u for any natural number n.

    But if you change your unit of measurement so that the triangle’s 3u side is now 2u’ (ie, a u’ is 3/2 of a u) then you have a problem, because now the hypotenuse will be an irrational number, and no size of chunk can possibly do the job.

    So a pair of triangles, one 2/3 the size of the other, proves that space is a continuum, since no finite Weyl-style “tiling” of space can account for them as long as you use the same units to measure both. (Maverick Philospher has a recent post about the Weyl Tile argument BTW).

    I guess the response is “well, perfect right-angled triangles are ideal objects, not physical ones”… but the outcome is counter-intuitive, isn’t it?

  2. Thanks for the comment Ornette!

    That is very interesting relation you draw.

    I share your doubt that it will convince people who think that space is made from chunks (or that there is just specific positions that objects could take, where positions are taken to be something which is there to “be taken”) – I guess people with such views will not have problem in ‘digitalizing’ everything including the distances and physical laws.

    For the rest of us, that outcome is counter-intuitive, but I think so is the idea of chunks of space too.

    Of course it might have to do with the spirit of the times. As through computers people learn more to think in terms of digitalized notions, the intuition might go in that direction. (AFAIK Newton talked in Principia that he is going against the intuitive view of distances being relative with his idea of absolute space and movement. Today the intuition seems to be other way around.)

  3. Anonymous said

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  4. Ramious Zoriastar said

    Square Root of 2 = 1.4142135623730950488016887242097…

  5. Ramious Zoriastar said

    What follows are the first 10 million digits of the square root of 2. Actually, slightly more than 10 million digits are given here. These digits were computed by Robert Nemiroff (George Mason University and NASA Goddard Space Flight Center) and checked by Jerry Bonnell (University Space Research Association and NASA Goddard Space Flight Center). They were computed during spare time on a VAX alpha class machine over the course of several weeks. We do NOT guarantee the accuracy of these digits – in fact, these digits past 6 million have not been checked. We believe these are the most digits ever computed for the square root of two on or before 1 June 1994. If anyone is aware of more digits we ask them to please alert us of their existence. We have also computed several digits of the number e and the square roots of other numbers. These are available on this mosaic server (URL: http://antwrp.gsfc.nasa.gov/htmltest/rjn.html). We welcome comments. – Robert Nemiroff and Jerry Bonnell The square root of two = 1.4142135623730950488016887242096980785696718753769480731766797379907324784621 07038850387534327641572735013846230912297024924836055850737212644121497099935831 41322266592750559275579995050115278206057147010955997160597027453459686201472851 74186408891986095523292304843087143214508397626036279952514079896872533965463318 08829640620615258352395054745750287759961729835575220337531857011354374603408498 84716038689997069900481503054402779031645424782306849293691862158057846311159666 87130130156185689872372352885092648612494977154218334204285686060146824720771435 85487415565706967765372022648544701585880162075847492265722600208558446652145839 88939443709265918003113882464681570826301005948587040031864803421948972782906410 45072636881313739855256117322040245091227700226941127573627280495738108967504018 36986836845072579936472906076299694138047565482372899718032680247442062926912485 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  6. Anonymous said

    Go away mother @#$%^&!

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