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This may be a silly question, but I saw this diagram on wikipedia and was intrigued:

https://en.wikipedia.org/wiki/File:Square_root_of_2_triangle.svg

How would such a triangle work in real life? It's in theory possible to draw two side lengths of exactly one meter. That would mean that if you connected them, the hypothenue would be root 2.

Or would it? The square root of two is an irrational number. Can we get that level of precision in practice, in the real world? And how would you know if you had drawn a side length that was exactly root 2, since it seems like it would be impossible to measure?

Hope that makes sense.

Graeme
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  • I don't think we can, a reason being is that if we had a measurement of exactly root 2 we would have infinite precision. Infinite precision is impossible in real life. We can draw something that looks like root 2, but when we zoom in really close, it will be off, if you adjusted it, you could just zoom in further again and it will be off again. Hope this answers your question – VikeStep Sep 09 '13 at 03:59
  • Define real world... – Salech Alhasov Sep 09 '13 at 03:59
  • Here's a question: how do you know if you have drawn a side length that is exactly $1$ centimetre long? How do you know that it isn't actually $0.999997289$ centimetres or $1.00000102003$ centimetres? – Adriano Sep 09 '13 at 04:00
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    There's nothing special about $\sqrt{2}$ in the real world. There isn't enough precision to EXACTLY measure 1 meter either. – rajb245 Sep 09 '13 at 04:00
  • It would be possible to measure sides of $1$ and $4\over 3$ and connect the ends and verify a hypotenuse of $5\over 3$, then measure to $1$ and connect for a hypotenuse of $\sqrt 2$... All of that would be subject to normal precision limits of the instruments used. – abiessu Sep 09 '13 at 04:03
  • Since a metre is defined by the distance travelled by light in 1/299792458 seconds, and then a second is defined by some multiple period of atomic transition, how would you be confident that you can measure exactly that duration of time, and thus an exact metre? – peterwhy Sep 09 '13 at 04:06
  • Unless something forces a discrete choice, the best you can do in 'real life' is to perform a measurement within some degree of tolerance. In addition, there are many approximations made - is the paper flat, is the line straight, is the line infinitesimally thin, etc, etc. So, the answer is you cannot be sure. But for most 'applications', it doesn't really matter. – copper.hat Sep 09 '13 at 04:18

4 Answers4

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Any real measurement has a range of error. You can't measure $1.0$ exactly any more than you can measure $\sqrt 2$ exactly. If you use a tape measure you might have an error range of $\pm \frac1{32}$ or maybe $\pm \frac 1{64}$ inch, but there are still infinitely many rational and irrational numbers within the range. Other measurement techniques have better accuracy, but any interval has infinitely many rational and irrational numbers within the range. Only in mathematics can you say a segment is exactly $1$ or $\sqrt 2$ and not by measurement, but by proof.

Ross Millikan
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There is not really something special or something irrational for $\sqrt2$ in real world.

Imagine, if a country decides to use non-SI unit and defines a unit length to be $\sqrt2$ m, people there can still measure lengths in terms of their unit up to their desired precision. And then, when they see your triangle above, they might equally wonder how you drew your sides in an irrational multiple of unit length, while seeing the hypotenuse at their precise unit length!

peterwhy
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In the real world, everything is made up of molecules, which, in turn, are made up of atoms. In that sense, any ruler could only measure a discrete number of quite small items.

Of course, these items are not packed together linearly, so the resulting length is problematical.

Probably the best way to measure would be the number of wavelengths of light at a certain frequency. With modern techniques, the measurements could be to a fraction of a wavelength.

In other words, I don't really know, but it makes for a fun discussion.

Remember, there will be a quiz next week!

marty cohen
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Measuring a line with infinite precision is impossible because the measuring tools we employ have certain amount of error(though small). We can think like this. A line is made of infinite number of points so does a line segment and so does a point(on a microscopic level, even a point can be futher divided into collection of points). It is an unending process. So theoretically we may prove, but practically it is impossible to measure a line with infinite precision.

Ranjan Yajurvedi
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