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If I would draw a right triangle with legs of length 1 centimeter with a ruler then its hypotenuse should be equal to $\sqrt2$ which is an irrational number - therefore its decimal representation, which is the limit of the sequence $\lim_{n \to\infty}\sum_{i=0}^\infty{\frac{a_i}{10^i}}$, has infinitely many numbers after the interger part of$ \sqrt2$.

What exactly can the ruler measure when drawing such a triangle, considering the fact that we draw irrational number (which by definition is infinitely long)?

I hope the question is clear.

Thanks.

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1  
Define ruler. –  lhf May 4 '12 at 16:37
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Also, consider how would you measure $1/3$ with a metric or imperial ruler. –  lhf May 4 '12 at 16:42
    
Why do you insist to discuss a number, which <i>cannot</i> be described by a finite sum of digit-multiples of powers of ten in terms of "long"? –  Gottfried Helms May 19 '12 at 7:26
    
not long, <b>infinitely long</b>. –  Anonymous May 19 '12 at 11:27

2 Answers 2

If the ruler has marks at 0 cm, 1 cm, 2 cm,... then you will measure that the length of the hypotenuse is between 1 and 2 cm .

If the ruler has marks at 0 mm, 1 mm, 2 mm,... then you will measure that the length of the hypotenuse is between 14 and 15 mm .

So everything depends on where are the marks in the ruler. If it is a special ruler with a mark at $\sqrt2$ cm, then it will measure the exact length of the hypotenuse.

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$\sqrt 2$ is irrational and thus the decimal period is infinite, however let's go back to calculus for a moment. Consider the following logical conundrum:

If I know it is 8ft to the door, and let's say I take a step halfway each time. Then before the first step, I have to cover 4ft first. Then 2ft, then 1ft, 0.5ft, 0.25ft.... etc. Movement should be impossible since there are an infinite number of points between each step. However, in calculus we learned that an infinite series converges to a point or diverges.

$\sqrt 2$ is approximated to be 1.4142, so as this decimal becomes longer, it eventually will converge to simply 1.4. The decimal period is infinite, however as it approaches $\infty$ the measurement becomes too small to comprehend.

Your ruler, undoubtedly contained infinite-many points on it, you just cannot see them.

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2  
The decimal expansion of $\sqrt 2$ never repeats! –  lhf May 4 '12 at 17:11
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@lhf, actually, the decimal expansion of $\sqrt2$ repeats a lot, just not periodically. –  Gerry Myerson May 19 '12 at 6:51

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