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About Irrational Lengths

The question asks: Can an irrational number depict length? (Question details: For example in a right angled triangle, if the length of both the sides are 1 respectively, the length of the hypotenuse is irrational. The number goes on forever. But the length of the side is finite. Isn’t this contradictory? How can this be explained? Are any of my assumptions or understanding of the concepts incorrect?)

My answer: (To vote and comment on Quora, visit the link here.)

A number with infinitely many decimal places is not necessarily infinite.

Rather, the number of decimal places in the numerical length of a line segment is a measure of precision with which you know the length.

An “infinite number” is such that it has its “left side” on the number line, but no “right side”. You keep going towards the right, but you never reach the number, because there is no end towards the right side.

A “number with infinitely many decimal places” is one which has a left side as well as a right side on a number line. You keep approaching it from the left as well as the right, and hence you can find that point somewhere on the number line! If you know that square root of 2 has decimals starting with 1.414…, then you find 1.4 on the number line and come right, and leftwards from 1.5, and gradually taking more precision, you keep reaching towards the actual point. However, you never reach the actual point, because there are infinitely many decimal places.

In both cases, you do not reach the actual point but the reasons are different. In the second case, you can narrow down and can actually feel the location of the point. Finite, hence proved.

Here, Pythagoras’ Theorem comes in handy to plot the actual length. If you believe in Pythagoras’ Theorem, you will believe that the length plotted by it as square root of 2 is indeed accurate.

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