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Pythagoras Theorem and Irrational Numbers

It is interesting to see how the Pythagorean theorem helps identify the location of an irrational number on the number line. Consider a number x that is a rational number but not a perfect square. It follows that the square root of x must be irrational, that is, a non-terminating, non-recurring decimal number. Now, our interest is in determining where this is on the number line. To do this, consider a right triangle whose base is equal to (x-1)/2 and whose hypotenuse is equal to (x+1)/2. What would be the height of this right triangle, we mean, the other arm of the right triangle? Suppose it is y. The Pythagorean theorem tells us that the sum of the squares of the arms of a right triangle is equal to the square of the hypotenuse. Thus, in the triangle we have just considered, we can write:

[(x – 1)/2]^2 + i^2 = [(x + 1)/2]^2

Or y^2 = [(x + 1)/2]^2 – [(x – 1)/2]^2

= [(x^2 + 2x + 1) / 4] – [(x^2 – 2x + 1) / 4]

= [(x^2 + 2x + 1) – (x^2 – 2x + 1)] / 4

= [x^2 + 2x + 1 – x^2 + 2x – 1)] / 4

= 4x ​​/ 4

= x

that is, y^2 = x ⟹ y = √x

This is exactly what we were looking for, the square root of x which is irrational. Now the length of the third arm of the triangle we have constructed can be marked on the number line with a compass.

So if you’re looking for the size of √x, this is how we do it. Mark a point A. Mark B so that AB = x units. Mark C so that BC = 1 unit. That is, AC = x+1. Bisect AC. If D is the point of bisection of AC, AD = DC = (x+1)/2. Now what would be the length of DB? Since DC = (x+1)/2 and BC = 1, DB = DC – BC = [(x+1)/2] – 1. That is (x+1-2)/2 or (x-1)/2.

Now let’s build the triangle. Draw a line perpendicular to AC at B. From D, cut the vertical line at E so that DE = AD. Now we have a right triangle in which the base is (x-1)/2 and the hypotenuse is (x+1)/2. Can you see what the measure of BE will be? Of course, as we showed above, it will be √x. You can transfer this length from BE to the number line now, using the compass.

Try representing √5, √7, √11, √6.8, and √9.5 on the number line. Each of them should barely take a couple of minutes or less.

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