We can construct a right triangle with its each leg as 1 unit,then the hypotenuse would be √2 units,and then we can point √2 on the number line.
But √2 has a non-terminating and non-recurring decimal representation.We always approximate the value of √2 up to certain decimal places.
What is the need for approximation,as we already know the correct lenght of √2 on the number line.
So my question is –
√2 can be plotted on the number line, and we know its exact length
So how √2 has a non-terminating decimal and non-recurring decimal representation,It must have have a fixed value ,as hypotenuse of the triangle has a fixed value.
Best Answer
Ruler and compass are not concerned with "units", but just with straight lines and circles.
Yes, but - per the previous point - leave out the units for now. You can construct an arbitrary isosceles right triangle, then yes, you can mark the length of its hypotenuse on the line definining one of the legs.
Why? So far, it's all been a geometric ruler-and-compass construction, which defined a few points.
Now, take the leg where the length of the hypotenuse was marked, and choose one (arbitrary) point of it to be the "unit".
If you choose that point to be the endpoint of the leg, then that would be $1$ (rational) and the point marking the length of the hypotenuse would be $\sqrt{2}$ (irrational).
If however you choose the unit point to be the one marking the length of the hypotenuse, then that would be $1$ (rational) and the leg would be $1 / \sqrt{2}$ (irrational).
Why would choosing the unit after the fact affect the legitimity of the construction itself? Of course, it doesn't. All that rationality tells is whether the ratio of two lengths can be expressed as the ratio of two integer numbers or not. It doesn't make either length less "measurable" than the other.