which of the following can be expressed by exact length but not by exact number?

Question

which of the following can be expressed by exact length but not by exact number?

(i) $ \sqrt{10} $

(ii) $ \sqrt{7} $

(iii) $ \sqrt{13} \ $

(iv) $ \ \sqrt{11} \ $

Answer:

I basically could not understand th question.

What is meant by expressing by exact length ?

Does we need to satisfy Pythagorean law?

Help me with hints

Answer 1

I think "cannot be expressed by exact number" means they are irrational so the decimal does not terminate, which is true of all of them.

I think "can be expressed by exact length" means you can construct it. You are expected to notice that $10=3^2+1^2$ so you can draw a segment of length $1$, a perpendicular segment of length $3$, make the hypotenuse, and that will be a segment of length $\sqrt{10}$. Similarly $13=3^2+2^2$ so it is easy to construct.

$\sqrt 7$ is constructible as well, but not in such a simple way. You can construct $\sqrt 5=\sqrt {2^2+1^2}$, then $\sqrt 6=\sqrt{\sqrt{5}^2+1^2}$ and finally $\sqrt 7$. From $\sqrt 7$ (or in other ways) you can construct $\sqrt {11}$.

Answer 2

The spiral of Theodorus constructs the square roots of the positive integers.

Answer 3

Correct me if wrong .

$√n$ is constructible, $n \in \mathbb{Z^+}$.

$n=1$, ok.

Assume $√n$ is constructible.

Step:

Show that $\sqrt{n+1}$ is constructible.

Pythagorean Theorem:

$(\sqrt{n})^2+1= n+1= (\sqrt{n+1})^2$, i.e.

construct a right triangle with leg lengths, $1$ and $√n$,

then the length of the hypotenuse is $\sqrt{n+1}$.