How can I find a rational number between sqrt(37) and sqrt(39)?

Question

Naturally, a calculator cannot be used. I need a proof based answer to this as to why such a rational number exists, and what it is. For clarity's sake, I need a rational number between the square root of 37 and the square root of 39.

Answer 1

Greetings,

It is not diificult to prove the existence of such a number.

Look at the decimal expansions of two irrational numbers a and b (a < b). Suppose they first differ in the nth digit.

Consider the number bn obtained from b by cutting all the digits after the nth.

Then a < bn < b. bn being a finite decimal expansion it is also a rational number.

Finding such a number without a calculator is a bit more involved...

sqrt37 < x < sqrt39, square all terms

37 < x^2 < 39, multiply by 100

3700 < 100x^2 < 3900

since we know 60^2 = 3600 try 61^2 or 62^2

Let 100x^2 = 61^2 = 3721

x^2 = 3721/100

x = 61/10

The proof goes backwards

3700 < 3721 < 3900, divide by 100

37 < 3721/100 < 39, take square roots

sqrt(37) < 61/10 < sqrt(39)

Regards