Explanation needed for the theorem of irrational/rational number

Question

This is the first theorem from 'Differential and Integral Calculus' written by N Piskunov: rational/irrational number theorem I guess? It's stated that given the irrational number, $\alpha$ is between rational numbers $N$ and $N+1$, $\alpha$ will lie between $N+\frac{m}n$ and $N+\frac{m+1}n$ when the segment between $N$ and $N+1$ is divided into $n$ parts. My question is, what does it mean by $N+\frac{m}n$ and $N+\frac{m+1}n$? What does the variable $m$ represent? Can anyone explain this in simple and understandable terms? (I'm really poor at math)

Answer 1

$N$ is an integer here.

Let's look at the example if you want to write a rational approximation to the irrational $\sqrt{2}$ accurate to $\frac1{100}$, i.e to two decimal places.

• first you note $1 < \sqrt{2} < 2$ so $N=1$;
• then you take $n=100$ since that is reciprocal of the desired accuracy;
• then you note $1+\frac{41}{100} <\sqrt{2} < 1+\frac{42}{100}$ so $m=41$;
• and, since $\left( 1+\frac{42}{100}\right)-\left( 1+\frac{41}{100}\right) = \frac1{100}$, both $1+\frac{41}{100}$ and $1+\frac{42}{100}$, i.e. $1.41$ and $1.42$, are sufficiently accurate rational approximations to $\sqrt{2}$.

You can use floor or round-down notation to say $N=\lfloor x \rfloor$ and $m=\lfloor n(x-N) \rfloor$.