Question

The rational number between 1/2 and π

Answer 1

Huh, that's an interesting question.

Both 1 and 2 are rational number (because they can be expressed in terms of ratios ex: 1=5/5 and 2=24/12 e.t.c)

Now our job is to see how many irrational number there are between two rational number (here it is 1 and 2)

Let's say a=1 and b=2 then our numbers should be between a and b i.e. if the number is x

a<x<b.

Now let's see what's an irrational number. Do you Know what does irrational mean? Something that doesn't make sense right? Off course.

If you explore the digits of √2 you see that decimal expansion is non ending non repeating, and something that doesn't end and doesn't repeat, does that make any sense? How could you possibly write it in terms of ratios?

And we know that that value of √2 goes like this 1.41421356… .. and that's between 1 and 2. Therefore √2 is one such irrational number.

Now if I write number like this 1.010010001 … that doesn't repeat nor end ( what should we call this number?) irrational right? (Yes).

But as you can see you can find many such pattern, for eg. 1.2001000100001 …, 1.2020020002 …. e.t.c

Also note that if your number are a and b and you find c and d between them, then any numbers between c and d would be between a and b and so forth and so on….

It is very clear by now that you can find infinite such numbers.

Any questions? Ask me in the comment section.