Question

What are three irrational number between -11 and -12?

Answer 1

`(-11)-(pi-3), (-11)-(sqrt2-1), (-11)-(sqrt3-1)`

Explanation:

The difficult part to this question is not in identifying irrational numbers between `-11` and `-12` (there are an infinite number of them) - it's in expressing them. What is an easy way to express an unending, non-repeating set of digits that by definition cannot be expressed as a fraction composed of integers?

One way is to take a few well-known irrational numbers, alter them slightly, and present those as solutions. For instance, `pi` is a well known irrational number:

3.1415...

So what if we simply change the "whole number" part of `pi` but keep the unending string of digits:

`-11.1415...`

and if we want to express it a bit more clearly, we can say:

`(-11)-(pi-3)=-11-0.1415...=-11.1415...`

We can do the same thing with `sqrt2=1.4142...` and `sqrt3=1.732...`:

`(-11)-(sqrt2-1)=-11-0.4142...=-11.4142...`
`(-11)-(sqrt3-1)=-11-0.732...=-11.732...`