Question

What is a real number and can you explain why the inequality x<2 or x>1 has every real number as a solution?

Answer 1

Let's handle the second part first:
what values of `x` must be included if `x<2` or `x>1`?
Consider two cases:

Case 1: `x<2`
`x` must be included

Case 2: `x >= 2`
if `x>=2` then `x>1`
and therefore it must be included

Note that the results would be quite different if the condition had been `x<2` and `x>1`

One way to think about Real numbers is to think of them as distances, comparable measure of length.
Numbers can be thought of as an expanding collection of sets:

1. Natural numbers (or Counting numbers): 1, 2, 3, 4, ...

2. Natural numbers and Zero

3. Integers: Natural numbers, Zero, and Negative version of Natural numbers ....-4, -3, -2, -1, 0, 1, 2, 3, 4, ....

4. Rational numbers: Integers plus all values that can be expressed as the ratio of two integers (fractions).

5. Real numbers: Rational numbers plus Irrational numbers where Irrational numbers are values which exist as lengths but can not be expressed as fractions (for example `sqrt(2)`).

6. Complex numbers: Real numbers plus numbers with components that include `sqrt(-1)` (called Imaginary numbers).