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

Apr 9, 2015

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: $x \ge 2$
if $x \ge 2$ 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).