Learn – Algebra foundations – Irrational numbers

$\mathbb{Q}$ denotes the set of rational numbers, and $\mathbb{Q} = \left\{\frac{p}{q}:p,q \mbox{ are integers and } q\ne 0\right\}$
Between any two rational numbers, no matter how close they are, one can find another rational number. In fact, between any two rational numbers, one can find an infinite number of rational numbers.
No matter how dense the rational numbers are on the number line, they do not completely fill up the number line. The number line still has holes
Real numbers is the set of all numbers on the number line.
Numbers that are real, but not rational – that is, numbers that cannot be expressed in the form $\frac{p}{q}$ where $p,q$ are integers and $q\ne 0$ , are called irrational numbers.
$\sqrt{2}$ is an example of irrational number.
If $b$ is an irrational number, then $-b$ is also irrational.
If $a$ is rational (with $a\ne 0$ ) and $b$ is irrational, then $a+b, a-b, b-a, ab, a/b, b/a$ are all irrational.
If $a,b$ are both irrational, then $a+b, a-b,ab, a/b$ could be rational or irrational.