Language:

There are a bunch of different numbers, and they can be divided into different sets. The set that consists of all the numbers on the real number line are called the real numbers. But the real line also includes many other sets of numbers. Here’s a description of them.

Theory

$\mathbb{N}=\phantom{\rule{-0.17em}{0ex}}\left\{1,2,3,\dots \right\}$ is the set of natural numbers. These are the numbers you learned to count as a small child. It consists of all the positive integers.

$\mathbb{Z}=\phantom{\rule{-0.17em}{0ex}}\left\{\dots ,-2,-1,0,1,2,\dots \right\}$ is the set of all integers. These are the numbers you learned when you were little with both pluses and minuses. It consists of all positive and negative integers.

$\mathbb{Q}=\phantom{\rule{-0.17em}{0ex}}\left\{\frac{a}{b}\mid b\ne 0,\phantom{\rule{0.33em}{0ex}}a,b\in \mathbb{Z}\right\}$ (the symbol $\mid $ is read “such that”) is the set of rational numbers. This is the set of all fractions where the numerator and denominator are integers. Remember that an integer itself is also a fraction, as any integer is equal to itself divided by $1$ ($5=\frac{5}{1}$).

$\mathbb{R}\setminus \mathbb{Q}$ ( the symbol $\setminus $ is read as “without”) $=\pi ,e,\sqrt{2},\dots $ is the set of irrational numbers. These are numbers like $\pi $, $e$, $\sqrt{2}$ and all numbers that have an infinite number of decimals without any repeating pattern. Irrational numbers can’t be written as fractions.

$\mathbb{R}=$ is the set of real numbers, which is all the numbers on the real number line.

The sets of numbers above are included like this: The natural numbers $\mathbb{N}$ are included in the integers $\mathbb{Z}$, which again are included in the rational numbers $\mathbb{Q}$, which finally are included in the real numbers $\mathbb{R}$.