Sets of Numbers

$\mathbb{N}=\left\{1,2,3,\dots \right\}$ 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.

$\mathbb{Z}=\left\{\dots ,-2,-1,0,1,2,\dots \right\}$ 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.

$\mathbb{Q}=\left\{\frac{a}{b}\mid b\ne 0,a,b\in \mathbb{Z}\right\}$ (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 $1$ ($5=\frac{5}{1}$).

$ℝ\setminus \mathbb{Q}$ ( 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.

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