Closure and Sets of Numbers

Sets of Numbers

Number Theory is - not surprisingly - devoted to the study of the properties and patterns of numbers. There are different kinds of numbers, of course: whole numbers, integers, rational and irrational, real and complex, etc... Some of these sets are described in the list below, along with the symbol we normally use to denote them. More often than not, however, if you are doing something with number theory, you will be working with integer values in some form or another.

  • $\mathbb{N}$ (or $\mathbb{Z}^+$), the natural numbers - which consist of the counting numbers

    $$\{1, 2, 3, \ldots\}$$
  • $\mathbb{Z}$, the integers - which include all of the natural numbers, their negatives, and zero

    $$\{\ldots -3, -2, -1, 0, 1, 2, 3, \ldots\}$$
  • $\mathbb{Q}$, the rationals - which include all quotients of integers

    $$\{\frac{1}{2}, -\frac{3}{4}, 92, \frac{312}{517}, -\frac{1239}{41}, \ldots \}$$
  • $\mathbb{R}$, the real numbers - which include the limiting values of all sequences of rational values

    $$\{-3, \, 0, \, \frac{2}{3}, \, \pi, \, \sqrt{2}, \, 0.1011011101111\cdots, \, \sqrt[3]{287}+e, \ldots\}$$
  • $\mathbb{C}$, the complex numbers - defined by

    $$\{a + bi \, : \, a \textrm{ and } b \textrm{ are real values, and } i = \sqrt{-1} \}$$

Closure

We say that a given (binary) operation is closed with respect to some set if, when we apply the operation to any two elements from the set, the result must also be within the set. For example, the sum of two integers is always an integer, so we say integers are closed with respect to addition. The product of two integers is also always an integer, so we can say the integers are also closed with respect to multiplication. However, the quotient of integers need not always be an integer (even though it sometimes is). In this case, we say the integers are not closed with respect to division.

We can write these facts more succinctly with the following symbols

$$\begin{array}{cl} \exists &:& \textrm{there exists ...}\\\\ \exists ! &:& \textrm{there exists a unique ...}\\\\ \forall &:& \textrm{for all ...}\\\\ \in &:& \textrm{is in, or is an element of ...}\\\\ \notin &:& \textrm{ is not in, or is not an element of ...} \end{array}$$

For example, we can say that the integers are closed under multiplication as

$$\forall a, b \in \mathbb{Z}, \quad ab \in \mathbb{Z}$$

while the integers are not closed under division as

$$\exists a, b \in \mathbb{Z} \textrm{ such that } \frac{a}{b} \notin \mathbb{Z}$$