Number sets

In mathematics, there are several standard number sets that are frequently used for variables in expressions and statements. such as:

• Natural Numbers($\mathbb{N}$)
  • Definition: The set of all positive whole numbers starting from 0 or 1.
  • Notation: $\mathbb{N}$ = {$0,1,2,3,\dots$}(most of the time)
• Whole Numbers($\mathbb{W}$ or ${\mathbb{N}}_0$)
  • Definition: Includes all non-negative integers (i.e. the natural numbers plus $0$).
  • Notation: $\mathbb{W}$ = {$0,1,2,3,\dots$}
• Integers($\mathbb{Z}$)
  • Definition:The set of all positive and negative whole numbers, including 0.
  • Notation: $\mathbb{Z}=\{\dots ,-3,-2,-1,0,1,2,3,\dots \}$
• Rational Numbers($\mathbb{Q}$)
  • Definition: The set of all numbers that can be expressed as a fraction $\frac{a}{b}$ , where $a\text{ and }b\in \mathbb{Z}$ and $$\begin{gathered} b \\ \ne 0 \end{gathered}$$
  • Notation: $\mathbb{Q}$ = {$\frac{a}{b}\mid a,b\in \mathbb{Z},b\ne 0$}, Encompasses all numbers that have finite or repeating decimal expansions.
• Irrational Numbers
  • Definition: Numbers that cannot be expressed as a fraction of two integers (non-repeating, non-terminating decimals).
  • Notation: There isn’t a standard single-letter notation for irrationals, but they are typically considered the complement of $\mathbb{Q}$ in $\mathbb{R}$.
  • Examples: $\sqrt{2}$, $\pi$, $e$.
• Real Numbers($\mathbb{R}$)
  • Definition: The set of all rational and irrational numbers; essentially, any number that can be located on a continuous number line.
  • Notation: $\mathbb{R}$.
• Complex numbers($\mathbb{C}$)
  • Definition: The set of all numbers that can be expressed in the form $a+bi$, where$a$ and $b$ are real numbers and $i$ is the imaginary unit($i^2=-1$).
  • Notation: $\mathbb{C}$ = {$a+bi\mid a,b\in \mathbb{R}$}.

Subset Notations and Extensions

we can get the subset of these number sets by adding some symbols:

• Positive subset (${\mathbb{Z}}^{+}$)
  • all positive numbers in the set
• Negative subset (${\mathbb{Z}}^{-}$)
  • all negative numbers in the set
• conditional subset (${\mathbb{Z}}_{{\ge }_0}$) Not actual name
  • all numbers that fit the condition in the set