Real Number R in Math

Real Number $ \mathbb{R} $

The symbol $ \mathbb{R} $ (or $ R $) denotes the set of all real numbers.

• LaTeX Command: \mathbb{R}

Rational Numbers $ \mathbb{Q} $

• Integers: $ \mathbb{Z} = {\ldots, -2, -1, 0, 1, 2, \ldots} \subset \mathbb{Q} $
• Fractions: $ \frac{a}{b} $ where $ a, b \in \mathbb{Z} $ and $ b \neq 0 $
• Terminating decimals: e.g. $ 0.5, 1.75, -2.0 $
• Repeating decimals: e.g. $ 0.333\ldots, 0.123123\ldots $

Irrational Numbers

• Non-repeating, non-terminating decimals: e.g. $ \sqrt{2}, \pi, e $

Non-Real Numbers

• Complex numbers: $ \mathbb{C} = { a + bi \mid a, b \in \mathbb{R}, i = \sqrt{-1} } $
• Imaginary numbers: e.g. $ 3i, -2i $
• Infinity: e.g. $ +\infty, -\infty $

Note:

Complex Numbers
├── Real Numbers: a + 0i (real part ≠ 0)
└── Imaginary Numbers: (imaginary part ≠ 0)
    ├── Pure Imaginary: 0 + bi
    └── Non-real Complex: a + bi (a≠0, b≠0)

With Dimension $ D $

The symbol $ \mathbb{R}^D $ denotes the set of all real numbers in $ D $-dimensional space.

Examples:

• $ \mathbb{R}^1 $: 1-Dimensional space (a line).
• $ \mathbb{R}^2 $: 2-Dimensional space (a plane, coordinates (x, y)).
• $ \mathbb{R}^3 $: 3-Dimensional space (space, coordinates (x, y, z)).
• $ \mathbb{R}^D $: $ D $-Dimensional space (coordinates ($ x_1, x_2, \ldots, x_D $)).