Question

formula for rational numbers, natural numbers, whole numbers

Answer 1

Here are the definitions and formulas for rational numbers, natural numbers, and whole numbers:

### Natural Numbers (ℕ)
Natural numbers are the set of positive integers that start from 1 and go on indefinitely. The formula can be expressed as:

\[
\mathbb{N} = \{1, 2, 3, 4, \ldots\}
\]

### Whole Numbers (W)
Whole numbers include all natural numbers along with zero. The formula is:

\[
\mathbb{W} = \{0, 1, 2, 3, 4, \ldots\}
\]

### Rational Numbers (ℚ)
Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. The formula can be expressed as:

\[
\mathbb{Q} = \left\{ \frac{p}{q} \mid p \in \mathbb{Z}, q \in \mathbb{Z}, q \neq 0 \right\}
\]

where \( \mathbb{Z} \) is the set of all integers (positive, negative, and zero).

### Summary:
- **Natural Numbers (ℕ)**: \(\{1, 2, 3, 4, \ldots\}\)
- **Whole Numbers (W)**: \(\{0, 1, 2, 3, 4, \ldots\}\)
- **Rational Numbers (ℚ)**: \(\left\{ \frac{p}{q} \mid p \in \mathbb{Z}, q \in \mathbb{Z}, q \neq 0 \right\}\)

These sets are fundamental categories in mathematics that help classify numbers based on their properties.