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.