The subsets of real numbers can be categorized into several key groups that encompass different kinds of numbers. The main subsets include:
- Naturals (ℕ): These are the counting numbers starting from 1, 2, 3, and so on. They do not include zero or any negative numbers.
- Whole Numbers (ℤ+): This set includes all natural numbers along with zero (0, 1, 2, 3, …). Thus, it is a bit broader than the set of natural numbers.
- Integers (ℤ): Integers comprise all whole numbers, both positive and negative, including zero (…, -3, -2, -1, 0, 1, 2, 3, …).
- Rational Numbers (ℚ): These are numbers that can be expressed as a fraction, where both the numerator and the denominator are integers and the denominator is not zero (e.g., 1/2, -3/4). Rational numbers include all integers as they can be written as themselves divided by 1.
- Irational Numbers: This set contains numbers that cannot be expressed as a fraction of two integers, meaning they have non-terminating and non-repeating decimal expansions. Examples are √2 and π.
- Real Numbers (ℝ): This is the broadest subset that includes all the above sets. It encompasses both rational and irrational numbers, representing all points on the number line.
These subsets effectively categorize the real numbers based on their properties and characteristics, helping to understand their usage in various mathematical contexts.