What are the classification criteria for real and rational numbers?
To understand the classification criteria for real and rational numbers, let's start by discussing the definitions of these types of numbers.
Real Numbers: Real numbers include all the numbers on the number line, both the rational and irrational numbers. They can be positive, negative, or zero and can be expressed as decimals, fractions, or integers.
Rational Numbers: Rational numbers are numbers that can be expressed as fractions, where both the numerator and denominator are integers. The denominator cannot be zero. Rational numbers can include integers, repeating decimals, and terminating decimals.
Now, let's talk about the classification criteria for these numbers:
1. Real Numbers:
- All integers, including positive and negative whole numbers (e.g., -3, 0, 5).
- Fractions, where both the numerator and denominator are integers (e.g., 2/3, -4/5).
- Irrational numbers, which cannot be expressed as fractions and have non-repeating and non-terminating decimal expansions (e.g., √2, π).
2. Rational Numbers:
- Integers, as they can be expressed as fractions where the denominator is 1 (e.g., -2 = -2/1).
- Fractions, where both the numerator and denominator are integers (e.g., 1/2, -3/4).
- Repeating decimals, which are decimal representations of fractions with repeating patterns (e.g., 0.333..., 0.656565...).
- Terminating decimals, which are decimal representations of fractions with a finite number of digits after the decimal point (e.g., 0.25, -0.6).
To classify a number, you need to determine if it fits the criteria mentioned above. If a number can be expressed as a fraction, it is a rational number. If it cannot be expressed as a fraction (non-repeating and non-terminating decimal), it is an irrational number. And remember, all rational numbers are real numbers, but not all real numbers are rational.