Real numbers are a broad and fundamental set of numbers that include all rational and irrational numbers.
They form the foundation of the number system used in mathematics and have various properties that make them essential for a wide range of mathematical operations and applications.
Real numbers can be represented on the number line, which is a horizontal line where each point corresponds to a specific real number. The number line extends infinitely in both positive and negative directions.
Real numbers can be categorized into three main subsets:
- Rational Numbers: Rational numbers are numbers that can be expressed as the quotient of two integers (fractions). They can be written in the form p/q, where p and q are integers and q is not equal to zero. Rational numbers include integers, terminating decimals, and repeating decimals.Examples of rational numbers:
- 2 (an integer, as it can be written as 2/1)
- 0.5 (a terminating decimal, written as 1/2)
- 0.333… (a repeating decimal, written as 1/3)
- Irrational Numbers: Irrational numbers are numbers that cannot be expressed as the quotient of two integers. They cannot be written in the form p/q, where p and q are integers, and q is not equal to zero. Irrational numbers have non-repeating and non-terminating decimal representations.Examples of irrational numbers:
- √2 (approximately 1.41421356…)
- π (pi, approximately 3.14159265…)
- √3 (approximately 1.73205081…)
- e (approximately 2.71828183…)
- Integers: Integers are whole numbers (positive, negative, or zero) and do not include fractional or decimal parts.Examples of integers: