Real number

 Real numbers are a fundamental concept in mathematics that represent the set of all numbers that can be found on the number line. They include both rational numbers (fractions) and irrational numbers (numbers that cannot be expressed as fractions of integers). Real numbers encompass a wide range of values and are used extensively in various mathematical and scientific contexts.

Here are some key characteristics and subsets of real numbers:

1. Rational Numbers: Rational numbers are numbers that can be expressed as a quotient or fraction of two integers, where the denominator is not zero. Examples include integers (whole numbers), fractions, and decimals that terminate or repeat.

2. Irrational Numbers: Irrational numbers cannot be expressed as fractions of integers. They have non-terminating, non-repeating decimal expansions. Famous examples include the square root of 2 (√2), pi (π), and the mathematical constant "e."

3. Integers: Integers are whole numbers (positive, negative, or zero) that do not have fractional or decimal parts. They include positive integers, negative integers, and zero.

4. Natural Numbers: Natural numbers, also known as counting numbers, are the set of positive integers starting from 1 and increasing without bound.

5. Whole Numbers: Whole numbers include zero and all positive integers. They do not include negative numbers.

6. Ordering: Real numbers can be ordered on the number line, meaning that any two real numbers can be compared to determine which one is greater, lesser, or equal to the other.

7. Density: Real numbers are dense on the number line, which means that between any two distinct real numbers, there are infinitely many other real numbers. This property is in contrast to integers, where there are gaps (no integers) between consecutive integers.

8. Arithmetic Operations: Real numbers support the standard arithmetic operations of addition, subtraction, multiplication, and division. These operations follow familiar rules and properties.

9. Decimal Expansion: Every real number can be represented as a decimal expansion, which can be finite (terminating) or infinite (non-terminating and possibly repeating).

10. Real Line: The real numbers are represented on a number line, with each point on the line corresponding to a unique real number. This visual representation helps in understanding the ordering and relative magnitudes of real numbers.

Real numbers have applications across various branches of mathematics, as well as in fields such as science, engineering, economics, and more. They provide a foundation for calculus, analysis, and other advanced mathematical concepts.