Real Number

A real number is a number that can be represented on the number line. It includes all natural numbers, integers, rational numbers, and irrational numbers.

Key Properties of Real Numbers:

• Closure: The sum and product of two real numbers is a real number.
• Associativity: The operations of addition and multiplication can be grouped in any order.
• Distributivity: Multiplication distributes over addition.
• Commutativity: The order in which numbers are added or multiplied does not matter.
• Identity elements: There are additive and multiplicative identity elements, which are zero and one, respectively.
• Inverses: Each real number has an additive inverse (negation) and a multiplicative inverse (reciprocal).

Types of Real Numbers:

• Natural numbers: Counting numbers starting from one (1, 2, 3, …).
• Integers: Whole numbers, including zero and negative numbers (-2, -1, 0, 1, 2, …).
• Rational numbers: Numbers that can be expressed as fractions or ratios of integers (e.g., 1/2, 3/4).
• Irrational numbers: Numbers that cannot be expressed as fractions or ratios of integers (e.g., √2, π).

Examples of Real Numbers:

• 0
• 1/2
• 3
• √5
• π

Additional Notes:

• The set of real numbers is denoted by R.
• Real numbers are used to represent a wide range of quantities, including length, mass, time, and temperature.
• The concept of real numbers is fundamental to many mathematical and scientific disciplines.