REAL NUMBERS

COMPLETE REAL NUMBER REVISION IN JUST ONE CLASS. IN THIS SESSION WE WILL BE HAVING ALL THE IDEAS OF REAL NUMBER SYSTEM ALONG WITH PYQS

A real number is any value that can represent a quantity along a continuous line, encompassing both rational and irrational numbers. This includes integers, fractions, and numbers with non-repeating, non-terminating decimals like √2 (approximately 1.4142135623730951), π (approximately 3.14159), and e (approximately 2.71828) .

Real numbers are foundational in mathematics and are characterized by several key properties:

• Closure: The sum or product of any two real numbers is always a real number.

• Commutative: The order of addition or multiplication does not affect the result.

• Associative: Grouping of numbers does not change the outcome of addition or multiplication.

• Distributive: Multiplication distributes over addition.

• Identity Elements: Zero is the additive identity (a + 0 = a), and one is the multiplicative identity (a × 1 = a).

• Inverses: Every real number has an additive inverse (a + (-a) = 0) and a multiplicative inverse (a × (1/a) = 1, for a ≠ 0) .

These properties ensure that real numbers are well-behaved under standard arithmetic operations, making them essential for modeling and solving real-world problems in fields like physics, engineering, economics, and everyday life.