Real Number System Overview

Real Number System Overview

• The real number system encompasses all numbers that can be represented on a number line.
• It includes natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

Natural Numbers

• Natural numbers are positive integers starting from 1 (1, 2, 3, ...).
• They are used for counting.
• A subset of whole numbers and integers.

Whole Numbers

• Whole numbers include all natural numbers and zero (0, 1, 2, ...).
• Zero represents the absence of quantity.
• A subset of integers.

Integers

• Integers are whole numbers and their negative counterparts (... -3, -2, -1, 0, 1, 2, 3, ...).
• Include positive and negative whole numbers.
• Zero is an integer.

Rational Numbers

• Rational numbers are expressible as a fraction p/q where p and q are integers, and q ≠ 0.
• Examples include 1/2, 3/4, -2/5, and 7.
• Any terminating or repeating decimal can be expressed as a rational number.

Irrational Numbers

• Irrational numbers cannot be expressed as a fraction of two integers.
• Their decimal representations are non-terminating and non-repeating.
• Examples include π (pi), √2 (the square root of 2), and e (Euler's number).

Real Number Properties

• Closure: The sum or product of two real numbers is always a real number.
• Commutative Property: The order of addition or multiplication doesn't change the result (a + b = b + a, a × b = b × a).
• Associative Property: The grouping of numbers in addition or multiplication doesn't change the result (a + (b + c) = (a + b) + c, a × (b × c) = (a × b) × c).
• Distributive Property: Multiplication distributes over addition (a × (b + c) = a × b + a × c).
• Identity Property: The additive identity is 0 (a + 0 = a), and the multiplicative identity is 1 (a × 1 = a).
• Inverse Property: Every real number has an additive inverse (opposite) and a multiplicative inverse (reciprocal) (a + (-a) = 0, a × (1/a) = 1, a ≠ 0).
• Order Property: Real numbers are ordered; for any two real numbers a and b, either a = b, a < b, or a > b.

Real Number Line

• The real number line visually represents the real numbers.
• Each point on the line corresponds to a unique real number.
• Positive numbers are to the right of zero, and negative numbers are to the left.

Subset Relationships

• Natural numbers are a subset of whole numbers.
• Whole numbers are a subset of integers.
• Integers are a subset of rational numbers.
• Rational and irrational numbers together form the set of real numbers.