Irrational Numbers

Discovery and Properties - A Complete Mathematical Journey

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Welcome to the complete journey through irrational numbers! Click any section below to begin your detailed exploration.

Introduction: What are Irrational Numbers?

Irrational numbers are one of the most fascinating discoveries in mathematics. Unlike rational numbers, which can be expressed as simple fractions, irrational numbers cannot be written as p/q where p and q are integers.

Irrational Number: Cannot be written as p/q (where p, q ∈ β„€, q β‰  0)

Common Examples:

  1. √2 β‰ˆ 1.41421356... - The diagonal of a unit square
  2. √3 β‰ˆ 1.73205080... - Can be constructed geometrically
  3. Ο€ β‰ˆ 3.14159265... - Ratio of circumference to diameter
  4. e β‰ˆ 2.71828182... - The natural logarithm base

The Discovery That Shook Mathematics

Around 500 BCE, the Pythagorean school believed that all numbers could be expressed as ratios of integers. This belief was fundamental to their philosophy that "all is number."

Hippasus of Metapontum discovered that √2 could not be expressed as a fraction while studying the diagonal of a unit square. This discovery created a crisis in Pythagorean philosophy.

Proof by Contradiction: Assume √2 = p/q β†’ Leads to logical impossibility

Detailed Definition and Properties

An irrational number is a real number that cannot be expressed as a ratio of integers. Their decimal expansions never terminate and never repeat.

Key Properties:

  1. Decimal Expansion: Non-terminating, non-repeating
  2. Algebraic vs Transcendental: √2 is algebraic, Ο€ is transcendental
  3. Density: Between any two real numbers, infinite irrationals exist
  4. Uncountability: More irrational numbers than rational numbers

Geometric Construction of √2

Step-by-Step Process:

  1. Draw a unit square OABC with side length 1
  2. Draw the diagonal OB of the square
  3. Apply Pythagorean theorem: OBΒ² = 1Β² + 1Β² = 2
  4. Therefore: OB = √2
  5. Transfer this length to number line using compass
OB = √(1Β² + 1Β²) = √2 β‰ˆ 1.41421356...

Geometric Construction of √3

Building on √2:

  1. Start with √2 construction (diagonal OB = √2)
  2. From point B, draw perpendicular line of length 1
  3. Call the endpoint D
  4. Draw line OD
  5. By Pythagorean theorem: OD² = (√2)² + 1² = 3
  6. Therefore: OD = √3
OD = √((√2)Β² + 1Β²) = √3 β‰ˆ 1.73205080...

Number Line and Density

Irrational numbers fill the "gaps" between rational numbers on the number line. Both rational and irrational numbers are dense in the real numbers.

Density Properties:

  1. Between any two real numbers, infinite rationals exist
  2. Between any two real numbers, infinite irrationals exist
  3. Irrationals are uncountably infinite
  4. Complete the real number system