🔢 Irrational Numbers

What are Irrational Numbers? Irrational numbers are numbers that cannot be written as a simple fraction (ratio) of two integers. In other words, they cannot be expressed in the form \(p/q\) where \(p\) and \(q\) are integers and \(q ≠ 0\).

What is the Real Number System? The real number system is the collection of all rational and irrational numbers together. Every point on the number line represents a real number.

Key Differences between Rational and Irrational Numbers:

Rational NumbersIrrational NumbersCan be written as \(p/q\)Cannot be written as \(p/q\)Decimal expansion terminates or repeatsDecimal expansion never terminates and never repeatsExamples: \(1/2=0.5\), \(1/3=0.3\)Examples: \(√2=1.414213...\), \(π=3.141592...\)
Famous Irrational Numbers:

• \(√2\) (square root of 2)
• \(√3\), \(√5\), \(√6\), \(√7\), \(√8\), \(√10\) (square roots of non-perfect…

What is the Decimal Expansion of an Irrational Number? The decimal expansion of an irrational number is non-terminating (never ends) and non-repeating (no pattern repeats). You can keep calculating digits forever, and they will never settle into a repeating cycle.

Examples:

• \(√2 = 1.4142135623730950488...\) (never ends, no repeating pattern)
• \(π = 3.14159265358979323846...\) (never ends, no repeating pattern)
• \(√3 = 1.7320508075688772935...\)

Contrast with Rational Numbers:

• Terminating decimals: \(1/4 = 0.25\) (ends)
• Repeating decimals: \(1/3 = 0.333333...\) (repeats 3)
• Irrational: \(1.414213...\) (never ends and never repeats)

How do we know an irrational number's decimal never repeats? If a decimal expansion eventually repeats, the number can be written as a fraction (rational number). Since irrational numbers cannot be written as fractions, their…

How to Represent \(√2\) on a Number Line: We can use the Pythagorean theorem to construct irrational numbers geometrically.

Method for \(√2\):

• Draw a number line. Mark 0 and 1.
• At point 1, draw a perpendicular line of length 1 unit.
• Join 0 to the top of this perpendicular. The length of this line = \(√1² + 1² = √2\).
• Using a compass, transfer this length to the number line.

Method for \(√3\):

• First construct \(√2\) as above.
• At point \(√2\), draw a perpendicular of length 1.
• Join 0 to the top. Length = \((√2)² + 1² = √2 + 1 = √3\).

Approximation Methods for Irrational Numbers:

MethodDescriptionExample for \(√2\)Guess and checkFind perfect squares around the number\(1²=1\), \(2²=4\), so \(√2\) is between 1 and 2Long division methodSystematic digit-by-digit calculation1.414213...Using calculatorQuick approximation1.414213562
Approximating \(π\):

•…