AM

Lecture 1a. NumberSets (1) (1)

Computing with Numbers

  • Author: Dr. Rameez Asif

  • Institution: University of East Anglia, School of Computing Science

Contents

  1. Number Sets

  2. Computing with Real Numbers

  3. Number Systems

  4. Computer Representation of Numbers

Number Sets

  • N: Natural Numbers

  • Z: Integer Numbers

  • Q: Rational Numbers

  • R: Real Numbers

  • C: Complex Numbers (not really numbers in conventional sense)

Natural Numbers

  • Defined as: N = {1, 2, 3, 4, ...}

  • Countable: Can be listed infinitely with a defined position.

  • Example: 10 is the 10th natural number.

Definition Variations

  • Some definitions include 0: N0 = N ∪ {0}.

  • Disagreement exists between sources (e.g., French vs. Americans).

Integer Numbers

  • Defined as: Z = {..., -5, -4, ..., 0, 1, 2, ...}

  • Countable list includes both positive and negative integers.

  • Positive, negative, non-negative, and non-positive classifications.

  • Some authors prefer Z+ (positive) instead of N.

Rational Numbers

  • Defined as: Q = {x : x = a/b, a ∈ Z, b ∈ N}

  • Includes all numbers that can be expressed as fractions.

  • Subset relation: Z ⊂ Q.

  • Between any two rational numbers exists another rational number (dense).

Countability of Rational Numbers

  • Can be listed through specific rules, showing they are countable.

  • Example sequence of positive rational numbers is established.

Decimal Representations of Rational Numbers

  • Can be finite or infinite (e.g., -1/8 = -0.125, 11333/4510 = 2.5).

  • Infinite decimal representations can consist of cyclical sequences.

The Euclidean Algorithm

  • Method to find the greatest common divisor (gcd) of two numbers.

  • Steps include repeated division and remainder comparison.

  • Applicable to natural numbers making it efficient compared to prime factorization.

Real Numbers

  • Include all previously mentioned numbers plus irrational numbers.

  • Irrational numbers: Cannot be expressed as fractions and have infinite non-repeating decimals (e.g., √2, π).

Cantor's Findings

  • Georg Cantor proved that the set of real numbers is uncountable.

  • Example for irrationality: Proof of √2 being irrational through contradiction.