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.