Number Sets Evolution

Overview

The lecture explores the evolution of number sets from natural numbers to complex and beyond, explaining each set's defining features and relationships.

Natural Numbers and Integers

• Natural numbers are used for counting objects and are denoted by N.
• Zero (0) is considered a natural whole number and the starting point of natural numbers.
• Negative numbers expand the number system to include values less than zero, called integers.
• The set of all integers, including positive, negative, and zero, is denoted by Z.

Decimal and Rational Numbers

• Dividing or sharing integers leads to decimal numbers, such as 1.5, forming the set D.
• Some divisions produce infinite decimal sequences, like 1/3 = 0.333..., which cannot be written finitely as decimals.
• Rational numbers can be written as fractions A/B, where A and B are integers, and are denoted by Q.

Irrational and Real Numbers

• Some numbers, like √2, cannot be written as fractions or repeating decimals and are called irrational numbers.
• Irrational numbers have infinite, non-repeating decimals and are part of the set of real numbers, denoted by R.
• Algebraic numbers, a subset of real numbers, are solutions to polynomial equations with integer coefficients.

Transcendental Numbers

• Transcendental numbers, such as π, cannot be solutions to any polynomial equation with integer coefficients.

Complex Numbers and Beyond

• Complex numbers solve equations like x² = -1, which have no real solution.
• The imaginary unit i is defined such that i² = -1, and complex numbers are of the form A + iB, where A and B are real numbers.
• The set of complex numbers is denoted by C.
• Number sets are nested: Integers ⊂ Decimals ⊂ Rationals ⊂ Reals ⊂ Complex numbers.
• More advanced number systems include quaternions (H), octonions (O), and p-adics (QP).

Key Terms & Definitions

• Natural numbers (N) — Counting numbers 0, 1, 2, 3, ...
• Integers (Z) — All positive, negative whole numbers, and zero.
• Decimal numbers (D) — Numbers expressed using a decimal point.
• Rational numbers (Q) — Numbers written as fractions A/B with integer A, B (B ≠ 0).
• Irrational numbers — Numbers with infinite, non-repeating decimals, not expressible as fractions.
• Real numbers (R) — All rational and irrational numbers.
• Algebraic numbers — Solutions to polynomial equations with integer coefficients.
• Transcendental numbers — Numbers not algebraic, such as π.
• Complex numbers (C) — Numbers of the form A + iB, where i² = -1.
• Quaternions (H), octonions (O), p-adics (QP) — Advanced number systems beyond complex numbers.

Action Items / Next Steps

• Review the different number sets and their relationships.
• Prepare to identify numbers as natural, integer, decimal, rational, irrational, real, or complex.
• Optional: Read more about quaternions, octonions, and p-adics for advanced study.