🔶Intro to Abstract Math Unit 2 – Number Systems: Foundations and Structures

What's This Unit About?

• Explores the foundations and structures of various number systems
• Covers the properties, operations, and relationships within and between number systems
• Introduces key concepts such as sets, functions, and algebraic structures
• Examines the axioms and rules that govern different number systems
• Discusses the historical development and evolution of number systems
• Highlights the importance of number systems in mathematics and their real-world applications
• Provides a solid foundation for advanced mathematical concepts and theories

Key Concepts and Definitions

• Number systems: Sets of numbers along with defined operations and properties
• Natural numbers (ℕ): Positive integers starting from 1 $\{1, 2, 3, ...\}$
• Whole numbers (ℕ₀): Non-negative integers starting from 0 $\{0, 1, 2, 3, ...\}$
• Integers (ℤ): Whole numbers and their negatives $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$
• Rational numbers (ℚ): Numbers that can be expressed as fractions $\frac{a}{b}$, where $a, b \in \mathbb{Z}$ and $b \neq 0$
• Irrational numbers: Real numbers that cannot be expressed as fractions (√2, π)
• Real numbers (ℝ): The union of rational and irrational numbers
• Complex numbers (ℂ): Numbers of the form $a + bi$, where $a, b \in \mathbb{R}$ and $i = \sqrt{-1}$

Different Types of Number Systems

• Unary system: Represents numbers using only one symbol (tally marks)
• Binary system: Base-2 system using digits 0 and 1, widely used in computing
• Decimal system: Base-10 system using digits 0 through 9, most common in everyday life
• Hexadecimal system: Base-16 system using digits 0-9 and letters A-F, used in computing and color representation
• Octal system: Base-8 system using digits 0-7, used in computing and digital systems
• Balanced ternary: Base-3 system using digits -1, 0, and 1, has some advantages in computation
• Positional notation: Represents numbers using the position of digits to determine their value (decimal, binary)
• Non-positional notation: Represents numbers without relying on the position of digits (Roman numerals)

Properties and Operations

• Closure: A set is closed under an operation if the result of the operation is always an element of the set
• Associativity: $(a * b) * c = a * (b * c)$ for elements $a, b, c$ in a set under operation $*$
• Commutativity: $a * b = b * a$ for elements $a, b$ in a set under operation $*$
• Identity element: An element $e$ such that $a * e = e * a = a$ for all elements $a$ in a set under operation $*$
• Inverse element: An element $b$ such that $a * b = b * a = e$ for an element $a$ in a set under operation $*$, where $e$ is the identity element
• Distributivity: $a * (b + c) = (a * b) + (a * c)$ for elements $a, b, c$ in a set under operations $*$ and $+$
• Ordering: Properties such as trichotomy, transitivity, and completeness that define the order relations on a set

Proofs and Theorems

• Well-ordering principle: Every non-empty set of positive integers contains a least element
• Fundamental theorem of arithmetic: Every positive integer greater than 1 can be uniquely represented as a product of prime numbers
• Euclidean algorithm: A method for finding the greatest common divisor (GCD) of two integers
• Bezout's identity: For integers $a$ and $b$, there exist integers $x$ and $y$ such that $ax + by = \gcd(a, b)$
• Fermat's little theorem: If $p$ is prime and $a$ is not divisible by $p$, then $a^{p-1} \equiv 1 \pmod{p}$
• Euler's theorem: A generalization of Fermat's little theorem for any positive integer $n$, $a^{\phi(n)} \equiv 1 \pmod{n}$ if $a$ and $n$ are coprime
• Chinese remainder theorem: A system of linear congruences with coprime moduli has a unique solution modulo the product of the moduli

Real-World Applications

• Cryptography: Number theory is the foundation of many encryption and security systems (RSA, Diffie-Hellman)
• Error-correcting codes: Used in data transmission and storage to detect and correct errors (Reed-Solomon codes)
• Computer science: Number systems and their properties are essential in algorithm design and analysis
• Finance: Number theory is used in financial modeling, such as interest rate calculations and amortization
• Music theory: Mathematical relationships between frequencies and harmonies are based on number theory
• Crystallography: The study of crystal structures and symmetries relies on number theory concepts
• Quantum mechanics: Number theory is used in the description and analysis of quantum systems

Common Pitfalls and Misconceptions

• Confusing number systems: Mixing up the properties and rules of different number systems
• Forgetting to check for closure: Assuming an operation is closed without verifying the result is in the set
• Misapplying properties: Incorrectly assuming a property holds for a given set or operation (commutativity of matrix multiplication)
• Dividing by zero: Attempting to divide by zero, which is undefined in most number systems
• Confusing equality and congruence: Mistaking the concepts of equality and congruence modulo a number
• Misinterpreting the role of zero: Failing to recognize the unique properties and behaviors of the number zero in different contexts
• Overlooking edge cases: Not considering special cases or extreme values when proving theorems or solving problems

Practice Problems and Examples

1. Prove that the sum of any three consecutive integers is divisible by 3.
2. Find the greatest common divisor (GCD) of 1071 and 1029 using the Euclidean algorithm.
3. Solve the system of congruences: $x \equiv 2 \pmod{3}$, $x \equiv 3 \pmod{5}$, $x \equiv 4 \pmod{7}$.
4. Prove that for any integer $n$, $n^3 + 2n$ is divisible by 3.
5. Convert the decimal number 42 to binary, octal, and hexadecimal.
6. Determine whether the set of integers is closed under the operation $a * b = a + b - 1$.
7. Prove that the product of any four consecutive integers is one less than a perfect square.
8. Find the multiplicative inverse of 7 modulo 26, if it exists.