Number Systems and Types of Numbers

Number Systems

• A number system is a writing system for expressing numbers, utilizing symbols or digits to represent quantities.
• Different number systems possess distinct properties, enabling various applications.

Natural Numbers

• Natural numbers are the counting numbers (1, 2, 3, 4...).
• Also known as positive integers.

Whole Numbers

• Whole numbers encompass natural numbers and zero (0, 1, 2, 3...).
• Also known as non-negative integers.

Integers

• Integers incorporate zero, positive natural numbers, and negative natural numbers (-3, -2, -1, 0, 1, 2, 3...).
• Integers consist of the set of whole numbers and their negations.

Rational Numbers

• Rational numbers are expressible as a fraction p/q, where p and q are integers, and q is non-zero.
• Examples include fractions like 1/2, 3/4, and decimal representations such as terminating or repeating decimals (-2/5).

Irrational Numbers

• Irrational numbers are not expressible as a fraction of two integers.
• Their decimal representations are non-terminating and non-repeating.
• Examples include √2, π (pi), and e.

Real Numbers

• Real numbers are the combination of rational and irrational numbers.
• They are represented as points on a number line.

Complex Numbers

• Complex numbers are denoted as a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit (√-1).

Properties of Number Systems

• Commutative property: The order of operands does not affect the result (e.g., 2 + 3 = 3 + 2).
• Associative property: Grouping operands does not affect the result (e.g., (2 + 3) + 4 = 2 + (3 + 4)).
• Distributive property: Multiplication distributes over addition (e.g., 2 × (3 + 4) = (2 × 3) + (2 × 4)).
• Identity property: The addition identity is zero (e.g., 2 + 0 = 2); the multiplication identity is one (e.g., 2 × 1 = 2).
• Inverse property: Every number has an additive inverse (e.g., the inverse of 2 is -2). Every non-zero number has a multiplicative inverse (reciprocal) (e.g., the inverse of 2 is 1/2).
• Closure property: The outcome of an operation on numbers within a set remains within that set (e.g., the sum of two integers is an integer).

Different Bases

• Decimal (base 10) is the prevalent number system.
• Other systems include binary (base 2), octal (base 8), and hexadecimal (base 16).
• These bases utilize distinct sets of digits for numerical representation.

Number Line Representation

• A visual representation of numbers on a straight line.
• Numbers increase from left to right.
• Facilitates ordering and comparison of numbers.

Important Concepts

• Prime numbers: Whole numbers greater than 1, divisible only by 1 and themselves (e.g., 2, 3, 5, 7).
• Composite numbers: Whole numbers greater than 1, having more than two factors (e.g., 4, 6, 8, 9).
• Divisibility rules: Methods for determining if one number is divisible by another without executing division.

Applications

• Number systems are foundational to mathematics, science, and various other disciplines.
• Applied in computer science, engineering, and finance.
• Crucial for problem-solving.