Number System: Real, Irrational, and Rational Numbers

Number System

Real Numbers

• A set of numbers that include all rational and irrational numbers
• Can be represented on a number line
• Examples: 0, 1, 2, 3, ..., π, e, √2, ...

Irrational Numbers

• Cannot be expressed as a finite decimal or fraction (p/q)
• Have an infinite number of digits that never repeat
• Examples: π, e, √2, ...
• Irrational numbers cannot be expressed exactly, only approximated

Rational Numbers

• Can be expressed as a finite decimal or fraction (p/q)
• Have a finite number of digits that may repeat
• Examples: 1/2, 3/4, 22/7, ...
• Rational numbers can be expressed exactly

Operations On Real Numbers

• Addition and subtraction are commutative and associative
• Multiplication and division are commutative, but not associative
• The distributive property holds for real numbers: a(b + c) = ab + ac
• The order of operations (PEMDAS/BODMAS) applies to real numbers

Decimal Expansion

• A way to represent real numbers in a base-10 system
• Finite decimals represent rational numbers
• Infinite non-repeating decimals represent irrational numbers
• Examples:
  • 1/2 = 0.5 (finite decimal)
  • π = 3.14159... (infinite non-repeating decimal)

Rationalization

• The process of eliminating radicals (irrational numbers) from the denominator of a fraction
• Can be achieved by multiplying the numerator and denominator by a suitable radical
• Examples:
  • 1/√2 = (√2)/(√2 × √2) = √2/2
  • 1/(2 + √3) = (2 - √3)/(4 - 3) = 2 - √3

p/q Formation

• A way to express rational numbers in the form p/q, where p and q are integers and q ≠ 0
• Can be achieved by finding the common denominator of two fractions and adding/subtracting them
• Examples:
  • 1/2 + 1/3 = (3 + 2)/6 = 5/6
  • 2/3 - 1/4 = (8 - 3)/12 = 5/12

Real Numbers

• Integrates both rational and irrational numbers.
• Represented on a number line, illustrating their locations.
• Examples include integers (0, 1, 2), and notable irrationals such as π, e, and √2.

Irrational Numbers

• Cannot be expressed as finite decimals or fractions (p/q).
• Possess infinite, non-repeating digits.
• Examples encompass π, e, and √2, which cannot be conveyed precisely, only approximated.

Rational Numbers

• Can be represented as finite decimals or fractions (p/q).
• Possess a finite number of digits, which may repeat.
• Examples include 1/2, 3/4, and 22/7, which allow exact expression.

Operations On Real Numbers

• Addition and subtraction showcase commutativity and associativity properties.
• Multiplication and division are commutative but not associative.
• Distributive property applies: a(b + c) = ab + ac.
• The order of operations (PEMDAS/BODMAS) must be followed.

Decimal Expansion

• Represents real numbers through a base-10 system.
• Finite decimals correspond to rational numbers.
• Infinite non-repeating decimals represent irrational numbers.
• Examples include 1/2 expressed as 0.5 (finite) and π approximated as 3.14159... (infinite).

Rationalization

• Involves the elimination of radicals from the denominator of a fraction.
• Achieved by multiplying both numerator and denominator by an appropriate radical.
• Illustrative examples include:
  • 1/√2 rationalized to √2/2.
  • 1/(2 + √3) simplified to 2 - √3.

p/q Formation

• Expresses rational numbers in the form p/q with integers p and q (q ≠ 0).
• Involves finding common denominators for the addition or subtraction of fractions.
• Examples include:
  • 1/2 + 1/3 = 5/6 through common denominator 6.
  • 2/3 - 1/4 = 5/12 with a common denominator of 12.

Natural Numbers

• Defined as counting numbers starting from 1 and extending to infinity.
• Examples include 1, 2, 3, etc.
• Properties include being always positive and never zero or negative.
• Operations possible: addition, subtraction, multiplication, and division.

Whole Numbers

• Consist of all natural numbers plus zero.
• Examples are 0, 1, 2, 3, etc.
• Properties allow for operations such as addition, subtraction, multiplication, and division.
• Whole numbers can be positive and include zero.

Rational Numbers

• Defined as numbers that can be expressed as the ratio of two integers (p/q).
• Examples include 3/4, 22/7, and 1/2.
• Can manifest as finite or repeating decimals.
• Rational numbers can be positive, negative, or zero and allow for standard arithmetic operations.

Integers

• Comprise whole numbers, including both positive and negative numbers.
• Examples consist of ..., -3, -2, -1, 0, 1, 2, 3, etc.
• Properties permit addition, subtraction, multiplication, and division of integers.

Irrational Numbers

• Defined as numbers that cannot be expressed as the ratio of two integers (p/q).
• Examples include π, e, and √2.
• They cannot be written as finite decimals, and may be positive or negative.
• Can be manipulated with operations, but results may remain irrational.

Rationalization

• A mathematical process aimed at eliminating radicals (like square roots) from the denominator of a fraction.
• An example includes the rationalization of the denominator in the expression 1/√2.

p/q Formation

• The standard form of expressing a rational number is p/q, where both p and q are integers and q is not zero.
• Examples of this formation are 3/4 and 22/7.

Simplification

• The technique of reducing a fraction to its simplest form by dividing both the numerator and denominator using their greatest common divisor (GCD).
• For instance, the fraction 6/8 can be simplified to 3/4.