Real Numbers and Their Properties

• Real numbers encompass all rational and irrational numbers
• They can be positive, negative, or zero
• Real numbers can be represented on a number line

Properties of Real Numbers

• Closure property dictates that for real numbers a and b, a + b and a × b are also real numbers
• Commutative property means that for real numbers a and b, a + b = b + a and a × b = b × a
• Associative property states that for real numbers a, b, and c, (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
• Distributive property shows that for real numbers a, b, and c, a × (b + c) = (a × b) + (a × c)
• Identity property: A real number 0 exists such that for any real number a, a + 0 = a (additive identity); a real number 1 exists such that for any real number a, a × 1 = a (multiplicative identity)
• Inverse property: For any real number a, a real number -a exists such that a + (-a) = 0 (additive inverse); for any non-zero real number a, a real number 1/a exists such that a × (1/a) = 1 (multiplicative inverse)

Real Number Line

• Every point on the line represents a unique real number, and vice versa
• It extends infinitely in both positive and negative directions
• Rational and irrational numbers can be located on it
• It provides a visual representation of the order and completeness of real numbers

Irrational Numbers

• These cannot be expressed in the form p/q, where p and q are integers and q ≠ 0
• Decimal representation is non-terminating and non-repeating
• Examples include √2, √3, π, and e
• Operations (+,-,*,/) on irrational numbers can result in a rational or irrational number

Operations on Real Numbers

• Addition, subtraction, multiplication, and division are defined for real numbers
• The result of these operations between two rational numbers is always a rational number
• Performing these operations on irrational numbers can result in either rational or irrational numbers
• Real numbers follow the standard order of operations (PEMDAS/BODMAS)

Number Systems

• Natural Numbers (N):

  • Positive integers starting from 1
  • N = {1, 2, 3, 4, ...}

• Whole Numbers (W):

  • Non-negative integers starting from 0
  • W = {0, 1, 2, 3, 4, ...}

• Integers (Z):

  • Includes all whole numbers along with their negative counterparts
  • Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

• Rational Numbers (Q):

  • Numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0
  • Their decimal representation is either terminating or repeating
  • Includes integers, fractions, and terminating/repeating decimals
  • Examples: 1/2, -3/4, 0.5, 0.333...

• Irrational Numbers:

  • Numbers that cannot be expressed in the form p/q, where p and q are integers and q ≠ 0
  • Their decimal representation is non-terminating and non-repeating
  • Examples: √2, π, e

• Real Numbers (R):

  • The set of all rational and irrational numbers
  • Can be represented on a number line

• Complex Numbers (C):

  • Numbers are in the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1)
  • Includes real numbers as a special case (when b = 0)

Euclid's Division Lemma

• For any two positive integers a and b, unique integers q and r exist such that a = bq + r, where 0 ≤ r < b
• Where a is the dividend, b is the divisor, q is the quotient, and r is the remainder
• This lemma is the basis for Euclid's division algorithm

Euclid's Division Algorithm

• A method to find the highest common factor (HCF) of two positive integers
• Euclid's division lemma is applied repeatedly until the remainder is zero; the divisor at this stage is the HCF
• Example: To find the HCF of 455 and 42:
  • 455 = 42 × 10 + 35
  • 42 = 35 × 1 + 7
  • 35 = 7 × 5 + 0
  • The HCF of 455 and 42 is therefore 7

The Fundamental Theorem of Arithmetic

• Every composite number can be uniquely expressed as a product of prime numbers, disregarding the order of the prime factors
• Example: 140 = 2 × 2 × 5 × 7 = 2² × 5 × 7
• Used to find the HCF and LCM of two or more numbers

HCF and LCM

• HCF (Highest Common Factor): The largest positive integer that divides two or more integers without any remainder
• LCM (Least Common Multiple): The smallest positive integer that is divisible by two or more integers
• The product of two numbers equals the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b)

Revisiting Irrational Numbers

• Proof of irrationality often involves contradiction; for example, to prove √2 is irrational:
  • Assume √2 is rational, i.e., √2 = p/q, where p and q are co-prime integers and q ≠ 0
  • Squaring both sides, 2 = p²/q²
  • Rearranging, p² = 2q² implying that p² is divisible by 2, and thus p is divisible by 2
  • Let p = 2k, where k is an integer; substituting, (2k)² = 2q², which simplifies to 4k² = 2q², or 2k² = q²
  • This means q² is divisible by 2, and thus q is divisible by 2
  • Both p and q are divisible by 2, which contradicts the assumption that p and q are co-prime
  • Therefore, √2 is irrational

Revisiting Rational Numbers and Their Decimal Expansions

• Terminating Decimal Expansions: A rational number p/q possesses a terminating decimal expansion if q can be expressed in the form 2ⁿ × 5ᵐ, where n and m are non-negative integers
• Non-Terminating Repeating Decimal Expansions: A rational number p/q has a non-terminating repeating decimal expansion if q cannot be expressed in the form 2ⁿ × 5ᵐ, where n and m are non-negative integers

Converting Decimals to Rational Numbers

• Terminating Decimals: Readily converted to rational numbers by expressing them as fractions with a power of 10 in the denominator; example: 0.25 = 25/100 = 1/4
• Non-Terminating Repeating Decimals: Converted to rational numbers using algebraic manipulation; for example, to convert 0.¯3 (0.333...) to a rational number:
  • Let x = 0.333...
  • 10x = 3.333...
  • Subtracting the first equation from the second; 9x = 3
  • Therefore, x = 3/9 = 1/3

Logarithms

• The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number
• If aˣ = y, then logₐ(y) = x, where a is the base, y is the argument, and x is the logarithm

Laws of Logarithms

• Product Rule: logₐ(mn) = logₐ(m) + logₐ(n)
• Quotient Rule: logₐ(m/n) = logₐ(m) - logₐ(n)
• Power Rule: logₐ(mⁿ) = n × logₐ(m)
• Change of Base Rule: logₐ(b) = logₓ(b) / logₓ(a)

Applications of Real Numbers

• Measurement: Used in measurements like length, area, volume, and time
• Computation: Used in mathematical calculations, scientific computations, and engineering applications
• Representation: Used to represent quantities and values in fields including economics, finance, and statistics
• Problem Solving: Essential for solving mathematical problems and real-world scenarios