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Maths Class 01

CSAT Mathematics: Number System and Classification of Numbers

Classification of Numbers

  • Numbers are categorized under multiple sets: complex, real, rational, irrational, integers, fractions, whole, and natural numbers.
  • Complex numbers are represented as a ± ib, where i is the square root of –1, divided into real and imaginary parts.
  • Real numbers can be split into rational and irrational numbers.
  • Rational numbers include both integers and fractions
    • Integers: negative, zero, positive
    • Fractions: proper, improper, mixed
  • Whole numbers are non-negative integers starting from zero.
  • Natural numbers start from one and are used for counting.
  • Subsets of natural numbers include even, odd, prime, and composite numbers.
  • Unique or specialized classes include happy numbers, boring numbers, and the loneliest number, though these are not central to the curriculum.

Table: Overview of Classification

Main Set Subset(s) Examples Key Features
Complex Numbers Real, Imaginary a+ib i² = –1; not required for basic CSAT
Real Numbers Rational, Irrational 2, 3.5, √2, π Rational can be p/q, Irrational: non-terminating, non-recurring
Rational Numbers Integers, Fractions –2, 0, 1/2, 3/4 p, q are integers, q ≠ 0
Integers Negative, Zero, Positive –3, 0, 5 Defined on a number line
Fractions Proper, Improper, Mixed 3/5, 7/4, 1 1/2 Numerator < Denominator (proper), otherwise improper/mixed
Whole Numbers - 0, 1, 2, 3… Non-negative integers
Natural Numbers - 1, 2, 3… Counting numbers; excludes zero

Rational and Irrational Numbers

Rational Numbers

  • Defined as numbers expressible as p/q, where p and q are integers and q ≠ 0.
  • All integers and finite decimals qualify as rational because they can be expressed with denominator 1.
  • Non-terminating, recurring decimals (like 1.333…) can be converted to p/q form through algebraic manipulation.

Examples and Properties

  • Examples include 5/7, 1, 11/1, 100/10, and repeating decimals like 1.333… = 4/3.
  • Zero is rational since 0/1 = 0 is valid; note that 1/0 is not defined.
  • Non-terminating but recurring decimals can be expressed as ratios of two integers.

Irrational Numbers

  • Defined as real numbers not representable by p/q, with non-terminating, non-repeating decimal expansions.
  • Examples include √2, √3, √5, and π.
  • π is irrational, though approximated in practice as 22/7 for calculations. Its actual value is non-terminating and non-repeating.
  • Square roots of non-perfect squares are irrational, as they cannot be neatly written as p/q.
  • If a real number cannot satisfy the rational number definition, it is irrational.

Conversion of Recurring Decimals to Fractions

  • Recurring decimals with a repeating block (like 0.333…) can be converted into fractions via algebraic manipulation:
    • Let y = 1.333… then 10y = 13.333…
    • Subtract: 10y – y = 13.333… – 1.333… which gives 9y = 12, hence y = 12/9 = 4/3.

Identifying Rational and Irrational Numbers

  • Numbers like 3/7, √3/√3 (which simplifies to 1), and 7/10 are rational.
  • Expressions like π/22 are irrational, as π itself is irrational.
  • 100 is rational.

Subcategories of Rational Numbers

Integers

  • Three types:
    • Negative: –1, –2, …
    • Zero
    • Positive: 1, 2, …
  • Non-negative integers include zero and positive integers.
  • Non-positive integers include zero and negative integers.
  • Often used in questions stating “x is a non-negative integer” or “x is a non-positive integer” to determine value ranges.

Fractions

  • Proper Fractions: Numerator < Denominator, e.g., 1/2, 2/3.
  • Improper Fractions: Numerator > Denominator and denominator ≠ 1, e.g., 5/4, 7/3. (If denominator is 1, the value represents an integer.)
  • Mixed Fractions: A combination of an integer and a proper fraction that represents the same value as an improper fraction. For example, 3/2 can be written as 1 1/2.

Table: Types of Fractions

Type Form Example Note
Proper a/b, where a < b 2/3 Numerator less than denominator
Improper a/b, where a > b 5/2 Denominator not 1 for improper; else it is an integer
Mixed Integer + a/b 1 1/2 Equivalent to an improper fraction; different notation

Whole Numbers and Natural Numbers

  • Whole Numbers: Start from zero and include all positive integers (0, 1, 2, 3, …).
  • Natural Numbers: Counting numbers starting from one (1, 2, 3, …). Note that zero is not considered a natural number.

Subsets of Natural Numbers

Even Numbers

  • Defined as numbers divisible by two.
  • Sequence example: 2, 4, 6, 8, 10, …
  • Zero is divisible by two and is thus classified as an even integer.

Odd Numbers

  • Defined as natural numbers not divisible by two.
  • Examples include: 1, 3, 5, 7, 9, …
  • Zero is not considered an odd integer.

Prime Numbers

  • Prime numbers have exactly two distinct factors: one and the number itself.
  • The sequence starts from two (the lowest prime).
  • Examples include: 2, 3, 5, 7, 11, … (Note: Two is the only even prime.)
  • One is not a prime number because it has only one factor.

Composite Numbers

  • Numbers with more than two factors.
  • Sequence starts from four.
  • Examples include: 4, 6, 8, 9, 10, …
  • One is not classified as composite.

Special Sets

  • The Loneliest Number: One, as it has only a single factor.
  • Happy Numbers: Defined recursively. A happy number is one for which the sum of the squares of its digits, when iterated repeatedly, eventually leads to one.
    Example sequence: Start with 19 → 1² + 9² = 82 → 8² + 2² = 68 → 6² + 8² = 100 → 1² + 0² + 0² = 1.
  • Boring Numbers: Numbers where digits at odd positions are odd and at even positions are even.

Table: Prime Numbers Between 1 and 50

Range Prime Numbers (Count)
1–50 15
1–100 25

Properties of Even and Odd Numbers

Addition/Subtraction

  • Even ± Even = Even
  • Odd ± Odd = Even
  • Even ± Odd = Odd

Multiplication

  • Even × Even = Even
  • Odd × Odd = Odd
  • Even × Odd = Even

Division

  • No generalized property for even/even, odd/odd, or even/odd division results; the outcome depends on the specific values.

Combination Rules

  • The sum of an even number of odd numbers is always even.
  • The product is even if any factor is even.
  • For expressions such as y = 2a + b + 2c + d:
    • If a and c are any integers, then 2a + 2c is always even.
    • The parity of y depends on the sum of b and d:
      • If both b and d are even or both are odd, their sum is even.
      • If one is odd and one is even, their sum is odd.
  • For a product like y = a × b × c × d, if any factor is even, the entire product is even.

Prime and Composite Number Identification

To determine if a large number is prime (for example, a number with ten crore digits), algorithmic or programmatic checking of factors is required.

Zero and its Classification

  • Zero is an even integer as it satisfies the definition (divisible by two).
  • Zero is not considered odd, nor is it prime or composite.

Sample Questions on Even/Odd Properties

  • If y = a + b + c + d and two out of the four numbers are even while the other two are odd, both even and odd results are possible depending on the arrangement.
  • For larger sums such as y = 2a + b + 2c + d + 2e with three even and two odd numbers, the result can be either even or odd.
  • For expressions like y = 3a + 2b, the parity of y must be calculated based on whether a or b is even or odd.

Counting and Application-Based Questions

Example Problem: Cigarette Machine

  • A person smokes five cigarettes a day. With a machine that converts five waste parts into one new cigarette:
    • Over 25 days, they smoke 125 cigarettes (without the machine).
    • The 125 waste parts generate 25 new cigarettes, which in turn produce additional waste parts.
    • The 25 new waste parts generate 5 more cigarettes.
    • Those 5 waste parts yield one last additional cigarette.
    • The total maximum cigarettes smoked is 156.

Calculation Steps:

Step Cigarettes Smoked Waste Parts Generated
Regular (25 days, 5/day) 125 125
From 125 waste parts 25 25
From 25 waste parts 5 5
From 5 waste parts 1 —
Total 156 —

Sample Evaluations and Logical Deductions

  • Determine the rationality of decimals by examining their periodicity (recurring decimals are rational; non-repeating decimals are irrational).
  • For statements like “Are all odd numbers prime?” or “Are all prime numbers odd?”, counterexamples must be considered (for example, not all odd numbers are prime; 2 is prime but even).

Topic to be Discussed in the Next Class

  • Continuation with detailed examples and practice questions on number systems.
  • More questions on classification of numbers, especially application-based questions and previous years’ question patterns.