Quantitative Aptitude: Number Systems

• Quantitative aptitude is the ability to reason and solve problems involving numbers.
• It is a core skill assessed in various competitive exams.
• Number systems form the foundation of quantitative aptitude.

Types of Numbers

• Natural Numbers: Counting numbers starting from 1 (1, 2, 3...).
• Whole Numbers: Natural numbers including 0 (0, 1, 2, 3...).
• Integers: Whole numbers including negative numbers (...-3, -2, -1, 0, 1, 2, 3...).
• Rational Numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0 (e.g., 1/2, -3/4, 5).
• Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., √2, π).
• Real Numbers: All rational and irrational numbers.
• Complex Numbers: Numbers in the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1).
• Prime Numbers: Numbers greater than 1 that have only two factors: 1 and themselves (e.g., 2, 3, 5, 7, 11).
• Composite Numbers: Numbers greater than 1 that have more than two factors (e.g., 4, 6, 8, 9, 10).

Place Value and Face Value

• Place Value: The value of a digit based on its position in the number (e.g., in 345, the place value of 4 is 40).
• Face Value: The actual value of the digit itself (e.g., in 345, the face value of 4 is 4).

Divisibility Rules

• Divisibility by 2: The last digit is even (0, 2, 4, 6, 8).
• Divisibility by 3: The sum of the digits is divisible by 3.
• Divisibility by 4: The number formed by the last two digits is divisible by 4.
• Divisibility by 5: The last digit is 0 or 5.
• Divisibility by 6: The number is divisible by both 2 and 3.
• Divisibility by 8: The number formed by the last three digits is divisible by 8.
• Divisibility by 9: The sum of the digits is divisible by 9.
• Divisibility by 10: The last digit is 0.
• Divisibility by 11: The difference between the sum of digits at odd places and the sum of digits at even places is either 0 or divisible by 11.

Factors and Multiples

• Factor: A number that divides another number exactly (e.g., the factors of 12 are 1, 2, 3, 4, 6, and 12).
• Multiple: A number obtained by multiplying a given number by an integer (e.g., the multiples of 3 are 3, 6, 9, 12, 15).
• Prime Factorization: Expressing a number as a product of its prime factors (e.g., 12 = 2 x 2 x 3).

Highest Common Factor (HCF)

• The largest factor that divides two or more numbers exactly.
• Methods to find HCF:
  • Prime Factorization Method: Find the prime factors of each number and multiply the common factors.
  • Division Method: Divide the larger number by the smaller number, then divide the divisor by the remainder, continue until the remainder is 0. The last divisor is the HCF.

Least Common Multiple (LCM)

• The smallest multiple that is common to two or more numbers.
• Methods to find LCM:
  • Prime Factorization Method: Find the prime factors of each number and multiply the highest powers of all prime factors.
  • Division Method: Divide the numbers by a common prime factor, continue until no common factor exists. Multiply all the divisors and the remaining numbers.
• Relationship between HCF and LCM: For two numbers a and b, HCF(a, b) x LCM(a, b) = a x b.

Remainders

• Remainder Theorem: When a polynomial f(x) is divided by (x - a), the remainder is f(a).
• Cyclicity of Remainders: The remainders often follow a cyclic pattern when a number is raised to different powers and divided by a fixed number.

Number of Zeroes at the End of an Expression

• Count the number of 2s and 5s in the prime factorization of the expression.
• The number of zeroes is equal to the minimum of the count of 2s and 5s. Since 2s are usually more frequent than 5s, you often only need to count the number of 5s.

Fractions

• Types of Fractions:
  • Proper Fraction: Numerator is less than the denominator (e.g., 2/3).
  • Improper Fraction: Numerator is greater than or equal to the denominator (e.g., 5/2).
  • Mixed Fraction: A whole number and a proper fraction (e.g., 2 1/3).
  • Complex Fraction: A fraction where the numerator or denominator (or both) are fractions (e.g., (1/2)/(3/4)).
• Operations on Fractions: Addition, subtraction, multiplication, and division.

Decimals

• Types of Decimals:
  • Terminating Decimal: Decimal that ends after a finite number of digits (e.g., 0.25).
  • Non-Terminating Recurring Decimal: Decimal that repeats a pattern of digits infinitely (e.g., 0.333...).
  • Non-Terminating Non-Recurring Decimal: Decimal that neither ends nor repeats (irrational numbers like π).
• Converting Recurring Decimals to Fractions: Use algebraic methods to convert recurring decimals to fractions. For example, let x = 0.333..., then 10x = 3.333.... Subtracting the first equation from the second gives 9x = 3, so x = 3/9 = 1/3.

Surds and Indices

• Surds: Irrational numbers expressed as roots (e.g., √2, √3, ∛5).
• Indices: Powers or exponents (e.g., in 2³, 3 is the index).
• Laws of Indices:
  • a^m * a^n = a^(m+n)
  • a^m / a^n = a^(m-n)
  • (a^m)^n = a^(mn)
  • a^0 = 1
  • a^(-n) = 1/a^n
  • (ab)^n = a^n * b^n
  • (a/b)^n = a^n / b^n

Simplification

• BODMAS/PEMDAS Rule: Order of operations – Brackets, Of/Parentheses, Division, Multiplication, Addition, Subtraction.
• Approximation: Estimating values to simplify calculations.
• Algebraic Identities: Using standard algebraic identities to simplify expressions.

Number Series

• Identifying patterns in a sequence of numbers.
• Common patterns include arithmetic progressions, geometric progressions, squares, cubes, and combinations thereof.

Application in Problem Solving

• Quantitative aptitude involves applying these number system concepts to solve real-world problems.
• Proficiency in number systems is crucial for tackling questions on percentages, ratio and proportion, profit and loss, and other quantitative topics.