Understanding Number Systems

• The number system serves as a method to represent numbers, which are abstract mathematical constructs employed in both counting and measurement.
• Symbols and rules vary across different number systems for representing numbers.

Types of Numbers

• Natural Numbers: Counting numbers begin at 1 (e.g., 1, 2, 3,...).
• Whole Numbers: Consist of natural numbers along with zero (e.g., 0, 1, 2, 3,...).
• Integers: Include whole numbers and their negative counterparts (e.g., ... -3, -2, -1, 0, 1, 2, 3,...).
• Rational Numbers: Can be written as a fraction p/q with p and q as integers, where q is not zero (e.g., 1/2, -3/4, 5, 0.75).
• Irrational Numbers: Cannot be expressed as a simple fraction, possessing non-terminating and non-repeating decimal representations (e.g., √2, π).
• Real Numbers: Encompass all rational and irrational numbers.
• Complex Numbers: Formed as a + bi, where a and b represent real numbers, and i stands for the imaginary unit (√-1).

Prime and Composite Numbers

• Prime Number: A natural number greater than 1, divisible only by 1 and itself (e.g., 2, 3, 5, 7, 11).
• Composite Number: A natural number greater than 1 that has divisors other than just 1 and itself (e.g., 4, 6, 8, 9, 10).
• The number 1 is neither prime nor composite.

Divisibility Rules

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

Factors and Multiples

• Factor: An integer that divides the number evenly (without leaving a remainder).
• Multiple: An integer that results from multiplying the number by an integer.
• Highest Common Factor (HCF): The largest factor shared by two or more numbers; also known as the Greatest Common Divisor (GCD).
• Least Common Multiple (LCM): The smallest multiple shared by two or more numbers.

Finding HCF and LCM

• Prime Factorization Method: Express each number as a product of its prime factors.
  • HCF: Take the product of the common prime factors with the lowest powers.
  • LCM: Take the product of all prime factors with the highest powers.
• Division Method: Repeatedly divide the numbers by their common factors.

Relationship between HCF and LCM

• For two numbers a and b: HCF(a, b) × LCM(a, b) = a × b

Decimal System

• A base-10 number system.
• Utilizes ten digits (0-9).
• Each digit's position in a number corresponds to a power of 10 (e.g., ones, tens, hundreds).

Place Value and Face Value

• Place Value: The value of a digit contingent on its position within the number (e.g., in 345, the place value of 4 is 40).
• Face Value: The digit's inherent value (e.g., in 345, the face value of 4 is 4).

Fractions

• Proper Fraction: The numerator is less than the denominator (e.g., 2/5).
• Improper Fraction: The numerator is greater than or equal to the denominator (e.g., 7/3).
• Mixed Fraction: A combination of a whole number and a proper fraction (e.g., 2 1/3).

Operations on Fractions

• Addition/Subtraction: Fractions must have a common denominator.
• Multiplication: Multiply the numerators and the denominators.
• Division: Invert the second fraction and multiply.

Simplification

• BODMAS/PEMDAS rule dictates the order of operations:
  • Brackets / Parentheses
  • Orders / Exponents
  • Division
  • Multiplication
  • Addition
  • Subtraction
• Expressions should be simplified following this order.

Remainders

• The remainder represents the quantity "left over" when one number is divided by another.
• The Remainder Theorem can be applied to determine remainders when dividing polynomials.

Number of Zeros at the End of an Expression

• Count the pairs of 2s and 5s in the expression's prime factorization.
• The number of these pairs corresponds to the number of zeros at the end of the expression; typically, the count of factors of 5 is the determining factor, as 2s are usually more abundant.

Successive Division

• A number is divided by several divisors successively.
• To find the original number, work backwards from the last remainder.

Algebraic Formulae

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (a + b)³ = a³ + 3a²b + 3ab² + b³
• (a - b)³ = a³ - 3a²b + 3ab² - b³
• a² - b² = (a + b)(a - b)
• a³ + b³ = (a + b)(a² - ab + b²)
• a³ - b³ = (a - b)(a² + ab + b²)

Modular Arithmetic

• Focuses on remainders after division.
• a ≡ b (mod m) indicates that a and b yield the same remainder when divided by m.
• Calculations involving large numbers can be simplified by concentrating on remainders.

Progressions

• Arithmetic Progression (AP): A sequence with a constant difference between consecutive terms.
  • nth term: a + (n - 1)d, where a is the first term and d is the common difference.
  • Sum of n terms: (n/2)[2a + (n - 1)d] or (n/2)(a + l), where l is the last term.
• Geometric Progression (GP): A sequence where the ratio between consecutive terms remains constant.
  • nth term: ar^(n-1), where a is the first term and r is the common ratio.
  • Sum of n terms: a(1 - r^n) / (1 - r) if r < 1, or a(r^n - 1) / (r - 1) if r > 1.
  • Sum to infinity (for |r| < 1): a / (1 - r)
• Harmonic Progression (HP): A sequence formed by reciprocals that constitute an arithmetic progression.

Surds and Indices

• Surd: An irrational number expressed as the nth root of a positive integer (e.g., √2, ∛5).
• Index: The power to which a number is raised, also known as an exponent.
• Laws of Indices:
  • a^m * a^n = a^(m+n)
  • a^m / a^n = a^(m-n)
  • (a^m)^n = a^(mn)
  • (ab)^n = a^n * b^n
  • (a/b)^n = a^n / b^n
  • a^0 = 1
  • a^(-n) = 1 / a^n

Approximation

• Rounding off numbers to estimate the answer.
• Useful for quick calculations and checking the reasonableness of answers.

Word Problems

• Translate the word problem into mathematical expressions.
• Identify the unknowns and the given information.
• Use relevant formulas and concepts to solve for the unknowns.
• Check if the answer makes sense in the context of the problem.