IGCSE-Mathematics-Formula-Booklet

Natural Numbers

• Definition: The set of positive integers that are used for counting purposes, starting from 1 onwards without end.

• Examples: 1, 2, 3, 4, ..., 100, ...

Whole Numbers

• Definition: The set of natural numbers that includes zero, thus making it a complete set of non-negative integers.

• Examples: 0, 1, 2, 3, 4, ...

Integers

• Definition: This set includes all positive and negative natural numbers as well as zero, allowing for the representation of quantities that can be negative.

• Examples: ..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...

Rational Numbers

• Definition: Numbers that can be expressed in the form (p/q), where p and q are integers and q is not equal to zero, allowing for fractions and ratios.

• Examples: (3/4), (-5/2)

Irrational Numbers

• Definition: Numbers that cannot be expressed as a fraction of two integers, their decimal representations are non-terminating and non-repeating.

• Examples: √2, π, √5, √17, ...

Terminating Decimals

• Definition: Decimal numbers that have a finite number of digits after the decimal point, representing fractions with denominators that are powers of 10.

• Example: 7/8 = 0.875, terminating after 3 decimal places.

Recurring Decimals

• Definition: Decimal numbers that have one or more digits that repeat infinitely after the decimal point, often arising from divisions that cannot resolve into terminating decimals.

• Example: 0.528957, where 528957 keeps repeating.

BODMAS Rule

• Definition: A rule that dictates the order of operations used in mathematics, ensuring accurate calculations.

• Order of Operations:

  1. Brackets

  2. Orders (powers and roots)

  3. Division

  4. Multiplication

  5. Addition

  6. Subtraction

Even and Odd Numbers

• Even Numbers: Integers divisible by 2, which include all negative and positive even integers.

  • Examples: 2, 4, 6, 8, ...

• Odd Numbers: Integers not divisible by 2, meaning they have a remainder of 1 when divided by 2.

  • Examples: 1, 3, 5, 7, ...

Real Numbers

• Definition: A broad category in mathematics that encompasses both rational and irrational numbers, representing all possible numerical values on the number line.

Prime Numbers

• Definition: Natural numbers greater than 1 that have no divisors other than 1 and themselves, making them the building blocks of the integers.

• Note: The number 1 is not considered a prime number.

Square Numbers

• Definition: The product of an integer multiplied by itself, forming a perfect square.

• Examples: 1²=1, 2²=4, 3²=9, ...

Cube Numbers

• Definition: The result of multiplying an integer by itself twice, representing volume in geometric contexts.

• Examples: 1³=1, 2³=8, 3³=27, ...

Factors and Multiples

• Factors: Integers that can be evenly divided into another integer without leaving a remainder. They are the building blocks of numbers.

  • Example: Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18.

• Multiples: Quantities that result from multiplying a number by integers, showing all sizes in that number's series.

  • Example: Multiples of 6: 6, 12, 18, 24, 30, ...

Significant Figures

• Definition: The digits within a number that contribute to its precision, reflecting the certainty of the measurement.

• Examples:

  • 8064 = 8000 (1 significant figure)

  • 8064 = 8100 (2 significant figures)

  • 8064 = 8060 (3 significant figures)

Decimal Places

• Definition: The quantity of digits located to the right of the decimal point in a number, affecting its overall precision.

• Examples:

  • 0.0647 = 0.1 (1 decimal place)

  • 2.0647 = 2.065 (3 decimal places)

Standard Form

• Definition: A way of expressing very large or very small numbers in the form (a × 10^n), where a is a coefficient between 1 and 10 and n is an integer.

• Examples: 2400 = 2.4 × 10³, 0.0035 = 3.5 × 10⁻³.

Conversion Factors

• Length:

  • 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm.

• Mass:

  • 1 kg = 1000 gm, 1 gm = 1000 mg, 1 tonne = 1000 kg.

• Volume:

  • 1 litre = 1000 cm³ = 1000 litres, 1 kilolitre = 1000 litres.

• Time:

  • 1 hour = 60 minutes = 3600 seconds; 1 day = 24 hours.

Percentages

• Definition: A mathematical concept expressing a ratio as a fraction of 100, often used in statistics and finance to make comparisons easier.

• Calculation: To express a quantity as a percentage of another, convert the first quantity to a fraction of the second and multiply by 100.

Simple Interest

• Formula: ( I = (P x R x T) / 100 ) where I is interest, P is principal amount, R is rate of interest per annum, and T is the time in years.

Compound Interest

• Formula: ( A = P(1 + r/100)ⁿ ) where A is the amount, P is the principal, r is the interest rate, and n is the number of compounding periods.

Speed, Distance and Time

• Formulae:

  • Distance = speed x time

  • Speed = distance/time

  • Time = distance/speed

Quadratic Equations

• Definition: Equations of the form ax² + bx + c = 0 where the highest power of the variable (x) is 2, representing parabolic relationships.

• Solving Methods: Factorization, quadratic formula, completing the square.

Expansion of Algebraic Expressions

• Basic Identity: ( a(b+c) = ab + ac )

• Recognizing Forms:

  • ( (a+b)² = a² + 2ab + b² )

  • ( (a-b)² = a² - 2ab + b² )

Factorization of Algebraic Expressions

• Examples:

  • ( a² + 2ab + b² = (a + b)² )

  • ( a² - b² = (a + b)(a - b) )

Measurement: Area and Perimeter

• Rectangle: Area = length × breadth, Perimeter = 2(length + breadth).

• Square: Area = side × side, Perimeter = 4 × side.

• Triangle: Area = ½ × base × height.

• Circle: Area = πr², Circumference = 2πr.