Math IGCSE

Cube Numbers

• Definition: A cube number is any number raised to the power of three.
    - Example: 2 cubed, which is calculated as $2^3 = 8$.

Natural Numbers

• Definition: A natural number is any positive whole number.
    - Example: 243 is a natural number.

Square Numbers

• Definition: A square number is any number raised to the power of two.
    - Example: 2 squared, calculated as $2^2 = 4$.

Prime Numbers

• Definition: A prime number is a number that is only divisible by itself and 1.
    - Example: 7 is a prime number.

Common Multiples

• Definition: A common multiple of two numbers is a multiple that is shared by both.
    - Example: The first common multiple of 4 and 17 is 68.

• Finding Common Multiples:
    - Factors of 4: {4, 8, 12, 16, 20, …}
    - Factors of 17: {17, 34, 51, 68, …}
    - The least common multiple (LCM) involving 4 is 4 multiplied by 17 = 68.

Upper Bound Calculation

• Calculation Method: To calculate the upper bound of a measurement:
    - Example provided: Ner's millimeter is 0.1 divided by 2, yielding 0.05.
    - Upper bound of measurement: $6.5 + 0.05 = 6.55$ .
    - Context: If measuring height, multiplying by a factor (e.g., 3) gives a final answer: $6.55 imes 3 = 19.65$.

Standard Form

• Definition: In standard form, numbers are expressed as a product of a number between 1 and 10 and a power of 10.
    - Example: 4.56 in standard form is $4.56 imes 10^{-3}$ (moving back three decimal places).

Ratios

• Definition: A ratio shows the numerical relationship between two amounts.
    - Example: To find that Annie receives 330, calculate: $rac{11}{20} imes 600 = 330$.

Percentage Increase

• Definition: To calculate a percentage increase:
    - Formula: For a 12% increase, add this percentage to 100%.
    - Calculation: $rac{12}{100} + 1 imes 105$ gives increased value.

Original Price Calculation

• If an item is reduced by 16%, to find the original price:
    - Formula: $100 - 16 = 84$.
    - Calculation: $rac{105}{84} imes 100 = 1.25$.

Interest Calculations

• Simple Interest Formula:
    - $I = B imes R imes t$
      - Where:
        - I = Interest
        - B = Principal amount borrowed or invested
        - R = Rate of interest
        - t = Time

• Compound Interest Formula:
    - $A = B(1 + R)^t$
      - Where:
        - A = Total amount after interest
        - B = Principal amount borrowed or invested
        - R = Interest rate
        - t = Time period.
     

Exchange Rate Calculations

• To convert currencies using an exchange rate:
    - Example: If $1 = 124.3 Japanese yen, to find the value of $80, set up the equation and cross-multiply to find x.

Speed, Distance, and Time

• Formula: $ext{Speed} = rac{ ext{Distance}}{ ext{Time}}$
    - Example: Distance calculated as speed multiplied by time (converted to hours).
    - Given: Speed = 18, Time = 55 minutes.
    - Conversion of time: $rac{55}{60}$ hours.
    - Calculation: $ext{Distance} = 18 imes rac{55}{60} = 16.5 ext{ km}$.

Algebraic Equations

• Basic multiplicative and algebraic manipulation:
    - Example: $5 imes 3 = 15$, leading to an expression involving x's giving $N^2 + 2N + 15$.

Solving Linear Equations

• To solve for x in linear equations:
    - Rearrangement and consolidation of x terms results in:
    - Example: From $11x - 3x = -7 - 15$, results in $8x = -22$,
    - Final answer: $x = - rac{22}{8} = -2.75$.

Continuous Equations and Substitution Method

• For equations where one variable is expressed in terms of another:
    - Example used: If $y = rac{x}{2}$, then substitute into another equation and solve for x and y accordingly.
    - Resulting x is calculated as: $x = rac{2}{3}$ and substituting it back yields $y = rac{1}{3}$.

Radical Equations

• Method to eliminate radicals:
    - Example of operation: Square both sides to eliminate a square root which leads to an equation involving x.
    - Rearrangement gives: $y^2 - 1 = x^2$ and extracting roots provides solutions.

Inverse Proportion and Constant

• If y is inversely proportional to the square of x:
    - Equation format: $y = rac{k}{x^2}$ where k is a constant.
    - Provided examples lead to substituting values to find k and therefore y in terms of x.

Laws of Indices

• Fundamental properties of exponents:
    - Power of zero: $x^0 = 1$
    - Negative exponents yield the inverse: $x^{-n} = rac{1}{x^n}$
    - Multiplicative and divisive operations on powers:
    - $x^m imes x^n = x^{m+n}$ and $rac{x^m}{x^n} = x^{m-n}$.

Geometry: Polygons and Circle Theorems

• Polygon Angles:
    - Sum of angles: For an n-sided polygon, sum = $(n-2) imes 180$.
    - Each angle of a regular polygon: $rac{(n-2) imes 180}{n}$.
    - Exterior angles sum: $360^ ext{o}$ for any polygon.

Pythagorean Theorem

• Formula: For a right-angled triangle with sides a, b, and hypotenuse c:
    - $a^2 + b^2 = c^2$.
    - Solving for a side involves isolating it in the equation.

Symmetry and Shapes

• Definition: Lines of symmetry divide the shape into two identical halves.
    - Example: Rhombus has 2 lines of symmetry.

Similar Triangles

• In similar triangles, corresponding sides are proportional:
    - $rac{15}{5} = rac{16.5}{EF}$ leads to cross-multiplication and solving for EF yields $EF = 5.5$.

Trigonometry Basics

• SOCA TOA Mnemonic: This helps in identifying sine, cosine, and tangent ratios depending on the angles of right triangles:
    - $ext{Sine} = rac{ ext{Opposite}}{ ext{Hypotenuse}}$.
    - Example: If opposite is 8.6, hypotenuse is found using sine function and rearranging gives lengths.

Lines and Slopes

• Equation of a Line: Standard form is written as $y = mx + c$ where:
    - m = slope (gradient)
    - c = y-intercept

• Properties of parallel lines (same slope) and perpendicular lines (product of slopes = -1).

Set Theory

• Definitions:
    - Subset: A set completely contained within another.
    - Proper Subset: Elements of set A are also in set B, but B has at least one additional element.
    - Disjoint Sets: No common elements.
    - Intersection: Elements common to both sets, denoted as $A \bigcap B$.
    - Union: Combination of all elements from both sets, denoted as $A \bigcup B$.
    - Complement: All elements not in the specified set.

Vector Operations

• Subtraction of Vectors: Given vectors a and b, subtraction results in a new vector comprising their differences.
    - Example: For vectors (3,6) and (5,10), the resultant vector from a to b can be computed as: (5 - 3, 10 - 6) = (2, 4).

Transformations in Geometry

• Types of Transformations:
    - Reflection: Flipping over a line creating a mirror image.
    - Rotation: Turning shapes around a fixed point, could be at specific angles (90°, 180°, etc.).
    - Translation: Moving shapes without rotating or flipping.
    - Enlargement: Changing size while maintaining proportions, defined by a scale factor.

Probability Calculations

• To calculate probability:
    - Total given: 20 with 6 red, 9 blue, and 5 white.
    - Probability formula used: For red balls = $rac{6}{20}$; for white = $rac{5}{20}$; for yellow (none present) = 0.

Functions and Evaluations

• Evaluating functions:
    - Example for function application: Plugging in 3x into a function gives output = $3 + 4(3x) = 3 + 12x$.

Angles of a Sector and their Calculations

• Understanding arc lengths and areas: To solve problems related to circles, you need the formulas for arc lengths, areas, and sector perimeters, computed based on the inputs and relationships of angles in circles.
    - Example process detailed for different situations.