Mathematics: Important Concepts & Calculations
✨ Comprehensive Mathematics Concepts & Formulas
1. Numbers, Integers, & Rational Numbers
| Concept | Rule / Property / Formula |
|---|---|
| Product of Reciprocals | The product of a rational number and its reciprocal is: \[x \times \frac{1}{x} = 1\]. |
| Commutativity (Integers/Rational) | Rule: Addition is commutative \( (a+b = b+a) \), but subtraction is not \( (a-b \neq b-a) \). |
| Distributivity (Rational Numbers) | Rule: Multiplication distributes over addition: \[a \times (b + c) = a \times b + a \times c\]. |
| Rational Number Form | A number of the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). |
| Cube Root (Inverse Operation) | If \(N^3 = X\), then the cube root of \(X\) is \(\sqrt[3]{X} = N\). |
2. Arithmetic & Commercial Math
| Concept | Rule / Property / Formula |
|---|---|
| BODMAS Rule |
( ), { }, or [ ] first.3² or √9).÷) and Multiplication (×): Solve these next. They have equal priority, so you solve them from left to right.+) and Subtraction (-): Solve these last. They also have equal priority and are solved from left to right. |
| Simple Interest (SI) | Formula: \[\text{SI} = \frac{P \times R \times T}{100}\] (\(P\)=Principal, \(R\)=Rate, \(T\)=Time). |
| Compound Interest (CI) | Amount Formula: \[A = P\left(1 + \frac{R}{100}\right)^n\] (\(n\) = number of conversion periods). |
| Ratio (a:b) | Compares two quantities using division; expressed as \(a/b\). |
| Sales Tax/VAT/GST | Tax added to the Selling Price (SP): **Bill Amount = SP + Tax Amount**. |
3. Algebra & Exponents
| Concept | Rule / Property / Formula |
|---|---|
| Standard Identity 2 | Formula: \[(a – b)^2 = a^2 – 2ab + b^2\]. |
| Difference of Squares | Formula: \[(a – b)(a + b) = a^2 – b^2\]. |
| Laws of Exponents: Quotient Rule | Rule: \[\frac{a^m}{a^n} = a^{m-n}\]. |
| Laws of Exponents: Negative | Rule: \[a^{-m} = \frac{1}{a^m}\]. |
| Standard Form (Large Numbers) | A number \(N\) expressed as: \[a \times 10^k\] (where \(1 \le a < 10\) and \(k\) is an integer). |
4. Geometry & Mensuration
2D Shapes (Perimeter & Area)
| Shape | Perimeter Formula | Area Formula |
|---|---|---|
| Square | \[ P = 4s \] (\(s\) = side length) | \[ A = s^2 \] |
| Rectangle | \[ P = 2(l + w) \] (\(l\) = length, \(w\) = width) | \[ A = l \times w \] |
| Triangle | \[ P = a + b + c \] (\(a, b, c\) = side lengths) | \[ A = \frac{1}{2} \times b \times h \] (\(b\) = base, \(h\) = height) |
| Circle | \[ C = 2 \pi r \] (Circumference) (\(r\) = radius) | \[ A = \pi r^2 \] |
| Parallelogram | \[ P = 2(a + b) \] (\(a, b\) = adjacent sides) | \[ A = b \times h \] (\(b\) = base, \(h\) = vertical height) |
| Trapezium | \[ P = a + b + c + d \] (Sum of all four sides) | \[ A = \frac{1}{2}(a + b)h \] (\(a, b\) = parallel sides, \(h\) = height) |
| Rhombus | \[ P = 4s \] (\(s\) = side length) | \[ A = \frac{1}{2} \times d_1 \times d_2 \] (\(d_1, d_2\) = diagonals) |
| Kite | \[ P = 2(a + b) \] (\(a, b\) = adjacent side lengths) | \[ A = \frac{1}{2} \times d_1 \times d_2 \] (\(d_1, d_2\) = diagonals) |
Basic Concepts & Angles
| Concept | Definition / Property |
|---|---|
| Point, Line, Ray | A Point has no size. A Line extends infinitely in both directions. A Ray has one endpoint and extends infinitely in one direction. |
| Line Segment | A part of a line that is bounded by two distinct endpoints. |
| Pairs of Lines (Intersecting) | Two lines that have one common point. |
| Pairs of Lines (Parallel) | Two lines in a plane that never meet; they are always the same distance apart. |
| Types of Angles | Acute: Less than \(90^\circ\). Right: Exactly \(90^\circ\). Obtuse: Greater than \(90^\circ\) but less than \(180^\circ\). Straight: Exactly \(180^\circ\). |
Polygons (Triangles & Quadrilaterals)
| Concept | Definition / Property |
|---|---|
| Polygon | A simple closed curve made up entirely of line segments. |
| Types of Triangles (By Side) | Equilateral: All three sides are equal. Isosceles: Two sides are equal. Scalene: No sides are equal. |
| Types of Triangles (By Angle) | Acute-angled: All angles are acute. Right-angled: One angle is a right angle (\(90^\circ\)). Obtuse-angled: One angle is obtuse. |
| Triangle Angle Sum Property | The sum of the three interior angles of any triangle is always \(180^\circ\). |
| Parallelogram | A quadrilateral with two pairs of parallel sides. Opposite sides are equal in length, and opposite angles are equal. |
| Rhombus | A parallelogram with all four sides of equal length. |
| Trapezium | A quadrilateral with at least one pair of parallel sides. |
3D Shapes & Symmetry
| Concept | Definition / Property |
|---|---|
| Faces, Edges, Vertices | Faces: The flat surfaces of a 3D shape. Edges: The line segments where two faces meet. Vertices: The points (corners) where edges meet. |
| Euler’s Rule (for Polyhedra) | Relates Faces (F), Vertices (V), and Edges (E): \[ F + V – E = 2 \] |
| Nets for 3D Shapes | A 2D pattern that you can fold along its edges to form a 3D shape. For example, a net for a cube is made of 6 squares. |
| Line Symmetry | A shape has line symmetry if it can be folded along a line (the line of symmetry) so that the two halves match exactly. |
| Rotational Symmetry | A shape has rotational symmetry if it looks the same after being rotated less than a full turn (\(360^\circ\)) around a central point. |
5. Data Handling & Probability
| Concept | Rule / Formula |
|---|---|
| Arithmetic Mean (Average) | \[\frac{\text{Sum of Observations}}{\text{Number of Observations}}\]. |
| Mode (Un-grouped Data) | The **observation that occurs most frequently**. |
| Probability | Formula: \[P(E) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}\]. |
| Median (Un-grouped Data, Odd Count) | Position after ordering: \[(\frac{n+1}{2})^{\text{th}}\text{ term}\] (\(n\) = number of observations). |