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Mathematics · Chapter 1: Introduction to Mathematics
Branches of Mathematics
The branches of mathematics: pure and applied, including arithmetic, algebra, geometry, trigonometry, calculus, statistics and probability, with examples.
Key termRememberExampleWatch out
01Concept
Mathematics is divided into two big groups. Pure mathematics studies ideas for their own sake, such as numbers and shapes. Applied mathematics uses mathematics to solve real problems in science, business and daily life. The two overlap: an idea from pure maths is often used later in applied maths.
| Branch | What it studies | Example |
|---|---|---|
| Arithmetic | Numbers and the four operations | Working out a shopping bill |
| Algebra | Using letters for unknown or changing numbers | Solving 2x + 3 = 11 gives x = 4 |
| Geometry | Shapes, sizes, angles and positions | Area of a plot of land |
| Trigonometry | The sides and angles of triangles | Finding the height of a minaret from its shadow |
| Calculus | Rates of change and areas under curves | How fast a car's speed is changing |
| Statistics | Collecting, organising and analysing data | Average marks of a class |
| Probability | How likely an event is | Chance of winning the toss = ½ |
| Coordinate geometry | Shapes described with coordinates (x, y) | The straight line y = 2x + 1 |
| Mechanics | Forces and motion, using equations | Path of a thrown ball |
02Real World Examples
Building a house
Architects use geometry for the plan, and engineers use trigonometry for roof slopes and stairs.
Population census
The Pakistan Bureau of Statistics uses statistics to count people and plan schools and hospitals.
Insurance
Insurance companies use probability to work out how much to charge for a policy.
Space launches
Rocket paths are calculated with calculus and mechanics.
03Practice Questions
Try to answer in your own words first, then open the model answer.
Short questions
Q1.Differentiate between pure and applied mathematics.Show answer
Answer
- Pure mathematics studies ideas for their own sake. Algebra, geometry, number theory.
- Applied mathematics uses maths to solve real problems. Statistics, mechanics, financial maths.
Q2.Define algebra and geometry with one example each.Show answer
Answer
- Algebra: uses letters to stand for unknown numbers. 2x + 3 = 11, so x = 4.
- Geometry: studies shapes, sizes, angles and positions. The area of a rectangle.
Q3.What is statistics? Give one use.Show answer
Answer
Statistics is the branch that deals with collecting, organising, analysing and presenting data. Example: finding the average marks of a class, or a population census.
Q4.What does trigonometry study? Give a real use.Show answer
Answer
Trigonometry studies the relationship between the sides and angles of triangles. Example: finding the height of a tall building or minaret from the length of its shadow.
Long questions
Q5.Describe the main branches of mathematics with examples.Show answer
Answer
- Arithmetic: numbers and operations. A shopping bill.
- Algebra: letters for unknowns. Solving equations.
- Geometry: shapes and angles. Area of land.
- Trigonometry: sides and angles of triangles. Height of a minaret.
- Calculus: rates of change. Speed of a car.
- Statistics: data. Census.
- Probability: chance. Winning the toss.