Number Systems
Real numbers, rational and irrational numbers
The real number system includes all numbers you can think of — whole numbers, fractions, decimals, and more. Real numbers are divided into two main groups: rational numbers and irrational numbers.
A rational number is any number that can be written as a fraction p/q, where p and q are integers and q ≠ 0. Examples include 1/2, 3, 0.75, and −4. When written as decimals, rational numbers either terminate (like 0.5) or repeat (like 0.333...).
An irrational number cannot be written as a simple fraction. Its decimal goes on forever without repeating. Famous examples are √2 = 1.41421... and π = 3.14159.... You cannot write these exactly as p/q.
Sets are collections of distinct objects. A set of even numbers is {2, 4, 6, 8...}. Sets can be subsets of each other — for example, all natural numbers are a subset of real numbers. Key operations on sets include union (∪), intersection (∩), and complement.
A function is a rule that assigns exactly one output to every input. If you put x = 3 into f(x) = 2x + 1, you get f(3) = 7. Functions are like machines: one input gives one output.
Logarithms are the inverse of powers. If 10² = 100, then log₁₀(100) = 2. The key log laws are:
Key Points to Remember
- 1Real number system
- 2Rational and irrational numbers
- 3Logarithms
- 4Sets and functions
Pakistan Example
Counting Mangoes in Multan vs. Measuring π in Islamabad
When a fruit seller in Multan counts 50 mangoes, he uses a natural (rational) number. But when engineers designing the Islamabad Expressway calculate the circumference of a circular roundabout, they multiply diameter by π — an irrational number that never ends.