Chapter 1: Number Systems

1. Classification of Numbers

Natural Numbers (N): The counting numbers starting from 1 (1, 2, 3, ...).
Whole Numbers (W): The collection of natural numbers including zero (0, 1, 2, 3, ...).
Integers (Z): The collection of all whole numbers and the negative of natural numbers (..., -3, -2, -1, 0, 1, 2, 3, ...). The symbol 'Z' comes from the German word zahlen (to count).
Rational Numbers (Q): A number 'r' is rational if it can be written in the form p/q, where 'p' and 'q' are integers and q ≠ 0.
- The rational numbers include natural numbers, whole numbers, and integers.
- There are infinitely many rational numbers between any two given rational numbers.

2. Irrational Numbers

A number 's' is called irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.

Discovery: Discovered by the Pythagoreans in Greece (around 400 BC).
Examples: √2, √3, √5, √15, π, and numbers like 0.101101110...
Square Roots: The square roots of positive integers which are not perfect squares are irrational (e.g., √2 is irrational, but √4 is rational because it equals 2).

3. Real Numbers and the Number Line

The collection of all rational numbers and irrational numbers together forms the collection of Real Numbers (R).

Every real number is represented by a unique point on the number line.
Conversely, every point on the number line represents a unique real number. Therefore, the number line is often called the real number line.
Irrational numbers like √n (where n is a positive integer) can be located geometrically on the number line using the Pythagoras theorem.

4. Decimal Expansions

Real numbers can be distinguished by the nature of their decimal expansions:

Rational Numbers

The decimal expansion of a rational number is either:

Terminating: The remainder becomes zero after some steps (e.g., 7/8 = 0.875).
Non-terminating Recurring (Repeating): The remainder never becomes zero, and a block of digits repeats periodically (e.g., 10/3 = 3.333...).

Irrational Numbers

The decimal expansion of an irrational number is Non-terminating Non-recurring. (e.g., π = 3.141592... with no repeating pattern).

5. Operations on Real Numbers

Rational and irrational numbers satisfy the commutative, associative, and distributive laws, but specific rules apply to their combinations:

Addition/Subtraction: The sum or difference of a rational number and an irrational number is irrational.
Multiplication/Division: The product or quotient of a non-zero rational number with an irrational number is irrational.
Two Irrationals: If you add, subtract, multiply, or divide two irrational numbers, the result may be rational or irrational.

6. Identities Involving Square Roots

For positive real numbers a and b:

√(ab) = √a × √b
√(a/b) = √a / √b
(√a + √b)(√a - √b) = a - b
(a + √b)(a - √b) = a² - b
(√a + √b)² = a + 2√(ab) + b

7. Rationalising the Denominator

When the denominator of an expression contains a square root, the process of converting it into an equivalent expression with a rational denominator is called rationalising the denominator.

Method: Use the identity (x + y)(x - y) = x² - y².

Example: To rationalise 1 / (√a + b), multiply the numerator and denominator by the conjugate (√a - b).