📐 Number Systems - Class 9

1. Introduction to Number Systems

Number Systems form the foundation of mathematics. In this chapter, we will learn about different types of numbers, their properties, and how to work with them.

2. Types of Numbers

2.1 Natural Numbers (ℕ)

• Natural numbers are counting numbers that start from 1 and go on forever.
• Examples: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
• Symbol: ℕ = {1, 2, 3, 4, 5, ...}
• Natural numbers do NOT include zero, negative numbers, fractions, or decimals.
• They are the most basic numbers we use in daily counting.

2.2 Whole Numbers (W)

• Whole numbers are natural numbers plus zero.
• Examples: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
• Symbol: W = {0, 1, 2, 3, 4, 5, ...}
• The only difference between natural and whole numbers is the inclusion of 0.
• Whole numbers also do NOT include negative numbers, fractions, or decimals.

2.3 Integers (ℤ)

• Integers include all whole numbers and their negative counterparts.
• Examples: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Symbol: ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}
• Positive integers: 1, 2, 3, 4, ... (same as natural numbers)
• Negative integers: -1, -2, -3, -4, ...
• Zero (0) is neither positive nor negative.
• Integers do NOT include fractions or decimals.

2.4 Rational Numbers (ℚ)

• Rational numbers are numbers that can be written as a fraction p/q, where p and q are integers and q ≠ 0.
• Examples: 1/2, 3/4, -5/7, 2 (can be written as 2/1), 0.5 (equals 1/2), 0.75 (equals 3/4)
• Symbol: ℚ (represents rational numbers)
• All integers are rational numbers because they can be written as fractions (e.g., 5 = 5/1).
• Decimal forms: Rational numbers have either terminating decimals (like 0.5, 0.25) or repeating decimals (like 0.333..., 0.666...).
• Examples of repeating decimals: 1/3 = 0.3333... (written as 0.3̅), 2/3 = 0.6666... (written as 0.6̅)

2.5 Irrational Numbers

• Irrational numbers CANNOT be written as a simple fraction p/q.
• Their decimal representation is non-terminating and non-repeating.
• Examples: √2, √3, √5, π (pi ≈ 3.14159...), e (Euler's number ≈ 2.71828...)
• √2 = 1.414213562... (goes on forever without repeating)
• π = 3.141592653... (never repeats or terminates)
• Golden ratio (φ) = 1.618033988... is also irrational.

2.6 Real Numbers (ℝ)

• Real numbers include BOTH rational and irrational numbers.
• Symbol: ℝ = Rational Numbers + Irrational Numbers
• Every number on the number line is a real number.
• Real numbers can be positive, negative, or zero.
• All natural, whole, integers, rational, and irrational numbers are real numbers.

3. Representation on Number Line

3.1 Representing Integers

• Draw a horizontal line and mark a point as 0 (origin).
• Positive integers are marked to the right of 0.
• Negative integers are marked to the left of 0.
• Each point is equally spaced (unit distance).
• Example: ..., -3, -2, -1, 0, 1, 2, 3, ...

3.2 Representing Rational Numbers

• Rational numbers like 1/2, 3/4, etc. lie between integers.
• To represent 1/2: Mark the midpoint between 0 and 1.
• To represent 3/4: Divide the distance between 0 and 1 into 4 equal parts, mark the 3rd point.
• Rational numbers fill the gaps between integers on the number line.

3.3 Representing Irrational Numbers

• Irrational numbers like √2, √3 can be represented using geometric construction.
• Method for √2: Draw a right triangle with both sides = 1 unit. Hypotenuse = √2 (by Pythagoras theorem).
• Use compass to transfer this length to the number line.
• Similarly, √3, √5, and other square roots can be constructed.

4. Operations on Real Numbers

4.1 Addition and Subtraction

• When you add or subtract two rational numbers, the result is always rational.
• When you add or subtract a rational and an irrational number, the result is irrational.
• Example: 2 + √3 is irrational (2 is rational, √3 is irrational).
• Example: √2 + √3 is irrational (both are irrational).

4.2 Multiplication and Division

• Product of two rational numbers is always rational.
• Product of a non-zero rational and an irrational number is irrational.
• Example: 2 × √3 = 2√3 (irrational).
• Example: √2 × √3 = √6 (irrational).
• Special case: √2 × √2 = 2 (rational, because it's a perfect square).

5. Laws of Exponents for Real Numbers

Exponents help us write repeated multiplication in a shorter form. For any real numbers a and b, and positive integers m and n:

5.1 Basic Laws of Exponents

• Product Law: aᵐ × aⁿ = aᵐ⁺ⁿ (when bases are same, add the exponents)
• Quotient Law: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (when bases are same, subtract the exponents)
• Power of a Power: (aᵐ)ⁿ = aᵐˣⁿ (multiply the exponents)
• Power of a Product: (ab)ᵐ = aᵐ × bᵐ
• Power of a Quotient: (a/b)ᵐ = aᵐ/bᵐ
• Zero Exponent: a⁰ = 1 (any number raised to power 0 equals 1, except 0⁰ which is undefined)
• Negative Exponent: a⁻ᵐ = 1/aᵐ (negative exponent means reciprocal)

6. Rationalization of Denominators

Rationalization means converting an irrational denominator into a rational one by multiplying both numerator and denominator by a suitable number.

6.1 Why Rationalize?

• It makes calculations easier and more accurate.
• Standard form: We prefer not having square roots in the denominator.
• It helps in comparing and adding fractions with irrational denominators.

6.2 How to Rationalize?

• Single Term: For 1/√a, multiply both numerator and denominator by √a.
• Result: 1/√a × √a/√a = √a/a
• Example: 1/√2 = (1 × √2)/(√2 × √2) = √2/2

• Binomial Form: For 1/(a + √b), multiply by conjugate (a - √b).
• Conjugate pairs: (a + √b) and (a - √b)
• Using identity: (a + √b)(a - √b) = a² - b
• Example: 1/(2 + √3) = (2 - √3)/[(2 + √3)(2 - √3)] = (2 - √3)/(4 - 3) = 2 - √3

7. Converting Decimals to Fractions

7.1 Terminating Decimals

• Decimals that end after a finite number of digits.
• Method: Count decimal places, write denominator as 10ⁿ (where n = number of decimal places).
• Example: 0.25 = 25/100 = 1/4
• Example: 0.625 = 625/1000 = 5/8

7.2 Non-Terminating Repeating Decimals

• Decimals where one or more digits repeat infinitely.
• Method: Let x = repeating decimal, multiply by 10ⁿ to shift decimal, subtract to eliminate repeating part.

8. Important Properties of Real Numbers

8.1 Closure Property

• Addition: If a and b are real numbers, then a + b is also a real number.
• Multiplication: If a and b are real numbers, then a × b is also a real number.

8.2 Commutative Property

• Addition: a + b = b + a (order doesn't matter)
• Multiplication: a × b = b × a
• Note: Subtraction and division are NOT commutative.

8.3 Associative Property

• Addition: (a + b) + c = a + (b + c)
• Multiplication: (a × b) × c = a × (b × c)
• Grouping doesn't affect the result.

8.4 Distributive Property

• a × (b + c) = (a × b) + (a × c)
• Multiplication distributes over addition.
• Example: 3 × (4 + 5) = (3 × 4) + (3 × 5) = 12 + 15 = 27

8.5 Identity Property

• Additive Identity: a + 0 = a (zero is the additive identity)
• Multiplicative Identity: a × 1 = a (one is the multiplicative identity)

8.6 Inverse Property

• Additive Inverse: a + (-a) = 0 (negative of a number)
• Multiplicative Inverse: a × (1/a) = 1, where a ≠ 0 (reciprocal of a number)