Number System Class 9 Maths Notes | Chapter 1 Revision

Key Concepts: Must Remember!

• Rational Numbers (Q): Can be expressed as p/q, where p, q are integers and q ≠ 0. Their decimal expansion is either terminating or non-terminating repeating.
• Irrational Numbers (I): Cannot be expressed as p/q. Their decimal expansion is non-terminating non-repeating.
• Real Numbers (R): The collection of all rational and irrational numbers. Every real number corresponds to a unique point on the number line, and vice-versa.
• Between any two distinct rational numbers, there are infinitely many rational numbers (and irrational numbers).
• The sum/difference of a rational and an irrational number is always irrational.
• The product/quotient of a non-zero rational and an irrational number is always irrational.
• If we add, subtract, multiply, or divide two irrational numbers, the result can be either rational or irrational.
• Laws of Exponents for Real Numbers: For a > 0 (real number) and p, q (rational exponents): a^p · a^q = a^(p+q); (a^p)^q = a^(pq); a^p / a^q = a^(p-q); a^p · b^p = (ab)^p.

Essential Definitions

Natural Numbers (N)

The counting numbers: {1, 2, 3, ...}

Whole Numbers (W)

Natural numbers including zero: {0, 1, 2, 3, ...}

Integers (Z)

Whole numbers and their negatives: {..., -3, -2, -1, 0, 1, 2, 3, ...}

Rational Numbers (Q)

Numbers that can be written in the form p/q, where p and q are integers and q ≠ 0.

Irrational Numbers (I)

Numbers that cannot be expressed in the form p/q. Their decimal expansion is non-terminating and non-repeating. Examples: √2, π.

Real Numbers (R)

The set of all rational and irrational numbers. They can be represented as points on a number line.

Terminating Decimal Expansion

A decimal expansion that ends after a finite number of digits (e.g., 0.5, 2.75). These represent rational numbers.

Non-terminating Repeating Decimal Expansion

A decimal expansion that goes on forever, but a block of digits repeats periodically (e.g., 0.333..., 1.272727...). These represent rational numbers.

Non-terminating Non-repeating Decimal Expansion

A decimal expansion that goes on forever without any repeating pattern (e.g., 0.101101110...). These represent irrational numbers.

Rationalization

The process of converting an irrational denominator of a fraction into a rational number by multiplying both numerator and denominator by a suitable factor (rationalizing factor).

Understanding Decimal Expansions & Number Classification

One of the most crucial ways to distinguish between rational and irrational numbers is by examining their decimal expansions. Every real number has a decimal expansion. For rational numbers, there are two possibilities:

1. Terminating Decimal Expansion: These are decimals that end after a finite number of digits. For example, 1/2 = 0.5, 3/4 = 0.75, or 1/8 = 0.125. These fractions can always be simplified such that the denominator has only prime factors of 2 and/or 5.
2. Non-terminating Repeating (or Recurring) Decimal Expansion: These are decimals that continue indefinitely, but a specific block of digits repeats an infinite number of times. For instance, 1/3 = 0.333... (where 3 repeats), 2/7 = 0.285714285714... (where '285714' repeats). Such repeating decimals can always be converted back into the p/q form, proving their rational nature.

In contrast, irrational numbers have only one type of decimal expansion:

• Non-terminating Non-repeating Decimal Expansion: These decimals go on forever without any repeating pattern whatsoever. Take √2, for example, its decimal expansion is 1.41421356... and it never ends or repeats a sequence of digits. Similarly, π (pi) is approximately 3.14159265... and also shows no repeating pattern. This unique characteristic is the defining feature of irrational numbers, differentiating them clearly from rational numbers. Understanding this distinction is fundamental for classifying numbers encountered in various mathematical contexts.

Representing Irrational Numbers on the Number Line (e.g., √x)

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Solved Examples for Quick Understanding

• Example 1: Convert 0.7̅ to p/q form. Let x = 0.777... (1) Multiply by 10: 10x = 7.777... (2) Subtract (1) from (2): 10x - x = 7.777... - 0.777... 9x = 7 x = 7/9
• Example 2: Rationalize the denominator of 1/√7. Multiply numerator and denominator by √7: (1/√7) (√7/√7) = √7 / (√7 √7) = √7 / 7
• Example 3: Simplify (√5 + √2)² Using (a+b)² = a² + 2ab + b²: (√5)² + 2(√5)(√2) + (√2)² = 5 + 2√10 + 2 = 7 + 2√10

Exam Tip: Avoiding Common Pitfalls

When dealing with rationalization, always ensure you multiply by the conjugate when the denominator is of the form (a + √b) or (a - √b). For instance, to rationalize 1/(3+√2), multiply by (3-√2)/(3-√2). Don't forget to apply the laws of exponents correctly; a common mistake is adding exponents when bases are different (e.g., 2³ × 3² ≠ 6⁵). Double-check if the question asks for rational or irrational numbers between two given numbers, as methods differ. For representing √x on the number line, ensure your construction lines are clear and all points are accurately marked for full marks.

Practice Questions with Solutions

• Q: Is 3.1416 an irrational number? Justify. A: No, it is a rational number because its decimal expansion is terminating. It can be written as 31416/10000.
• Q: Give an example of an irrational number between 1/7 and 2/7. A: 1/7 ≈ 0.1428... and 2/7 ≈ 0.2857.... An irrational number between them could be 0.150150015000... (a non-terminating, non-repeating decimal).
• Q: What is the sum of a rational number and an irrational number? A: The sum is always an irrational number.
• Q: Simplify: (2^2/3) (2^1/3). A: Using a^p a^q = a^(p+q), we get 2^(2/3 + 1/3) = 2^(3/3) = 2^1 = 2.

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