Irrational Numbers - Algebra-1

1. Fundamental Concepts

Definition: Irrational numbers are real numbers that cannot be expressed as a ratio of two integers and have non-terminating, non-repeating decimal expansions, eg.,, , .
Properties: Irrational numbers cannot be represented as fractions and do not terminate or repeat in their decimal form.

2. Key Concepts

Characteristics: Cannot be written as : No integers and satisfy the equation (e.g., ). Decimal Expansion: Never repeats or terminates. π = 3.141592653... (no pattern/no repetition).

Common Examples: π = 3.141592653..., = 1.414213562..., e (Euler’s number) = 2.71828182845...

3. Examples

Example 1 (Basic)

Problem: Determine if is rational or irrational.

Step-by-Step Solution:

Express as .
The expression becomes .
Since is irrational and multiplying it by a rational number (3) does not make it rational, is irrational.

Example 2 (Intermediate)

Problem: Prove that is irrational.

Step-by-Step Solution:

Assume where and are coprime integers.
Squaring both sides gives which implies .
This shows is even, so must be even. Let for some integer .
Substituting into the equation gives or , simplifying to .
This shows is even, so must also be even.
Since both and are even, they share a common factor of 2, contradicting the assumption that they are coprime.
Therefore, cannot be expressed as a ratio of two integers and is irrational.

4. Problem-Solving Techniques

To identify irrational numbers:
Check if it’s a non-perfect square/cube root (e.g., irrational, rational).
Look for non-repeating, non-terminating decimals (e.g., 0.123456789101112...).
Operations with irrationals:
Rational + Irrational = Irrational (e.g., ).
Non-zero Rational × Irrational = Irrational (e.g., 3 × π).
Exception: 0 × Irrational = 0 (rational).