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.,$$\sqrt{2}$$, $$\pi$$, $$e$$.
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 $$\frac{p}{q}$$: No integers $$p$$ and $$q$$ satisfy the equation (e.g., $$\sqrt{2} \neq \frac{p}{q} $$). Decimal Expansion: Never repeats or terminates. π = 3.141592653... (no pattern/no repetition).

Common Examples: π = 3.141592653..., $$\sqrt{2}$$= 1.414213562..., e (Euler’s number) = 2.71828182845...

3. Examples

Example 1 (Basic)

Problem: Determine if $$\sqrt{8} + \sqrt{2}$$ is rational or irrational.

Step-by-Step Solution:

Express $$\sqrt{8}$$ as $$\sqrt{4 \cdot 2} = 2\sqrt{2}$$.
The expression becomes $$2\sqrt{2} + \sqrt{2} = 3\sqrt{2}$$.
Since $$\sqrt{2}$$ is irrational and multiplying it by a rational number (3) does not make it rational, $$3\sqrt{2}$$ is irrational.

Example 2 (Intermediate)

Problem: Prove that $$\sqrt{2}$$ is irrational.

Step-by-Step Solution:

Assume $$\sqrt{2} = \frac{p}{q}$$ where $$p$$ and $$q$$ are coprime integers.
Squaring both sides gives $$2 = \frac{p^2}{q^2}$$ which implies $$p^2 = 2q^2$$.
This shows $$p^2$$ is even, so $$p$$ must be even. Let $$p = 2k$$ for some integer $$k$$.
Substituting $$p = 2k$$ into the equation gives $$(2k)^2 = 2q^2$$ or $$4k^2 = 2q^2$$, simplifying to $$2k^2 = q^2$$.
This shows $$q^2$$ is even, so $$q$$ must also be even.
Since both $$p$$ and $$q$$ are even, they share a common factor of 2, contradicting the assumption that they are coprime.
Therefore, $$\sqrt{2}$$ 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., $$\sqrt{2}$$ irrational, $$\sqrt{4}=2$$ rational).
Look for non-repeating, non-terminating decimals (e.g., 0.123456789101112...).
Operations with irrationals:
Rational + Irrational = Irrational (e.g., $$2 + \sqrt{3}$$).
Non-zero Rational × Irrational = Irrational (e.g., 3 × π).
Exception: 0 × Irrational = 0 (rational).