1.2 Irrational Numbers & Advanced Proofs

Step 1

Assume the contrary, that $x = 3 + 2\sqrt{5}$ is rational.

Step 2

Rearrange to isolate the irrational term.

$x - 3 = 2\sqrt{5} \implies \frac{x-3}{2} = \sqrt{5}$

Step 3

Analyze the equation: Since $x$ is rational, $\frac{x-3}{2}$ is also rational (closure of rationals).

Step 4

Contradiction: We have Rational = Irrational ( $\sqrt{5}$ ). This is impossible.

Step 5

Therefore, the assumption is false.

Final Answer

$3 + 2\sqrt{5}$ is irrational.

Operation

Number

Quick Reference:

N = Natural Numbers: 1, 2, 3, ...

W = Whole Numbers: 0, 1, 2, ...

Z = Integers: ..., -2, -1, 0, 1, 2, ...

Q = Rational Numbers: p/q form (p, q ∈ Z, q ≠ 0)

R = Real Numbers: Q ∪ Irrational

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