Question

prove that is irrational

$\sqrt{2}$
​

Answer 1

Step-by-step explanation:

Let us assume √2 is a rational number. Their exist unique integers like p and q where p and q are co-prime (€) to the integer (Z) and q ≠ 0.

→ √2 = p/q.

→ √2. q = p → Consider Equation no 1.

Squaring on both sides.

→ (√2. q)² = p².

→ 2. q² = p².

2 divides the p² and 2 divides p.

→ p = 2k.

Substitute the value of p in Equation no 1.

→ √2. q = p.

→ √2. q = 2k.

Squaring on both sides.

→ (√2. q)² = (2k)².

→ 2q² = 4k².

→ q² = 4/2 k².

→ q² = 2k².

2 divides q² and 2 divides q.

p and q divides 2.

Therefore, our assumption is √2 is a rational number is wrong.

Hence √2 is irrational number.