1.

6. Prove that √3 is an irrational

Answer»

Step-by-step explanation:

√3 = p/Q

⇒ 3 = p²/q² (SQUARING on both the sides)

⇒ 3q² = p²………………………………..(1)

This means that 3 divides p2. This means that 3 divides p because each factor should appear two times for the SQUARE to exist.

So we have p = 3r

where r is some integer.

⇒ p² = 9r²………………………………..(2)

from equation (1) and (2)

⇒ 3q² = 9r²

⇒ q² = 3r²

Where q² is multiply of 3 and also q is multiple of 3.

Then p, q have a common factor of 3. This runs contrary to their being co-primes. Consequently, p / q is not a rational number. This demonstrates that √3 is an irrational number.


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