Q1.
Can √2 be written as a rational number p/q?
Answer
No. There is no pair of integers p, q with q ≠ 0 for which √2 = p/q.
A first look at why the answer must be no, before the formal proof of Section 3.5.1:
if √2 = p/q then 2q² = p²
count the factor 2 on each side
p² has an even number of 2s (each 2 in p appears twice)
q² also has an even number of 2s
so 2q² has an odd number of 2s
an even count can never equal an odd count
count the factor 2 on each side
p² has an even number of 2s (each 2 in p appears twice)
q² also has an even number of 2s
so 2q² has an odd number of 2s
an even count can never equal an odd count
Why it happens: squaring doubles every exponent in a prime factorisation, so a perfect square always carries each prime an even number of times. Multiplying by a single extra 2 breaks that parity. Since 2 is not a perfect square, its square root cannot be a ratio of integers. The number √2 is therefore irrational — the very first number ever proved to be so.
Did you know? The chapter reaches √2 through geometry: the diagonal d of a unit square satisfies 1² + 1² = d², so d² = 2. A perfectly ordinary length turns out to be a number no fraction can name.