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PROOF 09

Prove.

√2 is not rational

On the reals ℝ.

What you may use

A1q = a/b, gcd(a, b) = 1
representation in lowest terms (b > 0)
A2n = 2k or n = 2k + 1
parity of integers (never both)
A3(a + b) + c = a + (b + c)
associativity of addition
A4(ab)c = a(bc)
associativity of multiplication
A5ab = ba
commutativity of multiplication
A6a × 1 = a
multiplicative identity
A7a(b + c) = ab + ac
distributive law
A8ca = cb, c ≠ 0 ⟹ a = b
cancellation law of multiplication
Ea = b ⟹ apply the same operation to both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

a/b denotes the x satisfying b × x = a.

Proof.

Related labs

Irrational Numbers lab →

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