PROOF 85
Prove.
The sum of a rational and an irrational is irrational
On the real numbers ℝ.
A way to read this proof
First separate what is assumed from what is to be proved. At each step, ask which definition or earlier result permits the move. If it is unclear, return to that premise rather than skipping to the conclusion.
Definition D1
a − b denotes a + (−b).
Proof.
Subtraction is not a new operation; it is addition of the additive inverse.convention
∎
Definition D2
That α is irrational means α is real and not rational.
Proof.
There are two conditions: being real, and not being rational.convention
∎
Lemma L1
If q is rational, then −q is rational.
Proof.
(1)0 − q = 0 + (−q)D1
(2)0 + (−q) = (−q) + 0 = −qA5, A7
(3)0 is rational.A8
(4)q is also rational, so 0 − q is rational.assumption, (3), A3
(5)Therefore −q is rational.(1), (2), (4), E
∎
Theorem T1
If q is rational and α is irrational, then q + α is irrational.
Proof.
(1)Suppose q + α is rational.assumption, D2
(2)−q is rational.L1
(3)Therefore (q + α) + (−q) is rational.(1), (2), A2
(4)(q + α) + (−q) = (α + q) + (−q)A5
(5) = α + (q + (−q))A4
(6) = α + 0 = αA6, A7
(7)Therefore α is rational.(3), (4), (5), (6), E
(8)This contradicts α being irrational.(1), (7), D2
(9)q + α is real and not rational. That is, irrational.(8), A1, D2
∎
Corollary C
The sum of two irrationals need not be irrational.
Proof.
(1)√2 is real and not rational, hence irrational.A9, D2
(2)Suppose −√2 is rational.assumption
(3)Then (−√2) + √2 is irrational.(1), (2), T1
(4)(−√2) + √2 = √2 + (−√2) = 0A5, A6
(5)0 is rational.A8
(6)(3), (4), (5) cannot all stand together. Therefore −√2 is irrational.(3), (4), (5), D2, E
(7)√2 and −√2 are both irrational, and their sum is 0.(1), (6), (4)
(8)0 is rational, so their sum is not irrational.(7), (5), D2
∎
Remark
Can A3 be removed from this proof?
The proof subtracts q from q + α to recover α. Check what that recovering operation rests on.