PROOF 96
Prove.
The condition under which √(ab) = √a √b holds
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
For a ≥ 0, √a denotes the nonnegative x satisfying x × x = a.
Proof.
(1)x² means x × x.convention
(2)A1 guarantees the existence of such an x.A1
(3)Choosing the nonnegative one is a convention. It is what makes √ determine a single value.convention
∎
Lemma L1
If x ≥ 0, y ≥ 0 and x² = y², then x = y
Proof.
(1)Suppose x ≠ y.assumption
(2)x < y or y < x.(1), A3
(3)If x < y, then x² < y².(2), A2
(4)If y < x, then y² < x².(2), A2
(5)Either way, x² ≠ y².(3), (4), A3
(6)This contradicts x² = y², so x = y.(5), assumption
∎
Theorem T
If a ≥ 0 and b ≥ 0, then √(ab) = √a √b
Proof.
(1)ab ≥ 0.A4
(2)√(ab) is defined, √(ab) ≥ 0, and (√(ab))² = ab.(1), D1
(3)√a ≥ 0, √b ≥ 0, and √a √a = a, √b √b = b.D1
(4)√a √b ≥ 0.(3), A4
(5)(√a √b)² = (√a √b)(√a √b)D1
(6) = (√a √a)(√b √b)A5, A6
(7) = ab(3), (6), E
(8)Both numbers are nonnegative and their squares equal ab.(2), (4), (7)
(9)Therefore √(ab) = √a √b.(8), L1
∎
Corollary C
The condition a ≥ 0, b ≥ 0 cannot be dropped. The equality breaks at a = b = −1.
Proof.
(1)Set a = −1, b = −1.assumption
(2)ab = (−1)(−1) = 1.(1), A8
(3)1 × 1 = 1 and 1 > 0, so the left side is √(ab) = √1 = 1.(2), A9, A10, D1
(4)For every real x, x² ≥ 0.A7
(5)−1 < 0, so no real x satisfies x² = −1.(4), A10, A3
(6)Therefore √(−1) is undefined on this stage.(5), D1
(7)The left side is determined as 1, but the right side √a √b has no meaning.(3), (6)
(8)Suppose the stage is widened to the complex numbers and √(−1) is set to i.assumption
(9)Then the right side is i × i = −1 while the left side is 1.(3), (8), convention
(10)Before widening there is no meaning; after widening the values differ.(7), (9)
∎
Remark
Can A2 be removed from this proof?
Find the line that passes from equal squares to the conclusion that the numbers are equal.