Definition D1
For a > 0, √a denotes an x satisfying x² = a.
Proof.
(1)A square root asks squaring in reverse: what number, squared, gives a?convention
(2)If several such x exist, this definition does not say which one √a refers to.convention
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Definition D2
When x² = a has several solutions, √a denotes the nonnegative one among them.
Proof.
(1)For the symbol √a to refer to a single value, each input a must have a single output.F
(2)Therefore, when there are several solutions, a convention must choose one of them.(1), D1
(3)The nonnegative one is chosen.convention
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Lemma L1
If a > 0, the only reals satisfying x² = a are x and −x, one positive and the other negative.
Proof.
(1)There exists x > 0 with x² = a.A1
(2)(−x)² = x² = a, so −x is also a solution.(1), A2, E
(3)x > 0, so −x < 0.(1), A5
(4)x and −x are two distinct solutions.(1), (2), (3), A4
(5)Take y with y² = a and y ≥ 0.assumption
(6)If x ≠ y, then x < y or y < x.A4
(7)Either way, x² ≠ y².(6), A3, A4
(8)x² = y² = a, so this is a contradiction, and therefore x = y.(1), (5), (7), E
(9)If z < 0 and z² = a, then −z > 0 and (−z)² = a.A2, A5, E
(10)By (8), −z = x, so z = −x.(8), (9)
(11)Therefore the only solutions are x and −x.(4), (8), (10)
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Theorem T
If a > 0, then √a is uniquely determined, √a ≥ 0 and (√a)² = a.
Proof.
(1)The only solutions of x² = a are x and −x.L1
(2)x > 0 and −x < 0, so there is exactly one nonnegative solution.(1), A5, A4
(3)D2 fixes that one as √a.(2), D2
(4)Therefore √a ≥ 0 and (√a)² = a.(3), D1, D2
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Corollary C
The solutions of x² = 4 are x = 2 and x = −2, but √4 = 2.
Proof.
(1)2 × 2 = 4, so 2 is a solution of x² = 4.A6, D1
(2)The only solutions are 2 and −2.(1), L1
(3)The nonnegative one of the two is 2.(2), A5
(4)Therefore √4 = 2.(3), D2
(5)The equation's solutions are x = ±2, and the symbol √4 refers to only one of them. The two sentences are different.(2), (4)
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