PROOF 97
Prove.
√(a²) is the absolute value of a
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| means a if a ≥ 0, and −a if a < 0.
Proof.
(1)Every real number belongs to exactly one of the two cases.A7
(2)The absolute value is not an operation of stripping the sign; it is a value fixed by cases.convention
∎
Definition D2
For x ≥ 0, √x denotes the nonnegative y satisfying y² = x.
Proof.
(1)A2 guarantees the existence of such a y.A2
(2)If y is a solution, so is −y, so the solution is not unique.A3
(3)By convention, the nonnegative one is fixed as √x.A1
∎
Lemma L1
For every real a, |a| ≥ 0
Proof.
(1)If a ≥ 0, then |a| = a, so |a| ≥ 0.D1
(2)If a < 0, then |a| = −a.D1
(3)If a < 0, then −a > 0.A6
(4)The two cases cover all reals, so |a| ≥ 0 always.(1), (2), (3), A7
∎
Lemma L2
For every real a, |a|² = a²
Proof.
(1)If a ≥ 0, then |a| = a, so |a|² = a².D1, E
(2)If a < 0, then |a| = −a.D1
(3)(−a)² = a².A3
(4)Therefore |a|² = a² also when a < 0.(2), (3), E
(5)The two cases cover all reals, so |a|² = a² always.(1), (4), A7
∎
Theorem T
For every real a, √(a²) = |a|
Proof.
(1)a² ≥ 0, so √(a²) is defined.A5, D2
(2)√(a²) ≥ 0 and (√(a²))² = a².D2
(3)|a| ≥ 0.L1
(4)|a|² = a².L2
(5)Both numbers are nonnegative and their squares equal a².(2), (3), (4)
(6)Therefore √(a²) = |a|.(5), A4
∎
Corollary C
What follows from a² = b² is not a = b but |a| = |b|.
Proof.
(1)Suppose a² = b².assumption
(2)Taking √ of both sides gives √(a²) = √(b²).(1), E, D2
(3)Therefore |a| = |b|.(2), T
(4)With a = −3, b = 3, we have a² = b² but a ≠ b.A8
(5)Returning from a squared equation to the original, the sign must be checked separately.(3), (4)
∎
Remark
Can A1 be removed from this proof?
Find what decides whether √(a²) is a or −a.