Proof: Square Root of a Square

Let's prove the following theorem:

If we square a number and then take the square root of the result, then the output is the original number. Here are some examples:

High-Level Proof

(x⋅x) = x^{2, so x⋅x^1/2 = x21/2}

We claim that

x^{21/2 = x(2 ⋅ 1/2)}

2 ⋅ 1/2 is 1, so x^{(2 ⋅ 1/2) == x}

Proof:

Proof Table

#	Claim	Reason
1	x ⋅ x = x2	x ⋅ x = x2 (power_of_2_2)
2	(x ⋅ x)(1 / 2) = (x2)(1 / 2)	if x ⋅ x = x2, then (x ⋅ x)(1 / 2) = (x2)(1 / 2) (Substitution)
3	(x2)(1 / 2) = x(2 ⋅ (1 / 2))	(x2)(1 / 2) = x(2 ⋅ (1 / 2)) (Power of a Power)
4	2 ⋅ (1 / 2) = 1	2 ⋅ (1 / 2) = 1 (Fixed Value Statement)
5	x(2 ⋅ (1 / 2)) = x1	if 2 ⋅ (1 / 2) = 1, then x(2 ⋅ (1 / 2)) = x1 (Substitution)
6	x1 = x	x1 = x (Power By 1)
7	(x ⋅ x)(1 / 2) = x	if (x ⋅ x)(1 / 2) = (x2)(1 / 2) and (x2)(1 / 2) = x(2 ⋅ (1 / 2)) and x(2 ⋅ (1 / 2)) = x1 and x1 = x, then (x ⋅ x)(1 / 2) = x (Transitive Property Application 4)

Comments

Please log in to add comments