Given any r in positive R (reals), the number sqroot(r) is unique in the sense that, if x is a positive real number such that x^2 = r, then x = sqroot(r)

Any help with this proof would be appreciated.

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- May 2nd 2011, 11:32 PMjstarks44444Square Root Uniqueness
Given any r in positive R (reals), the number sqroot(r) is unique in the sense that, if x is a positive real number such that x^2 = r, then x = sqroot(r)

Any help with this proof would be appreciated. - May 3rd 2011, 04:47 AMtonio
- May 5th 2011, 07:42 PMTinyboss
It's also pretty straightforward to prove that for positive reals, a>b ==> a^2 > b^2. So you get the contrapositive: if x does not equal sqrt(r), then x^2 does not equal r.

- May 6th 2011, 08:16 PMDeveno
- May 7th 2011, 04:02 AMtonio

First, the question was about reals, so ordered fields and stuff is way too much for this EXCEPT for the part that we talk about POSITIVE numbers.

Second, the equalities x^2 = y^2 <==> (x-y)(x+y) = 0 are true in any commutative ring, and if we add some other condition, not necessarily of ordered

fields, we can decide whether x = y or x = -y (for example, if we take the usual representatives of the residue classes

modulo a prime p, we can decide that the solution to x^2 = has to be between

0 and (p-1)/2 (mod p), say...

Tonio