I'm not sure where I would fit this detail in the proof, but:
x2-2y2 would only equal zero if x-y√2 were zero.
I'll first let x-y√2=0
so, x=y√2.
Because x and y are rational, this can only occur if x=y=0. This means that when both x and y aren't zero, x2-2y2 will give a non-zero rational...
If the denominator were zero, then either (x+y√2) or (x-y√2) would have to equal zero, but this can't be since division by zero is impossible. (x+y√2)-1 or (x-y√2)/(x-y√2) would provide an undefined answer.
Should I have stated beforehand that both terms are non-zero?
If you wouldn't mind...
Ok, so I think I have the constructive proof:
(x+y√2)-1=(x+y√2)-1*[(x-y√2)/(x-y√2)]=(x-y√2)/(x2-2y2)=x/(x2-2y2) + (-y)√2/(x2-2y2)
where both x/(x2-2y2) and (-y)/(x2-2y2) are rational.
For the proof by contradiction, do I start by asserting (x+y√2)-1 is not a member of F? So x and/or y are...
Can I use the previously mentioned method for the other conditions?
For example:
(x1+y1√2)+(x2+y2√2)=(x1+x2)+(y1+y2)√2
The sum must be in F since the sum of rational numbers are rational.
Likewise, a rational number multiplied by negative one will still be rational, thus they still fit...
I'm taking my first proof heavy class, linear algebra. Unfortunately, I'm taking a long time picking up proofs in general, so I'm going to try and work through some of the material in Hoffman/Kunze.
Homework Statement
Prove that the set of all complex numbers in the form of x + y√2, where x and...
Apparently my college doesn't offer the same intro physics courses in both semesters, and they suggest that physics majors take physics starting in their first year. Well, I never really considered being a physics major and opted to have a more liberal curriculum my first semester. It's kind of...
Hi, do any of you know if MIT's Open CourseWare Calculus I (http://ocw.mit.edu/OcwWeb/Mathematics/18-01Fall-2006/Calendar/index.htm) is the equivalent of the high school AP BC Calculus / Calculus I + II at many other less math intensive colleges?
This forum is great for physics/math, but I suggest you try the collegeconfidential forums for some answers as well. The people there are more knowledgeable on the topic of colleges and I'm sure there are a decent amount of physics majors there as well. Try the "College Search and Selection"...
So it would it go:
x in R
x^2-2x+2x>0
.
.
.
or
x in R
(x-1)^2>-1
.
.
.
If the second one, then wouldn't I still have to go through the first one to get (x-1)^2>-1?
Homework Statement
Spivak's "Calculus" Chapter 1, Problem 4v
Find all numbers x for which
x^2-2x+2>0
Homework Equations
The Attempt at a Solution
x^2-2x+2>0
x^2-2x>-2
x^2-2x+1>-2+1
(x-1)^2>-1
x\in R
Would that be an adequate proof? Anything...
Are there any single variale calc video lectures based off of spivak book? I'm planning on self learning calculus over the summer and using his book to do so. It would be nice to have some kind of a backup just in case I don't understand something. By the way, I already bought the textbook...