- #1
silimay
- 26
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This isn't really a question about homework specifically, it's more just that I don't understand part of my chapter...I am just starting Principles of Mathematical Analysis by Ruben...
Here is what I don't understand:
It is proving that [tex]p^2 = 2[/tex] is not satisfied by any rational p. And it says that if there were such a p, we could write [tex]p = m/n[/tex] where m and n are integers that are not both even. This is probably a foolish question, but I don't understand that...why can't m and n both be even?
My next question is, further on in the proof, it says, let [tex]A[/tex] be the set of all positive rationals p such that [tex]p^2 < 2[/tex] and let [tex]B[/tex] consist of all positive rationals p such that [tex]p^2 > 2[/tex], and that they are going to show that [tex]A[/tex] contains no largest number and [tex]B[/tex] contains no smallest. For every p in [tex]A[/tex] we can find a rational q in [tex]A[/tex] such that [tex]p < q[/tex] (and similar for B). But then there is this equation:
[tex]q = p - \frac{p^2 - 2}{p+2} = \frac{2p + 2}{p + 2}[/tex]
I don't understand at all where this equation came from.
Thanks for any help in advance...Sorry if these are stupid questions :)
Here is what I don't understand:
It is proving that [tex]p^2 = 2[/tex] is not satisfied by any rational p. And it says that if there were such a p, we could write [tex]p = m/n[/tex] where m and n are integers that are not both even. This is probably a foolish question, but I don't understand that...why can't m and n both be even?
My next question is, further on in the proof, it says, let [tex]A[/tex] be the set of all positive rationals p such that [tex]p^2 < 2[/tex] and let [tex]B[/tex] consist of all positive rationals p such that [tex]p^2 > 2[/tex], and that they are going to show that [tex]A[/tex] contains no largest number and [tex]B[/tex] contains no smallest. For every p in [tex]A[/tex] we can find a rational q in [tex]A[/tex] such that [tex]p < q[/tex] (and similar for B). But then there is this equation:
[tex]q = p - \frac{p^2 - 2}{p+2} = \frac{2p + 2}{p + 2}[/tex]
I don't understand at all where this equation came from.
Thanks for any help in advance...Sorry if these are stupid questions :)