The document proves two mathematical statements:
1) If x is rational, then x must be an integer. It shows that if x is written as p/q in lowest terms, then q must be 1 or -1, making x an integer.
2) If x is not an integer, then x must be irrational. This is proved to be true as the contrapositive of the statement proved in part 1.
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a) Prove that if x is rational, then x is an integer.b) Prove that.pdf
1. a) Prove that if x is rational, then x is an integer.
b) Prove that if x is not an integer, then x is irrational.
(Please show in a well mathematical maner. Thank you so much~)
Solution
a) Suppose that x is rational.
So, x = p/q for some relatively prime integers with q nonzero.
Substitute this into the equation:
(p/q)^n + c(n-1) (p/q)^(n-1) +.....+ c(1) (p/q) + c(0) = 0
==> p^n + c(n-1) p^(n-1) q +.....+ c(1) pq^(n-1) + c(0) q^n = 0, clearing denominators
==> p^n = -q (c(n-1) p^(n-1) +.....+ c(1) pq^(n-2) + c(0) q^(n-1))
So, q | p^n.
However, p is prime and gcd(p, q) = 1.
So, we conclude that q = -1 or 1, and thus x is an integer.
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b) This is true, being the contrapositive of the result from part a.