Solving Difficult Math Problems: Proving Irrationality & More

[osnarf]

Okay I'm feeling extremely discouraged. I've already been through differential equations and I decided it would be a good idea to go read through a calc book again and make sure I have everything down really well before I move on to more advanced stuff. Courant's book sounded good because the second edition goes over more advanced calc, and the physics applications sounded interesting, so I started reading that. So far I think I can prove one of the problems (if i understood it correctly). I've been at this for five hours. I've never had a problem in any math class I don't understand why this is so difficult for me. I don't understand what half these questions are even asking.
If anybody has any suggestions on getting through this book, that would be great, or if you could at least point me in the right direction on some of these questions that would be awesome too. I guess there isn't a solutions manual for this book in print anymore :( I really want to do it now because I feel like I should definitely be able to do this stuff and its kind of upsetting.

Sorry for writing so many problems I'm just really really frustrated and I've been up for too long trying to do this.






1) a) If a is rational and if x is irrational, prove that a+x is irrational, and if a =/= 0, that a*x is irrational.
It makes sense, but I don't know how to prove it?
b) show that between any two rational numbers there exists at least one irrational number and, consequently, infinately many.

2) d) prove that the pth root of n is irrational, if n is not a perfect pth power.

* 3) a) Prove that for any rational root of an nth order polynomial with integer coefficients where the nth order variables coefficient is not equal to zero,

a(sub n)x^n + a(sub (n-1))x^(n-1) + ... + a(sub 1)x + a(sub 0)

if written in lowest terms as p/q, that the numerator p is a factor of a(sub 0) and the denominator q is a factor of a(sub n). (this criterion permits us to obtain all rational real roots and hence to demonstrate the irrationality of any other real roots.)
b) Prove the irrationality of [(square root of 2) + (cubed root of 2)] + [(sqrt of 3) + (cubed root of 2)] .
again this makes sense but how do you go about proving it?

next section
1) Let [x] denote the integer part of x; that is, [x] is the integer satisfying

x - 1 < [x] <= x

set c(sub 0) = [x] and c(sub n) = [(10^n)*(x - c(sub 0)) - (10^(n-1))*c(sub 1) - (10^(n-2))*c(sub 2) - ... - 10*c(sub (n-1))] for n = 1,2,3,... verify that the decimal representation if x is
x = c(sub 0) + 0.c(sub 1)c(sub 2)c(sub 3)...
and that this construction excludes the possibility of an infinite string of 9's.

What?

2) Define inequality x>y for two real numbers in terms of their decimal representations.
How is this a question? Why would it be defined any differently than for anything else?

3) Prove if p and q are integers, q > 0, that the expansion of p/q as a decimal either terminates (all the digits following the last place are zeroes) or is periodic; that is, from a certain point on the decimal expansion consists of the sequential repetition of a given string of digits. For example, 1/4 = .25 is terminating, 1/11 = .090909... is periodic. The length of the repeated string is called the period of the decimal; for 1/11 the period is 2. In general, how large may the period of p/q be?

How do you go about proving something like this?

I could go on. This is like trying to read mandarin chinese.

 

[rochfor1]

This looks much more like beginning analysis than the kind of calculus you get in introductory classes/diffeq. You may do well to learn a bit about elementary proof techniques before tackling such a book. That being said, a few hints:

1. a) Try proof by contradiction. Remember that a number z is rational if and only if there are integers m and n with n \neq 0 such that z = m/n.

b) Think about how (a) shows that if sufficies to find an irrational number between any two integers. This is much easier.

2. d) Try proving that \sqrt{2} is irrational. The general case should be similar. You may be able to google this proof...it's quite famous.

3. a) Google the Rational Roots Theorem. See if you can understand the proof.

Best of luck.

 

[ice109]

1a. assume that the statement is false and then prove that that leads to a contradiction. with thing like irrationals which are fairly vaguely defined this is the best way to proceed.

1b. statements like there exists are most easily proved by producing an example. so for example take to rationals and construct an irrational in between them. this problem would be easier if first your proved that between two rationals there lies another rational.