Rational Definition and 616 Threads

Conjecture Regarding Rotation of a Set by a Sequence of Angles. Consider the following sequence, where the elements are rational numbers mulriplied by \pi: (\alpha_{i}) = \hspace{2 mm}\pi/4,\hspace{2 mm} 3\pi/8,\hspace{2 mm} \pi/4,\hspace{2 mm} 3\pi/16,\hspace{2 mm} \pi/4,\hspace{2 mm}...

1. ∫\frac{dx}{x3 + 2x} We're suppose to evaluate the integral. Use Partial Fraction Decomposition: \frac{1}{x3 + 2x} = \frac{A}{x} + \frac{Bx + C}{x2 + 2} 1 = A(x2 + 2) + (Bx + C)(x) 1 = Ax2 + 2A + Bx2 + Cx 1 = x2( A + B) + Cx + 2A Solving for A gives \frac{1}{2} Solving for B...

Hello dears, My faculty is Cvil Engineering. We had a subject "Rational Mechanics". The professor didn't gave textbook. I have my written notes during lectures only. I have many doubts and misunderstandings, so I need good book. Can you help me with this? Actually, I have found some books...

Homework Statement Find the derivative of: y = [(2x - 5)^4][(8x^2 - 5)^-3] Homework Equations I get: y' = -[(48x(2x - 5)^4) / (8x^2 - 5)^4] + [(8(2x - 5)^3) / (8x^2 - 5)^3] Wolfram gets: http://www.wolframalpha.com/input/?i=derivative+%282x+-+5%29%5E4%288x%5E2+-+5%29%5E-3 The...

Homework Statement limx→2(x+2)/(x^3+8) 2. I only recently started learning calculus on my own so correct me if I'm wrong. When using direct substitution and the denominator equals 0, the limit is undefined, just like any fraction is when its denominator equals 0. However, it's limit can still...

Say we have a rational function P(x)=(x^2-3x-4)/(x-4)=[(x+1)(x-4)]/(x-4) I'm a little confused as to why the (x-4) doesn't cancel out. It graphs the same as y=x+1 for x≠4. I feel like I'm missing something from the order of operations.

I've been doing some exercises in introductory Galois theory (self-study hence PF is the only avaliable validator :) ) and a side-result of some of them is surprising to me, hence I would like you to set me straight on this one if I'm wrong. Homework Statement Let K(x) be the field of rational...

Homework Statement Simplify. Write as positive exponent c(c^5/6)/c^2 Homework Equations None really, just exponent laws. The Attempt at a Solution c(c^5/6) comes out as... c^5/6 Dividing exponents requires subtracting them so... 5/6 - 2/1 Find common denominator... 5/6 -...

Hi, I am struggling with this puzzle from a book. Puzzle : Can you find a number n such that, the numbers n-7, n, and n+7 have rational square roots (can be expressed as integers or fractions)? According to the book one of the solutions is n =113569 /14400 This is what I have done so...

Now if I have a function y=(ab+ac)/a, it can be further factorised, y=(a(b+c))/a. Now if we cancel off the a, we will have only y=b+c that will also give the same y-values as the original form of the function y with respect to the same x-value. This statement implies that cancellation or...

$\cos(1^0)$ and $\tan(1^0)$ are Rational or Irrational no. Where angle are in Degree help required

Hi All: I am new to the site, so I thought this would be a good time to post an interesting integral I ran across that I am having a time with. It is a miscellaneous problem in Schaum's Outline of Complex Variables, #86 in ch. 7. I have been self-teaching a little CA when I get time and this...

I have been an ugly rational function. I graphed it and put end behaviors and intercepts pods etc... Now after my teacher asked me to take the limit of this function and she has informed me that it is a much more simple way than taking the derivative of the entire peice...

Rational numbers form a group under addition. However, a sequence of rational numbers converges to irrational number. Presumably, group theory does not allow adding an infinite number of rational numbers. This is not indicated in the textbook definition of a group. I might be looking in vain...

F(x)= 6x^2-17x-3/3x+2, find horizontal asymptote

Homework Statement Prove that a rational root of a monic polynomial is an integer. Use this to prove that the \sqrt{2} is irrational. Homework Equations The Attempt at a Solution /// I am really not sure where to begin?

Homework Statement Prove or disprove that there is a rational bijective function f : R to (0; 1) Homework Equations i found a bijective map from (0,1) to R (y=(2x-1)/(2x^2-2x) The Attempt at a Solution Im just stuck and i was thinking since it has to be a rational function...

Homework Statement The set of squares of rational numbers is inductive Homework Equations definition of an inductive set The Attempt at a Solution sorry i know this is probably very easy to most but I am just learning analysis. Okay, so we can see that 1 is in the set because it is...

OKAY I have a trg test make up and due tomroow and I have no clue what I am doing, I've searched online chatrooms and they all want money. so this is basiaclly my last hope the test is over rATIONAL ROOTS ZEROS POLYNOMIALS ETC. please show all work if answering QUESTIONS i NEED HELP ON BELOW...

Homework Statement Prove |\frac{e^{2i\theta} -2e^{i\theta} - 1}{e^{2i\theta} + 2e^{i\theta} -1}| = 1 Homework Equations The Attempt at a Solution I feel like this should be fairly simple, anyone have any hints? Also this is just one step in an attempt to solve a much larger problem, so don't...

I'm trying to put a matrix into RCF, and I keep running into problems. I've checked my work a few times, so I think I must be making a conceptual error. Here's what I've got: $$A=\left( \begin{matrix}2 & -2 & 14 \\ 0 & 3 & -7 \\ 0 & 0 & 2\end{matrix}\right)\quad \text{ so }\quad xI-A=\left(...

it is possible to evaluate sums over the set of Rational so \sum_{q} f(q) with q= \frac{m}{n} and m and n are POSITIVE integers different from 0 ?? in any case for a suitable function is possible to evaluate \sum_{q} f(qx) with f(0)=0 ??

Homework Statement Express the repeating decimal as a series, and find the rational number that it represents 1) 3.2\overline{394} Homework Equations Geometric Series (a/1-r) The Attempt at a Solution I tried putting the value into a series by having n=1 as it goes to infinity Ʃ3.2...

Homework Statement Simplify. (p4 - q4)/(p + q)2 ÷ 1/(p2 + q2) Answer: (p - q)/(p + q) Homework Equations -- The Attempt at a Solution Transformed it to a multiplication problem. (p4 - q4)/(p + q)2 X (p2 + q2)/1 Difference of the squares in the numerator of the first expression...

Homework Statement Write as a single power, then evaluate: a) (-32)^3/5 x (-32)^-4/5 / (-32)^2/5 b) 4096^3/6 / 4096^2/3 x 4096^5/6 Homework Equations The Attempt at a Solution a) (-32)^3/5 x (-32)^-4/5 / (-32)^2/5 = (-32)^3/5+(-4/5)-2/5 = -32^-3/5 = -1/8 <- not sure...

Homework Statement ∫(x3+4)/(x2+4)dx Homework Equations n/a The Attempt at a Solution I know I have to do long division before I can break this one up into partial fractions. So I x3+4 by x2+4 and got x with a remainder of -4x+4 to be written as x+(4-4x/x2+4). Then I rewrote...

Hi, I have to approximate an irrational number x by rationals r = p/q. Let ε>0 in ℝ, then, for almost all x exist α and r in (x-ε,x+ε) such that q ≈ c(x) ε^-α, c(x) in ℝ? I know, from Hurwitz theorem (and a conseguence) that α>2, if exists.

Hi everyone and nice to meet you. I'm velox_xox a newbie to PF and a high school correspondence student currently taking Algebra II. Since I'm correspondence, I am basically teaching myself my subjects, which means if I don't understand something it's a big problem. I enjoy math, but I must...

This is the problem I'm trying to slove: Consider the sequence s_n = Sumation (from k=0 to n) p^k (i.e. s_n=p^0+p^1+p^2...+p^n) in Q(rationals) with the p-adic metric (p is prime). Show that s_n converges to a rational number.[/B] Now, I do get some intuition on showing that the...

Homework Statement ∫(7x^2+22x-54)/((x-2)(x+4)(x-1)) dx Homework Equations Partial functions The Attempt at a Solution ∫(7x^2+22x-54)/((x-2)(x+4)(x-1)) dx = ∫(Ax)/(x2+2x-8) + B/(x-1) = ∫A(x-1)dx + ∫B(x^2+2x-8)dx Or is it...

Question: When determining the coefficients of the partial fractions for say 5 or more coefficients... Do you find it easiest to set up linear equations and solving? Any advice would be appreciated... Next question.. look in paint doc... why would I3 not be equal to I21??

Homework Statement Prove that every positive rational number x can be written in ONE way in form x=a1+ a2/2! + a3/3! + ... + ak/k! where a1,a2,...,ak are integers and 0<=a1, 0<=a2<2,... ,0<=ak<k I wrote my solution below. Please check if it is correct and rewrite it for me in a neater way...

Homework Statement Evaluate the integral. (Remember to use ln |u| where appropriate.) ∫ds/s^2(s − 1)^2 Homework Equations The Attempt at a Solution I attempted a solution using the method of partial fractions, but it seems my answer is wrong. Here's what I did... 1=A/s...

Homework Statement Evaluate the integral. (Remember to use ln |u| where appropriate.) ∫(x^3 + 36)/(x^2 + 36) Homework Equations The Attempt at a Solution A little bit confused about arriving at the solution for this problem. I get stuck a little ways in. Any help would be...

Hello, I'm trying to do exercise number 20 from chapter 6 of this http://www.people.vcu.edu/~rhammack/BookOfProof/index.html, it asks to show that the curve x2 + y2 - 3 = 0 has no rational points. In the answer it has this tip: first show that a2 + b2 = 3c has no solutions, other than the...

Homework Statement Show that if a\in\mathbb{Q} and t\in\mathbb{I} then a+t\in\mathbb{I} and at\in\mathbb{I} as long as a≠0 The Attempt at a Solution Let a=\frac{x}{y} where x and y are integers. and t is an irrational number If I have a+t . since t cannot be written as...

Homework Statement Construct a sequence that has all rational numbers in it Homework Equations None. The Attempt at a Solution Here are my thoughts, though I have no solutions yet. If I construct a sequence Sn= n*sin(n)-1/n, will it work? Thanks guys!

Homework Statement Calculate the volume (V) generated by the given function: y^{2}=\frac{x^3}{2a-x} about the line x=2a I suppose you have to use cylindrical shells [because it's difficult to fin y], so: V=2\pi(2a-x)\sqrt{\frac{x^3}{2a-x}}Δx now...

Homework Statement I need to find the limit of this using L'Hopital Rule, if it exists. lim_{x\rightarrow0^{+}}(\frac{sinx}{x})^{1/x^{2}} Homework Equations All of our examples used lim f(x)/g(x)=f'(x)/g'(x) The Attempt at a Solution We have not dealt with any limits like this that have...

Homework Statement b^2 = a b is a rational number a is an integer prove that b is an integer. This is self assigned, but I think this is the appropriate place to put my question. Homework Equations see above The Attempt at a Solution Is this legitimate...? Since b is a...

Homework Statement Give an example of a bounded subset of Q which has no least upper bound in Q. Explain why your answer has this property. Homework Equations The Attempt at a Solution [1/8, 1/4, 3/8, 1/2, 5/8, 3/4...infinity] is this correct?

Homework Statement \int_0^{\infty} \ln \left( \frac{e^x+1}{e^x-1} \right) \mbox{d}x \int_0^{\infty} \frac{1}{x^n+1}\ \mbox{d}x\ \forall n >1 Homework Equations - The Attempt at a Solution I've tried IBP and separating the ln into two terms and failed. I've also tried a...

When solving 3x^3+4x^2-7x+2 the rational zeros theorem says there can only be a possibility of zeros at plus minus 1, 2, 1/3, and 2/3 but for some reason a zero appears at about -2.4 (or more accurately -1-squareroot of 2). How can this be? It states that if there are any zeros they will...

y=(5x^4 + 5^x -1)/(2x^3 +1) More specifically is 5^x a valid term in a polynomial?

Homework Statement Find all accumulation points of the following sequence: an enumeration of all rational numbers in (0,1) Homework Equations Boltzano-Weirstrass Theorem: Every bounded sequence has a convergent sequence (hence, an accumulation point) The Attempt at a Solution Because...

I know that Q (rational numbers) are countable and that the finite cartesian of Q with itself, Q^n is countable but is it true that the countably infinite cartesian product of Q with itself is countable? The set of all rational sequences are isomorphic to Q^∞ (here I am saying Q^∞ is the...

Homework Statement -I have the zero, which is x=-1, however its a squared zero {(x+1)^2)} -Vertical asymptote is at x=1 -Equation of oblique asymptote is y=x+4 Homework Equations The Attempt at a Solution I tried finding the numerator by multiplying the oblique asymptote by the...

Homework Statement Regard Q, the set of all rational numbers, as a metric space, with d(p, q) = |p − q|. Let E be the set of all p ∈ Q such that 2 < p2 < 3. Show that E is closed and bounded in Q, but that E is not compact. Is E open in Q? Homework Equations Definition of interior...

You're about to flip a quarter while your friend guesses which side will come up. You agree to switch turns after one incorrect guess each. He gets 2 in a row right before guessing wrong, and now it's your turn. You incorrectly guess the outcome of the first flip he does, and now it's his turn...

Give a formula for a rational function having the following properties: • Horizontal Asymptote: y = 2 • Vertical Asymptotes: x = − 3 and 1 • x-intercepts: -1 and 5 Here is what I've done so far \frac{2x^2}{(x+3)(x-1)} I basically get stuck when trying to come up with something for the x...