What is Rational: Definition and 626 Discussions

Rationality is the quality or state of being rational – that is, being based on or agreeable to reason. Rationality implies the conformity of one's beliefs with one's reasons to believe, and of one's actions with one's reasons for action. "Rationality" has different specialized meanings in philosophy, economics, sociology, psychology, evolutionary biology, game theory and political science.

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  1. melese

    MHB No Rational Roots of $x^n+\cdots+1=0$

    (BGR,1989) Prove that for any integer $n>1$ the equation $\displaystyle \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}\cdots+\frac{x^2}{2!}+\frac{x^1}{1!}+1=0$ has no rational roots.
  2. A

    Difficult rational expressions thinking question

    Homework Statement Find the Values of B and C given this: \frac{(3x-18)}{(2x+1)(7x-3)}= \frac{B}{2x+1} + \frac{C}{7x-3} Homework Equations The equation given: \frac{(3x-18)}{(2x+1)(7x-3)}= \frac{B}{2x+1} + \frac{C}{7x-3} The Attempt at a Solution My attempt: \frac{(3x-18)}{(2x+1)(7x-3)}=...
  3. W

    Limit rational function without L'H

    Homework Statement evaluate. Homework Equations lim_{x->0+} \frac{\sqrt{x}}{\sqrt{sinx}} The Attempt at a Solution i've tried l'hopital's and it is just endless cycle.
  4. W

    So what can you do for this integral?

    Homework Statement evaluate the integral. Homework Equations \displaystyle\int {\frac{1}{secx+tanx} dx} The Attempt at a Solution can i just do ln|secx + tanx| ??
  5. W

    Can This Integral Be Solved Using Factoring and Substitution?

    Homework Statement evaluate the integral.Homework Equations \displaystyle\int {\frac{3x+2}{\sqrt{1-x^2}} dx}The Attempt at a Solution - i tried factoring hoping for a perfect square that i could take the square root of, but that doesn't work. -u-sub won't work: u=1-x^2 ; du=2x -i don't know...
  6. W

    Evaluate Integral: 2 ln| \frac{v-1}{v}|

    Homework Statement evaluate the integral. Homework Equations \displaystyle\int_2^∞ {\frac{2}{v^2 -v} dv} The Attempt at a Solution how does this integrate into: 2 ln| \frac{v-1}{v}| i tried and got 2ln|v^2-v| but not above.
  7. P

    MHB Integral closure of rational polynomial ring

    Consider the ring $\mathbb{Q}[X]$ of polynomials in $X$ with coefficients in the field of rational numbers. Consider the quotient field $\mathbb{Q}(X)$ and let $K$ be the finite extension of $\mathbb{Q}(X)$ given by $K:=\mathbb{Q}(X)[Y]$, where $Y^2-X=0$.Let $O_{K}$ be the integral closure of...
  8. R

    Why is the domain of this rational function restricted to x > 0?

    When I plot y = 3x^{\frac{4}{3}}-\frac{3}{32}x^{\frac{2}{3}}, I get: https://dl.dropbox.com/u/5653705/pfgraph1.png but I don't see any reason for the domain to be restricted to x > 0 . There is only a cubic root, which is well defined for negative numbers ... I've tried on a few...
  9. G

    Analytic mapping from disk to disk must be rational

    Let f(x) be a function which is defined in the open unit disk (|z| < 1) and is analytic there. f(z) maps the unit disk onto itself k times, meaning |f(z)| < 1 for all |z| < 1 and every point in the unit disk has k preimages under f(z). Prove that f(z) must be a rational function. Furthermore...
  10. G

    Proving mapping must be a rational function (complex analysis)

    Homework Statement Let f(x) be a function which is defined in the open unit disk (|z| < 1) and is analytic there. f(z) maps the unit disk onto itself k times, meaning |f(z)| < 1 for all |z| < 1 and every point in the unit disk has k preimages under f(z). Prove that f(z) must be a rational...
  11. I

    Rational Completeing the Square

    Homework Statement Homework Equations The Attempt at a Solution Alright so the solution is in the above pic, but I can't get anywhere close. You can see from the green circles that the "squares" aren't matching up. So I'm not sure if I can't multiply fractions anymore or what.
  12. W

    How Do You Integrate the Rational Function ∫((x^2+x-7)/(x+3))dx?

    Homework Statement Integrate the following: Homework Equations ∫(x^2+x-7)/(x+3) The Attempt at a Solution The only way I can think of solving this would be to split up each term into a separate fraction.
  13. W

    Integrating Rational Functions: Solving ∫(x/(x-1)^3)

    Homework Statement integrate the following:Homework Equations ∫(x/(x-1)^3The Attempt at a Solution i've tried u-substitution, finding an inverse trig function that matched the formula, and still can't figure out how to solve this problem. u-subtitution for u=x gives the same problem...
  14. S

    How can the set of the rational numbers be countable if there is no

    rational number to count the next rational number from any rational number? We can count the next natural number, but we can't count the next rational numer.
  15. O

    MHB Ideals and Varieties, Rational Normal Cone

    I have the ideal I = <f1, f2, f3>, where f1 = x0x2-x12, f2 = x0x3 - x2x1, f3 = x1x3 - x22. I also have the parametrization of some surface given by \phi: \mathbb{C}^2 \rightarrow \mathbb{C}^4 defined by \phi(s, t) = (s^3, s^2t, st^2, t^3) = (x_0, x_1, x_2, x_3) . I want to show that V(I) =...
  16. M

    Prove , if x is a rational number , x ≠ 0 then , tan(x) is not rational

    prove , if x is a rational number , x ≠ 0 then , tan(x) is not rational this theorem was proved by a mathematician called Lambert , I search for the proof , anyone knows it ?!
  17. P

    Finding extrema when derivative has no rational roots.

    Homework Statement How may I find stationary points, increase/decrease intervals, concavity for f(x)=(x^3-2x^2+x-2)/(x^2-1)? Homework Equations The Attempt at a Solution I am familiar with how it should be done, except that here I get f'(x)=x^4-4x^2+8x-1 for the numerator of the...
  18. J

    Rational roots - standard form of equation

    Hi everybody! I've hit a blank with regards to this 1 equation on a old exam paper - think I've overloaded myself a bit and just feel a bit like a airhead at the moment! I understand the actual method and getting to the answer but it starts off with a equation which you then need to get to...
  19. H

    Singularities of two variables rational functions

    Homework Statement Let p(x,y) be a positive polynomial of degree n ,p(x,y)=0 only at the origin.Is it possible that the quotient p(x,y)/[absolute value(x)+absval(y)]^n will have a positive lower bound in the punctured rectangle [-1,1]x[-1,1]-{(0,0)}? Homework Equations The Attempt at a...
  20. T

    Every feild has a subset isomorphic to rational numbers?

    I am reading linear algebra by Georgi Shilov. It is my first encounter with linear algebra. After defining what a field is and what isomorphism means he says that it follows that every field has a subset isomorphic to rational numbers. I don't see the connection.
  21. L

    Understanding horizontal asymptotes of non-even rational functions

    I think the technique that is to be used for these types of problems, but I just am having trouble grasping why it is permitted. I have no problem with any homework, but it just doesn't seem right. Maybe my text is just not being clear (Larson, 9th, btw) Given a function...
  22. C

    MHB D's Rational Approximations Question from YAnswers

    Some bits are ok but I thought I would include them anyway as it is needed to answer the other parts of the question. I have labelled the parts which I need help with. (a) Recall why there are no integer solutions \(m, n \in \mathbb{N}\) to the equation \(m^2 = 2n^2\). ANSWER = an irrational...
  23. K

    Prove Continuous at Irrationals, Discontinuous at Rationals: Real Analysis

    Real Analysis--Prove Continuous at each irrational and discontinuous at each rational The question is, Let {q1, q2...qn} be an enumeration of the rational numbers. Consider the function f(x)=Summation(1/n^2). Prove that f is continuous at each rational and discontinuous at each irrational...
  24. A

    MHB Polynomial and Rational Functions

    For the years 1998-2009, the number of applicants to US medical schools can be closely approximated by: A(t)= -6.7615t4+114.87t3-240.1t3-2129t2+40,966 where t is the number of years since 1998. a) graph the number of applicants on 0<= t <= 11 b) based on the graph in part a, during what...
  25. S

    Finding the fuction of rational equations

    Homework Statement http://tinypic.com/r/14uet5i/6 just in case it didnt work http://tinypic.com/r/14uet5i/6 how do i find the equation of those two graphs? Also Homework Equations The Attempt at a Solution 1. asym is at -2 so bottom must have (x+2) what's the...
  26. S

    Nxmod(1) is a discrete subset of [0,1] iff x is a rational?

    Homework Statement Let \theta \in [0,1] and n \in \mathbb{Z} . Let n\theta mod(1) denote n\theta minus the integer part. Show n \theta mod(1) is a discrete subset of [0,1] if and only if theta is rational.The Attempt at a Solution I'm having a bit of trouble with the "only if" part...
  27. A

    MHB Help solving a rational inequality

    Solve the rational inequality (a-5)/(a+2) < -1.This is what I got so far: a-5/a+2 = -1 a-5= -a-2 0= -2a+3 subtract 3 from both sides: -3=-2a Divide by -2 3/2=a I know that the answer is (-2, 3/2), but I'm not sure where the -2 in the answer comes from. Thanks!
  28. P

    MHB Simplify Complex Rational Expression

    (3/x-2) - (4/x+2) / (7/x2-4) I got it down to... -x+14/7 but the book is showing x-14/7
  29. T

    Help evaluating a rational integral using residue

    Homework Statement I need to compute \int_0^\infty \frac{dx}{x^3+a^3}.Homework Equations If f = g/h, then Res(f, a) = \frac{g(a)}{h'(a)}. The Attempt at a Solution In the first I've used a semicircular contour in the upper plane that is semi-circular around the pole at -a. So I calculate the...
  30. O

    Solving Rational Inequalities: x < 2/(5x-1) | My Final Answer Matches Textbook!

    My final answer matches that of the textbook, but do I need to change the < to > at any point as I solve this? I ask because, if I assume x is 1 (until I solve for x), then the question statement and parts of the solution are made untrue. Homework Statement Solve the following for x...
  31. E

    Rational function hole at (0, 0)?

    The function is f(x) = 2x / x3 - 6x2 + 3x + 10 I was taught that any rational function with a numerator of smaller degree than the denominator has a horizontal asymptote at y = 0, which would apply in this case. This makes sense for the end behaviors because as x approaches +/- ∞, y...
  32. J

    Cubic asymptotes in rational functions?

    I just got this assignment for math and the question was is it possible to have a cubic asymptote in a rational function. If so explain how and where.
  33. K

    Limit continuous function of rational numbers

    Homework Statement Fix a real number a>1. If r=p/q is a rational number, we define a^r to be a^(p/q). Assume the fact that f(r)=a^r is a continuous increasing function on the domain Q of rational numbers r. Let s be a real number. Prove that lim r--->s f(r) exists Homework Equations...
  34. vrmuth

    Finding the range of rational functions algebraically

    how to find the range of rational functions like f(x) = \frac{1}{{x}^{2}-4} algebraically , i graphed it and seen that (-1/4,0] can not be in range . generally i am interested in how to find the range of functions and rational functions in particular
  35. J

    Help simplifying a rational expression?

    Homework Statement \frac{x^2}{(x^2 -1)} Homework Equations N/A The Attempt at a Solution I know the solution is supposed to be 1+\frac{1}{2(x-1)}-\frac{1}{2(x+1)} and when I did long division, I got 1+\frac{1}{(x^2-1)} , so I'm making progress, but I'm not quite there. What should...
  36. M

    Prove: Rational Number Squared Has Prime Factors w/ Even Exponent

    I have a theory that i need to prove but I am not quite sure how to mathematically prove that it is true. Theory: When you square a rational number, each of the prime factors has an even exponent. For example, 10 --> If i square 10, which is a rational number, =10^2 =(5^2 x 2^2)...
  37. R

    Integration of rational function by partial functions. The last step confuses me

    Ok all save y'all a little reading. Worked out the problem. Got X^2+2x-1=A(2x-1)(x+2)+bx(x+2)+cx(2x-1) Ok then you write it standard form for a polynomial. Then use can use there coefficients to write new equations at you get 2a+b+c=1 3a+2b-c=2 and finally -2a=1 Now you solve for...
  38. S

    Simplifying with rational and negative exponents

    Homework Statement Simplify the expression completely 3(y)(1/3)-3x(y)(-2/3)(2x) / [(y)(1/3)]2 The Attempt at a Solution My attempt is totally wrong. I can't quit figure out what to do or where to start 3(y)(1/3)-3x(y)(-2/3)(2x) / [(y)(1/3)]2 3(y)(1/3)-6x(y)(-2/3) / y(2/3)...
  39. B

    Solve cubic with no rational zeros

    Homework Statement x^3-8x+10=0 Homework Equations The Attempt at a Solution By dividing by x-1 (1 is a factor of the solution), I got (x-1)(x2+x-7)+3+10=0 which equals x^3-8x+20=0 I think the solution would be imaginary but I used a graphing calculator and there is one...
  40. P

    Prove √ is not a rational number

    Homework Statement "Prove m√n is not a rational number for any natural numbers with n,m > 1, where n is not an mth power" Homework Equations Natural numbers for us start at 1. Since we know n is not an mth power, then n \neq km for an arbitrary integer k. The Attempt at a Solution...
  41. D

    When the Rational Root Theorem Fails

    On a math test, one of the questions was to solve -\sqrt{7-x}=-\frac{x^2}{2}+12x-10. I solved graphically with a calculator, but later tried to solve algebraically, when I had more time. The equation is equivalent (with extraneous solutions) to x^4 + 48x^3 +536x^2 -956x + 372=0. This quartic has...
  42. D

    MHB Irrational and rational numbers

    Show that there are infinitely many rational numbers between two different irrational numbers and vice versa. So I started as such: WLOG let $a,b$ be irrational numbers such that $a<b$. By theorem (not sure if there is a name for it), we know that there exist a rational number $x$ such that...
  43. D

    MHB Decimal expansion to rational number

    I am given the number $.334444\ldots$ So we have $0+33\times 10^{-2} + \sum\limits_{n=3}^{\infty}4\times 10^{-n}$ Is there a way to put this all together in one geometric series?
  44. D

    I feel so dumb for asking this question but domain of a rational function help

    Hello, I have a simple question that has bothered me forever.When I am given a rational function, say, f(x) = (x2+x-6)/(x+3)say I want to look at the domain of this function. Well the first thing that catches my eye is the fact that if I plug in x=-3, the denominator will be 0, which would be...
  45. V

    Limit Proof with Rational Functions

    Homework Statement If r is a rational function, use Exercise 57 to show that ##\mathop {\lim }\limits_{x \to a} \space r(x) = r(a)## for every number a in the domain of r. Exercise 57 in this book is: if p is a polynomial, show that ##\mathop {\lim }\limits_{x \to a} \space p(x) = p(a)##...
  46. C

    MHB Rational exponents expressed as fractions

    can someone help me (4a 3/2)(2a1\2) (3x5/6)(8x2/3) (27a6)-2/3 the fractions are powers
  47. M

    Oblique asymptotes of a rational function

    Homework Statement To find the oblique asymptotes of a rational function (i) f(x)=\frac{P(x)}{Q(x)}=\frac{a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{0}x^{0}}{b_{m}x^{m}+b_{m-1}x^{m-1}+...+b_{0}x^{0}} where n=m+1 we exprese it in a form (ii) f(x)=ax+b+\frac{R(x)}{Q(x)} using long division (my book...
  48. M

    Finding the inverse of a rational function

    Homework Statement Find the inverse of a rational function: f(x)=\frac{2x}{x-2} The Attempt at a Solution 1. Verify the function is one-to-one. Question: is there a way to do that withought drawing the graph? Some algebraic method? I know that this means that the function crosess any given...
  49. J

    Why don't rational numbers fulfill the completeness axiom?

    Homework Statement Show that Q does not fulfill the completeness axiom. Homework Equations "Every non empty set of rational numbers that contains an upper bound contains a least upper bound" (show this is false) The Attempt at a Solution I've sat on this question for a few days...
  50. C

    Indefinite Integration of a Rational Expression

    Homework Statement <Indefinite integral sign here>[r^2 -2r] / [r^3 - 3r^2 + 1]dr or the second example in the "Substitution" section here: http://people.clarkson.edu/~sfulton/ma132/parfrac.pdf Homework Equations nada. The Attempt at a Solution Nothing to really attempt, I just...
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