Rational Definition and 628 Threads

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.

View More On Wikipedia.org
Recent contents Top users
  1. A

    How Is a Closed Set of Rational Numbers Defined?

    Hi, here is the question, if A is a closed set that contains every rational number r: [0,1], show that [0,1] is a subset of A. But, how could A be closed? If A is closed, R^n-A is open, so any point in R^n-A would have a open sphere around it and this open sphere wouldn't intersect A...
  2. R

    Solving Rational Functions: Rewriting Equation to Get R(z)=...

    http://planetmath.org/encyclopedia/CPlace.html, how do I rewrite (2) to get the third equation R(z)=... ? thank you
  3. C

    Integral of x^m/(x^n+a^n)^p: Contour Integration

    can anyone show me step by step of how to evaluate the integral of [x^m/(x^n+a^n)^p](dx) from negative infinity to positive infinity. all i know is that contour integration is required to solve this problem.
  4. A

    Integrate (dx)/(-4 + x^2): Guidelines

    Homework Statement Integrate (dx)/(-4 + x^2) Homework Equations Trig substitution? The Attempt at a Solution How would you integrate something like this? I don't need answers, I just need some guidelines to start off.
  5. L

    Rational and Irrational Number Set proof.

    Hello, here is my problem: how can i prove that if a\in\mathbf{Q} and t\in\mathbf{I}, then a+t\in\mathbf{I} and at\in\mathbf{I}? My original thought was to show that neither a+t or at can be belong to N, Z, or Q, thus they must belong to I. However I'm not certain if that train of thought...
  6. L

    Irrational or Rational [Newton-Raphson]

    Homework Statement How are you able to determine if a solution is rational or irrational Homework Equations - The Attempt at a Solution - :confused: I'm pretty sure it will be something basic I'm forgetting, thanks for any help.
  7. M

    Integral of x/(x^4+x^2+1): Solution Attempts

    Homework Statement the indefinite integral of x/(x^4+x^2+1) Homework Equations n/a The Attempt at a Solution I didn't see an obvious u-substitution and it didn't look like a partial fractions candidate to me since the bottom is not easily factored. It doesn't look like any of the...
  8. J

    Proof that sum of 3 roots of rationals is rational etc

    Excuse my typography - I'm new here... a, b, and c are rational numbers. I want to prove that * IF S = root(a) + root(b) + root(c) is rational THEN root(a), root(b) and root(c) are rational in themselves. Now I have done as follows: I reverse the problem and try to show that: * IF...
  9. R

    What is the rational series question for (n + 1) / (3n - 1)?

    Homework Statement (n + 1) / (3n - 1) Homework Equations A_n = L The Attempt at a Solution lim n-> infinity (n/n + 1/n) / (3/n - 1/n) = (1 + 0) / (3 - 0) = 1/3 Thats the solution, however i have questions.. 1.) If a series is in rational form like this, is it...
  10. G

    Proving the Existence of Rational Points on a Circle

    Does anyone have an idea how to prove the following (or prove that it is not true): For any positive integer k, you can find k points on a circle such that each point is a rational distance from every other point.
  11. K

    Closest approximated rational triangle

    Given 2 points on a plane, if you arbitrarily place a third, is there any way to determine the closest approximation to this triangle where all sides of the approximation are rationally related? The only thing I can think of would be to draw a small circle around the third point that...
  12. D

    Rational Varieties: Finding Birational Maps & Inverses

    I am trying to show that some given simple affine algebraic varieties are rational (i.e. birationally equivalent to some A^k). Are there any tricks or even nice algorithms for finding the birational maps and their inverses? Examples are the curve x^2 + y^2 = 1 and the surface x^2 + y^2 + z^2 =...
  13. R

    Integration with a rational expression

    Homework Statement The rate at which people enter an amusement park on a given day is modeled by the function E defined by: E(t) = 15600/(t² - 24t + 160). The rate at which people leave the same amusement park on the same day is modeled by the function L defined by: L(t) = 9890/(t² - 38t...
  14. C

    Adding and subtracting Rational Expressions

    I'm stuck on how to do this problem My attempt: the answer to the second one is 3x + 2/(x+2)(x-2) Can someone point out my mistake?
  15. M

    Can a Non-Factorable Function Have Rational Real Roots?

    can a non-factorable function has rational real roots?
  16. C

    Computing the range for a rational function involving absolute value

    Hi. I need help computing a range. The question is : Find the domain and range of y=\frac{|x+2|}{x}. The domain is obvious, x can't be 0, (-inf,0,) U (0,inf). But how do I find the range?? Can someone help me out? I have tried messing around with the definition of absolute value... if...
  17. B

    Dovall rational numbers under multiplication form a group.

    I know the set of positives rationals form a group under multiplication and that the negative irrationals do not form a group under multiplication because there is no identity or inverse. My question is does the set of all rational numbers under multiplication form a group.
  18. B

    Understanding Rational Exponents with Negative Bases

    Can anyone point me to a text or link that summarizes the rules when evaluating/simplifying an expression of the form (a^n)^(1/m) for a < 0. (a^n)^(1/m) yields different answers for a^(n/m) and (a^(1/m))^n . Ex: (-8)^(2/6) = (-8)^(1/3) = -2 (-8)^(2/6) = ((-8)^2)^(1/6) = 2...
  19. J

    Rational Numbers: Is 1 a Rational Number?

    Is the number 1, or any other whole/negative number a rational number?
  20. L

    Number if rational iff it has periodic decimal expansion

    My teacher gave us as excercices this: I'm pretty certain you have to prove it by contradiction, but I don't get how to represent to periodic decimal expension in a proof? Any hint is welcome, thanks in advance.
  21. N

    For which value of x is each rational expression not defined?

    I don't get it... For which value of x is each rational expression not defined? a) 3/x are you supposed to solve for x? or?
  22. S

    Are all irrational numbers rational?

    Since pie is the ratio of the circumference of the circle to its diameter, isn't it possible that there exist a fraction for all nonrepeating going on forever decimal values?
  23. A

    Modelling with polynomials and rational functions

    4 The height, x metres, of a diver above a swimming pool at time t seconds after he has bounced from the diving board can be modeled by the function x(t)= 3- 3t (3t^2/2) a How long, in seconds, after he has bounced from the diving board does the diver reach his maximum height? b What is the...
  24. S

    Exploring Variation in Slopes of Horizontal Asymptotes in Rational Functions

    Hello, Recently I have been trying to reason why certain rational functions such as (-2(2x^2+x-31))/((x-3)*(x+4)*(x+1)) have varying near horizontal asymptote slopes. I know that the direction of the horizontal asymptote can be varied by altering the parent function x+1/x, but several...
  25. electronic engineer

    Calculating the Integral of a Rational Function

    how to calculate the calculus of this rational function: \int \frac{dx}{{x+x^6}} could anyone get me through the solution?!
  26. M

    Rational Expression Word Problem

    In a motorcycle race, one lap of the course is 650m. At the start of the race, Genna sets off 4 seconds after Tom does, but she drives her motorcycle 5m/s faster and finishes the lap 2.5 seconds sooner than he does. a) Find the speed at which each of them is driving. b) Find the tim etaken...
  27. S

    How Does Binomial Expansion Work for Rational Indices?

    Hi I wanted to know what is the expansion of (1+x)^n when n is a rational number and |x|<1 ... Please let me know as soon as possible.. Thanks for your help Sincerely Sparsh
  28. W

    Multiplying Rational Expressions

    This is a question I have to do for tomorrow \frac{1+4x+4x^2}{x}\times\frac{3x^4}{12x^2-3}. I just factor the tops and then cancel where I can. This is what I came up with: \frac{3x^4}{12x^3-3x} But the answer key says: \frac{x^3(1+2x)}{2x-1} What have I done wrong?
  29. G

    Trigonometry made more rational

    Trigonometry made more "rational" I read an article the other day on "rational trigonometry" and decided to go exploring... and I feel as if a lightbulb has turned on. This seems like such a wonderful new technique to doing otherwise cumbersome problems, and I agree that it does feel like a...
  30. A

    Multiply and Divide Rational Expressions

    I thought I knew what I was doing (because I've done this before) but I can't see what's happening in this one example...I understood what was going on right up until the last step. If you are dividing two rational expressions, flip the second expression and multiply the two rational...
  31. A

    Is e^pi Rational? - Research Progress

    Is e^pi rational? I seem to have heard from one of my tacher that research was going. How far we have gone?
  32. P

    Proof of Rational Root Theorem

    I want to find a nice and elegant proof of the Rational Root theorem, but I get stuck. I read some stuff on the Internet, but I have not found a complete proof of the theorem. Here's my try: Say we have a polynomial: F(x) = \sum ^{n}_{r = 0} a_{n}x^{n} = a_{n}x^{n} + a_{n - 1}x^{n-1} +...
  33. N

    Which rational numbers between 0 and 1 have finite decimal expansions?

    The question I have been given is essencially this: Give a brief description, with some explanation, of which rational numbers between 0 and 1 have finite decimal expansions? [An example is 3/40 = 0.075, however 2/3 = 0.66666666666...] I am truly :confused: Please help. I have looked...
  34. J

    Proving or Disproving rational raised to rational is rational number

    Im trying to either prove or disprove that if a and b are rational numbers, then a^b is also rational. I tried doing it with a contradiction, but i can't seem to correctly arrive at a solution. this is how i started the problem defn of rational number: a,b = {m/n: m,n are all nonzero...
  35. S

    Rational expression limit problem

    Sligtly more complex than the average one, I'd assume. How would I go about proving that the limit of of a rational expression consisting of two polynomials of the same degree goes to one and the limit of one where the degree of the bottom is greater than the degree of the top goes to zero. I'd...
  36. K

    Rational functions(just need explanation)

    okay i need help on rational functions I am not sure if that what they are called but it all should be the same.What i need to find is the holes zeros vertical asymtotes horizontal asymtotes and the slant asymtotes can someone please tell me how to find each on of those and the...
  37. M

    Mastering Rational Expressions: Simple Tips and Tricks for Advanced Math Classes

    Right now in math class we are learning rational expressions. Since I am in an advanced math class, it seems like we learn a new lesson everyday. So if you don't understand something, you pretty much need to teach yourself. I don't really understand rational expressions, so can someone tell...
  38. N

    Representation of Rational Numbers by Formulas

    Dear Experts. Do you know some literature on representation of rational numbers by formulas? TIA.
  39. I

    Is this well-defined in the rational numbers

    Need help with proving: Show that (a,b) + (c,d) = (a+c, b+d) is not well-defined in the rational numbers. [Note: (a,b) + (c,d) = (ad+bc, bd) is well-defined because (a,b) is related to (c,d) when ad = bc.)]
  40. C

    Spline what is a b spline what is rational b spline

    what is a spline what is a b spline what is rational b spline what is a uniform rational b spline what is non uniform rational b spline(NURBS)
  41. S

    Rational Expressions Answers: Check Your Work

    I was wondering if someone could check these questions to make sure that I am doing these correctly. I have attached a few questions in word document as I could not type it properly on here.
  42. B

    List all Rationals in [a,b]: a + lim x->0 x{n}_0^(floor((b-a)/x))

    Given that {\left\{ {\left( {a,b,x} \right)|0 < x < \left( {b - a} \right),\left( {a,b,x} \right) \in \mathbb{Q}^3} \right\}} , Can I list all the rational numbers in \left[ {a,b} \right] as the sequence represented by a + \mathop {\lim }\limits_{x \to 0} x\left\{ n...
  43. A

    Proving the Existence of Rational Numbers Between Real Numbers

    prove that between any two real numbers there is a number of the form \frac{k}{2^n} where k is an integer and n is a natural number.
  44. E

    Proving Rational Solutions in Linear Systems with Rational Coefficients

    I need some closure on the following, Question: Prove that if a linear system of equations with only rational coefficients and constants has a solution then it has at least one all-rational solution. Must it have infinitely many? My Solution: If the RHS of the equations is a rational...
  45. A

    Understanding the Pattern of Positive Rational Numbers

    I am asked to list the positive rational numbers in one list so that the pattern of the order is clear and so that all the positive rational numbers would eventually appear on the list. Then I have to explain the pattern and why every positive rational number will eventually be on the list...
  46. A

    The Product of Rational Numbers

    I've been grinding my brain at this problem because I am trying to figure out if the product of two rational numbers is always, never, or sometimes rational. a rational number would either have to terminate, or be infinitely periodic, so i would say that the product of two rational numbers is...
  47. T

    Irrational + irrational = rational

    Can someone prove that there exists x and y which are elements of the reals such that x and y are irrational but x+y is rational? Certainly, there are an infinite number of examples (pi/4 + -pi/4 for example) to show this, but how would you prove the general case?
  48. T

    What would your rules be for rational debate?

    If you could make a set of rules that would be the constitution for law courts, scientific debates and parliaments around the world, what would they be? Here's mine. This purpose of these rules are to ensure that the conclusion of a debate is restricted only by the intelligence of those...
  49. K

    Integrals of Rational Functions

    Integrals of Rational Functions... The integral of:... (x-1)/x^4+6x^3+9x^2 , dx...i factored out the bottom getting: x^2(x+3)(x+3)...so, my new integral is: (x-1)/x^2(x+3)^2... now when i muiltlpy both sides by (x-1)/x^2(x+3)^2...i get... x-1= A(x+3)^2 + Bx^2(x+3) + C x^2...for A i got...
  50. A

    Finding Rational Numbers Satisfying a Quadratic Equation

    heres a problem i stumbled: find three rational numbers a,b,x such that: x^2 + 5 = a^2 x^2 - 5 = b^2
Back
Top