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.

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

    Accumulation point of rational numbers in (0,1)

    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...
  2. R

    Set of all rational sequences countable?

    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...
  3. I

    Getting equation from graph of a rational function with an oblique asymptote.

    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...
  4. J

    Regard Q, the set of all rational numbers, as a metric space,

    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...
  5. F

    At what probability is it rational to look for paranormal explanations?

    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...
  6. P

    Create a rational function with the following properties

    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...
  7. H

    Prove every rational number has a never ending decimal view

    Its all in the title: How can we prove every rational number has a never ending decimal view. Since I translated the question from another language, I didn't know how to put it. I know it's true because: 1=0.9999... and i.e.: 1/3=0.333... or 24.67=24.669999... The problem is...
  8. A

    Can You Solve These Rational Equations?

    Homework Statement i) x+6 / x+2 ≤ x k) 4x - 4/ x - 1 < x^2 + 3x + 22. The attempt at a solution i) http://i950.photobucket.com/albums/ad342/jeet999/photo2-1.jpg k) http://i950.photobucket.com/albums/ad342/jeet999/photo1-1.jpg 3. Answers i) x ε [-3,-2] or x ε (2, ∞) k) x ε (-∞, -3) or x ε...
  9. A

    Which Method is Correct for Solving Rational Equations?

    Homework Statement In the picture above, I solved the rational equation using two different methods. In the first method I got x = 1 and in the second method I got x = 4/3. Which answer is correct or are both of them incorrect? Thanks.
  10. I like Serena

    Rational solutions of equation 3x² + 2 = y²

    Homework Statement Show that the equation 3x2 + 2 = y2 has no integer solutions by calculating modulo 3. Proof that the equation has no rational solutions. This is a problem from an introductory chapter in algebra that I'm teaching.Homework Equations The Attempt at a Solution I've got the...
  11. J

    Degree of Rational Function Field

    Hi, I am trying to find the degree [F_{q^2}(x):F_{q}(x)]$. Intuitively, I thought it will be 2? but cannot see that
  12. N

    Solving Rational Inequalities: How to Determine the Interval of Solutions?

    Homework Statement 2x-1 _____ > 0 5x+3 Homework Equations The Attempt at a Solution Just wondering, my teacher taught us that youre only supposed to look at what makes the denominator = 0, and don't look at the numerator because it has no affect on anything. So, if i...
  13. N

    Solving Rational Inequalities: How to Find the Solution Set?

    Homework Statement x^2 +3x +2 __________ < 0 x^2-9 Homework Equations The Attempt at a Solution I factored the top to be (x+2) (x+1) ______________ < 0 (x-3)(x+3) Couldnt cancel anything out, So now i don't know what to do. I missed this lesson in class and...
  14. M

    F(x) = x if x is rational, 0 if x is irrational.

    Homework Statement F(x) = x if x is rational, 0 if x is irrational. Use the δ, ε definition of the limit to prove that lim(x→0)f(x)=0. Use the δ, ε definition of the limit to prove that lim(x→a)f(x) does not exist for any a≠0. Homework Equations lim(x→a)f(x)=L 0<|x-a|<δ...
  15. A

    Discrete Math irrational and rational numbers proof

    Homework Statement Prove by contradiction. Your proof should be based only on properties of the integers, simple algebra, and the definition of rational and irrational. If a and b are rational numbers, b does not equal 0, and r is an irrational number, then a+br is irrational. Homework...
  16. I

    Solving Limits Involving Rational Fractions

    Homework Statement May seem easy to people but I have no idea how to do this :frown: lim_{x\rightarrow0}\frac{sin x}{2x^{2}-x} Homework Equations The Attempt at a Solution I factored out an x from the bottom but that really makes me see nothing else to do.. So I tried the quotient...
  17. C

    Rational and Irrational numbers

    Homework Statement Let f be the function defined on the real line by f(x)= \begin{cases} \frac{x}{3} & \text{if $x$ is rational } \\ \frac{x}{4} &\text{if $x$ is irrational.} \end{cases} Let D be the set of points of discontinuities of f. What is D? Homework Equations None...
  18. C

    How to prove rational number is a commutative field

    I was wondering how to prove rational number is a commutative field. Personally, I didn't think the word "commutative" is necessary, how about others? Do I simply prove it is commutative under multiplication?
  19. P

    Homemorphism between the set of circles with rational points and rectangles

    I am trying to solve a problem in which we need to prove that the set of all circles with rational points and radii is homemorphic to the set of all rectangles with vertices at rational points with the length of the diagonals as rational number. I am not able to figure out what the approach...
  20. D

    Basic limits of rational functions: behavior near vertical asymptotes

    Homework Statement We are required to sketch a (reasonably accurate) picture of a rational function f(x) = P(x)/Q(x) with P, Q polynomials in x and Q nonzero. We know that the roots of Q(x) are, say, x1, x2, etc. and so f(x) is (typically) asymptotic to the vertical lines x = xk for each k...
  21. Shackleford

    Prove F isomorphic to the field of rational numbers

    Sorry for so many questions. This is the second-to-last problem in the last homework. I don't think I even understand the question. I know what a group isomorphism is. How does it relate to the field of quotients of ring E of all even integers. How do I show F is isomorphic to the field of...
  22. M

    Square root of a rational quantity: Algebra

    Why arnt the two terms eqaul even though the square root of m was solved properly? And is this a general rule?
  23. K

    Extension of Rational Roots Test to UFDs

    Homework Statement Show that p(x) = x^2 - \sqrt2 is irreducible in \mathbb Z[\sqrt 2] . The Attempt at a Solution I think I have this, but I just want to make sure my reasoning is correct. I'm sure there are other ways. Firstly, it is sufficient to show that p(x) is irreducible in the...
  24. J

    Working with rational polynomials in Maple

    Hi Could someone please explain how to best handle rational polynomials in Maple? I have matrix of rational polynomials and for some reason Maple keeps grumbling i.e. "error, (in, linearalgebra:- HermiteForm) expecting a matrix of rational polynomials" The matrix I am working with is...
  25. rcgldr

    Computers rational and irrational numbers

    I think this needs it's own thread. e and pi are transcendental numbers: http://en.wikipedia.org/wiki/Transcendental_number The square root of 2 is n irrational number: http://en.wikipedia.org/wiki/Irrational_number 1/3 is a rational number...
  26. G

    Simplifying Rational Imaginary Functions

    Why does (5 i)/(3 (1-i)) = -5/6+(5 i)/6
  27. M

    Finding the Limit of a Rational Function

    This is just a problem I came across while reviewing basic calculus. Homework Statement Find the limit as x approaches 0 of f(x)=(1/(x(x+1)^1/2)) - (1/x) Homework Equations The Attempt at a Solution My problem here is really more of an algebra problem than a calculus problem...
  28. R

    Supremum & infimum of the set of all rational numbers

    Hi everybody, Please help me to find supremum & infimum of the set of rational numbers between √2 to √3 (ie) sup & inf of {x/ √2 < x < √3 , where x is rational number}
  29. R

    Concavity of a rational function

    I have to curve sketch this function, ((x)3))/((x)2-1) I did all of the steps, and I got this as the second derivative: ((2x(x2+3))/((x2-1)3) I got concave up:(-1, 0)u(1, inifinity) Concave down:(-infinity, -1)u(0,1) Am I right? I use -1 and 1 as the interval since they are the...
  30. H

    Solve Second Degree Polynomial for Rational Function

    Homework Statement Find the second degree polynomial P(x) such that P(0)=1,P'(0)=0,and \intP(x)/{x3(x-1)2} dx is a rational function Homework Equations this chapter is about integration techniques,L'Hopital's Rule, and Improper Integral partial fraction,partial integration are...
  31. C

    Irrational sequence that converges to a rational limit

    Hi. I found some rational sequences that converge to irrational limits, but am not having any luck going the other direction, i.e., an irrational sequence that converges to a rational limit. Any suggestions?
  32. M

    Limit of a rational function with a square-rooted expression in numerator.

    lim as x -> 0, [(x+4)1/2-2]/x That's the limit I want to evaluate. I keep running into problems getting to the real limit (1/4). You don't have to give me the answer, but let me know if I'm missing something simple. Or you can just give me a hint.
  33. M

    Limit of a rational function for different values of K (check)

    Find the limit of (4kx + 9)/(45x + 3) as x->inf, when K=5, K<5, K>5. Answer: I divided each term by the highest power of 4, and was able to come up with 0 (K<5), 1 (K=5), inf (K>5). I'm pretty sure these are correct. Was this the right method? Am I good to assume that dividing by the highest...
  34. G

    Invent a suitable rational function with characteristics

    Homework Statement For each part, invent a suitable rational function, f(x), with the characteristics given and make a sketch of your function. a) y behaves like x/3 for large x and x=1 is a vertical asymptote b) x=2 and x=4 are vertical asymptotes, f(-1) = 0, f(3) = 0, and f(0) = 1...
  35. Y

    The set of rational numbers in [0,1]

    this set is neither closed nor open, correct? the boundary of this set is the closed interval [0,1] because every ball centered at 0<=t<=1 contains both rational numbers and irrational numbers, am I right?
  36. Y

    Understanding Measure Zero and Countable Additivity for Rational Numbers

    A book I'm reading says: If a set A has infinitely many points which can be arranged in a sequence a_1,a_2,\cdots,, then A has measure zero. What does it mean by "can be arranged in a sequence"? The book gives an example on the set A which is all the rational numbers between 0 and 1. Why...
  37. B

    Lines and circles having rational operations

    Hi all, I am not sure if this is the right place to ask but I have two problems which I require enlightenment. The questions are, 1) Show that the intersection of two lines can be computed by rational operations. 2) Show that the intersection of a line and a circle can be computed by...
  38. A

    Simplest Way of finding zeros of cubic eq when rational roots test fails

    Homework Statement Find the zeros of the cubic equation: y = x^3 -9x^2 + 15x + 30 How do we find the zeros of this? In this case, subbing in x-values that will make it equal 0 does not work.
  39. X

    Inequality rational polymonial

    Homework Statement (3x+1)/(x+4)>=1 Homework Equations The Attempt at a Solution (3x+1)>=(x+4) 2x>=3 x>=3/2 But this is wrong?? Why?
  40. B

    Solving Rational Operations Questions

    Hi guys, I'm having trouble solving the following questions. 1) Show that the line through two rational points has an equation with rational coefficients 2) show that a circle whose center is a rational point and whose radius is rational has an equation with rational coefficient. Cheers
  41. A

    Evaluating limits of rational functions

    Homework Statement Why does the limit as x approaches 0 of x^2 + 5 / 3x go to infinity (with 0 as an essential disc.) but without the +5, the function goes to 0? Homework Equations The Attempt at a Solution I tried approaching evaluating the limit of the function by comparing the...
  42. N

    Rational power of complex number-calculation

    Homework Statement Use definition (1), Sec. 28 of z^c to show that (-1 + i*sqrt(3))^(3/2) = +/- 2*sqrt(2) Homework Equations z^c = e^(c*log z) The Attempt at a Solution (-1 + i * sqrt(3))^(3/2) = e^[(3/2) * log(-1 + i * sqrt(3))] = e^[(3/2) *...
  43. S

    Find term of a rational partial (reduced) fraction

    Homework Statement Find the coefficient on the x^2 term in ∫[(x+9)/3] * [log(x+1)] * dx Answer: -1/12 Homework Equations Integration by parts ∫dv * u = u * v - ∫du * v The Attempt at a Solution 1. I integrated by parts to get ------------------------- ∫[(x+9)/3] * [log(x+1)] * dx] =...
  44. E

    Decimal / Rational Exponents BY HAND

    I'm reviewing for a test. One of my questions on the review (and incidentally a question I've had in my own mind for a long time) is how do you evaluate an ARBITRARY rational exponent with pencil and paper and no calculator? The specific problem i was givens is "Solve without a calculator...
  45. W

    Find the partial fraction decomposition of the rational function.

    5x^2 + 8/x3 + x2 I got a 5x^2 + 8 = A/x^2 + B/x + 1 A(x+1) + B(x^2) (Ax+A) + (Bx^2) (Ax + Bx^2) + A 5x^2 + 8 = (A + B) x + (A)x^2 5x^2+x+8=(A+B) + (A) This is about as far as I can get but I think I made a mistake somewhere but I don't know where? Can someone help me?
  46. Z

    Asymptotic of rational function

    given a rational function R(x,y,z) that has no poles on the R^3 plane my question is if for big (x,y,z) \rightarrow \infty it has the following asymptotic expansion R(x,y,z) \sym \sum_{m,n,l= -\infty}^{N}a_{m,n,l}x^{m}y^{n}z^{l}
  47. M

    Are A and B Similar Over a Subfield if They Are Similar Over an Extension Field?

    Hi, I'm just new here, I don't know if I'm on the right thread.:D Homework Statement Let F be a subfield of K. A, B be elements of Mn(F). Show that if A and B are similar over K, then A,B are similar over F. (Hint: what can be said about the rank of f(C(f(x)^m))^n? about the rank of...
  48. M

    Expressing a rational expression in partial fractions

    Trying to help out a friend. I appolagise if yet again this is in the wrong part of the forum, i haven't an idea what it is categorised as, I am an apprentice engineer and simply that lol Can someone explain how the following is done: x^2 – x + 1 / (x - 2)(x^2 + 1) ≅ A / (x - 2) + Bx +...
  49. A

    Simplifying rational powers after completing a derivative

    I have the following function to differentiate: f(x) = x^\frac{4}{5} * (x-4)^2 My derivative is: f'(x) = x^\frac{4}{5} * 2(x-4) + (x-4)^2 * (\frac{4}{5}) * x^\frac{-1}{5} My calculator gives me: f'(x) = \frac{2(x-4) * (7x-8)}{5x^\frac{1}{5}} How does my expression simplify into...
  50. M

    Integral of a rational function over the positive real axis

    Hi everyone, this isn't really a homework question- this integral has come up during project work- but this seems like a sensible place to ask it nontheless. Homework Statement Compute I(t)=\int^{\infty}_{0} dx \frac{x^5}{(t+x^2)(1+(t+x^2)^4)} where t is real and positive. Homework...
Back
Top