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. R

    [number theory] prove that x^2+y^2=3 has no rational points

    Homework Statement The actual problem is: "Does x^2+y^2=3 have any rational points? If so, find a way to describe all of them. If not, prove it."Homework Equations NoneThe Attempt at a Solution I found a book on Google Books (can't find it again) that said that this circle has no rational...
  2. V

    Integration of rational functions by partial fractions

    Homework Statement write out the form of the partial fraction decomposition of the function, do not determine the numerical values of the coefficients x^2/(x^2 + x + 2) Homework Equations The Attempt at a Solution since the numerator is not less of a degree than the...
  3. L

    Finding Rational Solutions for ψ in √(87600-1.44ψ^2) with ψ ≤ 10

    How can I find the values for ψ that the following sqaure root equals to a rational number?: √(87600-1.44ψ^2) Also, I don't want ψ to be greater than 10. What's the procedure to find the answer?
  4. S

    Domains of Rational Functions (standard notation)

    Im preparing for a CLEP test in precalculus. As part of my prep, I need to review identifying domains of functions. I have a question about writing domains in standard notation. I was hoping someone could explain a bit the style. For an example: x-2 / x^2 -2x -35 As a rational...
  5. B

    Integration of Rational Functions by Partial Fractions?

    Integration of Rational Functions by Partial Fractions? Ok I'm working on some homework problems and I don't even know how to do the first one, here is my problems and the steps that I did thus far ( I don't know if I did them right) 5x-13/(x-3)(x-2)= A/x-3 + B/x-2\rightarrow ...5x-13=...
  6. I

    Limit of function with rational

    Homework Statement the title says it all x \rightarrow 4 for f\left(x\right) = \frac{\sqrt{1+2x}-3}{\sqrt{x}-2} I have multiplied both top and bottom by conjugate, \sqrt{x}+2: f\left(x\right) = \frac{\sqrt{x(1+2x)}+2\sqrt{1+2x} -3\sqrt(x)-6}{x-4}but don't know how to take this further...
  7. S

    Graph/analyze a function of a rational /w complex root?

    graph/analyze a function of a rational /w complex root?!? Homework Statement The function is y=x^2/(x^2+3) Homework Equations First and Second Derivatives Chart to find intervals of increase/decrease and concavity. The Attempt at a Solution 1) Domain {XeR} 2) Intercepts If...
  8. J

    Need help creating rational function no clue where to start

    b]1. Homework Statement [/b] lim f(x) = 2 x -> infinty lim f(x) = -2 x-> - infinty lim f(x) = - infinty x-> -4 lim f(x) = - infinty x-> 2- lim f(x) = infinty x-> 2+ relative min of 0 at x=2 relative max of -0.900466 at x=0.442818 concave down (-infinty, -4) (-4,-2)...
  9. B

    Rational numbers and commutative law

    How can i prove that addition and mulitiplication of rational numbers follow commutative and associative law?
  10. M

    Prove that F is discontinuous at every rational number

    Homework Statement Let x_{1}, x_{2}, ... be a sequence of rational numbers in which each rational number in (0,1) occurs exactly once. Define the function, H(x) = 0 if x \leq 0, and 1 if x > 0. Next, define the function F(x)= \sum^{\infty}_{k=1} 2^{-k} H(x - x_{k}). Prove that F is...
  11. J

    If a and b are rational numbers satisfying

    Homework Statement Prove Proposition 1.15. Proposition 1.15. If a and b are real numbers satisfying a<b, then there are rational numbers and irrational numbers between a and b. Homework Equations Professor said to use the Archimedean property The Attempt at a Solution a < b...
  12. R

    Take the derivative of rational expression

    Homework Statement f(x) = \frac{x}{x-1} + \frac{x+1}{3x} Homework Statement need to take the first derivative of this expression... I can do it but I am curious as too why i cannot take the derivative of \frac{x}{x-1} and then just add it to the derivative of \frac{x+1}{3x}...
  13. C

    Proving Rational Solutions of ax^n + a0 = 0

    Homework Statement Suppose that r is a solution of the equation: anxn + a(n−1)x(n−1) + . . . + a1x + a0 = 0 where the coefficients ak belongs to Z for k = 0, 1, . . . n, and n is greater or equal to 1. If r is a rational solution r = p/q, where p, q belong to Z and p and q are...
  14. A

    Rational numbers and periodic decimal expansions

    A long time ago I took a number theory course and really enjoyed it. At one point we were shown the proof for the theorem that a number is rational if and only if it has a periodic decimal expansion. The (<=) direction is really easy if you know some Calculus, but I remember the (=>) direction...
  15. F

    Closure of the Rational Numbers (Using Standard Topology)

    Prove that Cl(Q) = R in the standard topology I'm really stuck on this problem, seeing as we haven't covered limit points yet in the text and are not able to use them for this proof. Can anybody provide me with help needed for this proof? Many thanks.
  16. W

    Proving Rational Sum of n Rational Numbers

    Homework Statement Prove via Mathematical Induction that, The sum of n rational numbers is rational. Homework Equations The Attempt at a Solution Let N = 1 The sum of one rational number is the number it's self, which is a rational number. Assume when n = n, the sum of n...
  17. P

    Pre-Cal; rational inequalities

    Homework Statement 1. x-2|x| < 3 Homework Equations The Attempt at a Solution Okay, I attempted this equation below but I don't know if I'm right. x-2|x|- 3 < 0 x^2 - 2x - 3 < 0 (x - 3) (x + 1) < 0 x = 3 x = -1
  18. T

    Show that the field of rational functions is not a complete ordered field

    Homework Statement Show that R(x) cannot be made into a complete ordered field, where R(x) is the field of rational functions. Homework Equations Definition of a complete ordered field: An ordered field O is called complete if supS exists for every non empty subset S of O that is...
  19. silvermane

    Rational numbers and Lowest terms proof:

    I've been recently reading a book on abstract algebra and number theory, and I stumbled upon a problem that at first glance looked obvious, but I can't seem to figure out how to formally write the proof. 1.)So, let's say we have 4 integers, r,s,t,u, all greater than or equal to 1. Suppose...
  20. Fra

    Constructing a rational action?

    I've been skimming the paper of A.N Schellekens (http://arxiv.org/abs/0807.3249) that discusses the notion of string theory landscape. He argues against uniqueness, and seems to think that the existence of a landscape of possibilities adds explanationary value and is a good thing. Surprised...
  21. Z

    How Do We Calculate an Integral of a Rational Function from Zero to Infinity?

    how could we calculate the follwing integral ?? \int_{0}^{\infty} \frac{ K(x)}{Q(x)}dx here K(x) and Q(x) are POLYNOMIALS , of course if we had an integral over all R instead of (0 , \infty ) we could apply Cauchy's residue theorem i think there is a 'closed circuit' to perform the...
  22. A

    Prove: Set of rational numbers cannot be expressed as intersection of open sets

    Homework Statement Show that the set of rational numbers in the interval (0, 1) cannot be expressed as the intersection of a countable collection of open sets. Homework Equations The Attempt at a Solution This sounds like something requiring proof by contradiction. There must be...
  23. K

    Show f(x) = { x/2 if x rational , x if x irrational is not differentiable at 0

    Homework Statement Show that the function f(x) = { x/2 if x is rational { x if x irrational is not differentiable at 0 Homework Equations If f is differentiable at 0 then for every e > 0 there exists some d > 0 such that when |x| < d, |(f(x)-f(0))/x - L | < e...
  24. J

    Rational Numbers: Is Any Whole Number a Rational?

    Are whole numbers rational numbers? If so, will any whole number divided by any other whole number result in a rational number?
  25. B

    Solving Rational Dependence in Vector Spaces

    Hi guys: I've got a problem I've been working on for some weeks and this might be the key to unlocking it. The question is: Given a vector in R^k, what is the measure of the set of vectors whose components are rationally dependent? Rationally dependent means for a given vector, you may...
  26. T

    Rational Representation of a Repeating Decimal

    Homework Statement Find the rational number representation of the repeating decimal. 1.0.\overline{36}Homework Equations The Attempt at a Solution I know it has something to do with infinite geometric sequences but I'm not sure what. what would your ratio be for a repeating decimal, I've...
  27. N

    Definite integral of rational function

    Homework Statement The definite integral of (t^3 + t -1)/(sin(t)) from 2 to x^2 Homework Equations The Attempt at a Solution First off, I don't have the solution anywhere, my teacher just gave this to us to work on for the final exam review. I can think of a few things. I...
  28. T

    Tricky rational function integral

    Homework Statement \int \frac{4x^5-1}{(x^5+x+1)^2} dx = ? Homework Equations The solution is - \frac{x}{x^5 + x + 1} The Attempt at a Solution Other than getting lucky and noticing immediately that this could be the derivative of a fraction, I do not see an easy way to solve this. The...
  29. stripes

    Integration of a rational function

    Homework Statement find \int\frac{x^{2}-2x-1}{(x-1)^{2}(x^{2}+1)}dx Homework Equations None The Attempt at a Solution \frac{x^{2}-2x-1}{(x-1)^{2}(x^{2}+1)} = \frac{A}{x-1} + \frac{B}{(x-1)^{2}} + \frac{Cx + D}{x^{2}+1} x^{2}-2x-1 = A(x-1)(x^{2}+1) + B(x^{2}+1) + (Cx + D)(x-1)^{2}...
  30. E

    Is this correct? (sum of two rational numbers is rational)

    Homework Statement "Prove that the sum of two rational numbers is a rational number." I just started on proof writing, so I'll just like to verify if I'm not missing anything here, and get some comments about the style. The attempt at a solution Theorem. If a,b \in \mathbb{Q} then a+b \in...
  31. J

    How Can I Graph Rational Functions More Systematically?

    Ok so, This summer I will be taking a Pre-calc/trig course intensive, to get ready to take calculus in the fall, to start up my track for physics. I got a Pre Calculus Workbook For Dummies and I have to say so far I'm not too pleased. I have already found a bunch of typos, and when there...
  32. n.karthick

    Irrational number approximation by a rational number

    Is there a way ( a theorem ) to find a rational number for a given irrational number such that it is an approximation to it to the required decimal places of accuracy. For example 22/7 is an approximate for pi for 2 decimal places.
  33. Z

    Rational Functions – Homework Help

    Hi there I have a Rational function y = 1 / x^2-1 . I have a good idea what the graph looks it, it will have vertical asymptotes at -1 and 1 and I can work out the y intercept (-1 concave down). However I'm not sure about the other parts to the question. Homework Statement dy/dx...
  34. T

    Finding critical numbers of function with rational exponent

    Homework Statement Find the critical numbers of the function: F(x) = x^(4/5) (x - 4)^(2) Homework Equations None. The Attempt at a Solution I differentiated and got to (1 / 5th root of x) (x - 4)(2x + 4/5(x-4)) but I don't know how I can simplify the expression to be able...
  35. M

    Solving Rational Equations: What is the Missing Step?

    Homework Statement Solve for x. Homework Equations 3/2 + 2/2x-4 = 1/x-2 The Attempt at a Solution LCD = 2(x-2) 3/2 + 2/2x-4 = 1/x-2 Mult. all terms by 2(x-2) 3x - 6 + 2 = 2 3x = 6 x = 2 What am I missing here?
  36. I

    Help with proof for rational number problem

    Homework Statement For all a in the set of real numbers, if a is rational, a + \sqrt{2} is irrational. You may use that \sqrt{2} is irrational and the sum and difference of rational numbers is rational. Homework Equations The Attempt at a Solution My proof seems way too simple, I don't trust...
  37. H

    0 when irrational, 1/q in lowest terms with rational not differentiable.

    Homework Statement Hi, I have this function: f(x ) = 0 (x is irrational) or f(x) = 1/q for rational p/q in lowest terms. show that this function is not differentiable anywhere The Attempt at a Solution This is the answer from the solutions book: consider [(f(a+h) - f(a ) ) /...
  38. B

    Rational and irrational numbers proof

    Hey all, I'm new here so I'm a little noobish at the formatting capabilities of PF. Trying my best though! :P Homework Statement Let a, b, c, d \in Q, where \sqrt{b} and \sqrt{d} exist and are irrational. If a + \sqrt{b} = c + \sqrt{d}, prove that a = c and b = d. Homework...
  39. N

    How to state an equation of rational functions that has Asymptotes?

    How to state the equations of a rational functions with the following asymptotes? (1)x=2, y=-3 (2)y=0, x=4 (3)y=0
  40. J

    How do you multiply rational equations with multiple numerators?

    How do you multiply an equation that has 2 or more numerators. such as : 8x + 8 x - 1 ______ X _______ X2 - 2x + 1 2x + 2 And don't say anything like, "I'm not doing your homework for you" or anything stupid like...
  41. M

    Partial fraction decomposition of the rational expression

    Homework Statement Write the partial fraction decomposition of the rational expression. Check your result algebraically. (x2 – 7x + 16)/[(x + 2)(x2 – 4x + 5)] The Attempt at a Solution [A/(x+2)] + [(Bx+C)/(x2-4x+5)] x2-7x+16= A(X2-4x+5)+(Bx+C)(x+2)...
  42. P

    Convert Numbers to Rational Bases: An Example

    I have been playing around with numbers in different bases and then I thought, what if they were in fractional bases. I found a way to convert numbers to fractional bases and have been searching on the internet and not found a similar way to do this. Anyway, here is an example of how I would do...
  43. N

    Delta-epsilon limit of a rational function with a jump discontinuity

    Hello, I am trying to prove the following... lim (x+3) \left|x+5\right|/x+5 x\rightarrow-5 from the left, L=+2 from the right, L=-2 I used delta-epsilon on the right hand limit and got \delta = \epsilon However, I'm not sure how to proceed when I get to this step while trying to prove the...
  44. K

    Show that the nested intervals property fails for the rational numbers

    In the field of formal rational functions, construct a nest of closed, bounded intervals whose intersection is empty. (That is, show that the Nested Intervals property fails in this field) I know it has to involve radical 2 but because that is the only number we know is irrational but other...
  45. L

    Legal proof that set of Rational numbers is countable?

    Hello, Just wondering if this is a correct way to say the set of RATIONAL numbers is countable: Rationals (Q) is countable , because for every Q = p/q , such that p & q are positive INTS (Z) and since the set of positive INTs (Z) is countable ( a 1:1 correspondence) Q is countable because it...
  46. B

    Determine the rational function that gives the annual cost of a security system

    Homework Statement A security system costs $9/month in costs (fixed cost). If a new security system costs $1500 to purchase and last for 8 years: a)Determine the rational function that gives the annual cost of a security system as a function of the number of years you own the system. Using...
  47. B

    Help in need : Rational functions problem

    Homework Statement A scientist predicted that the population of fish in a lake could be modeled by the function f(t)= 40t/(t^2+1), where t is given in days. The function that actually models the fish population is g(t)=45t/(t^2+8t+7). Determine where g(t)>f(t). Homework Equations...
  48. F

    What is the Range of a Rational Function?

    how can i find the range of a rational function for ex. y=1/x+1
  49. C

    Rational Function Application Question

    Homework Statement A home security system costs $9/month in electricity costs (you may assume this is a fixed cost over the duration of the time involved in this question). A new security system costs $1500 to purchase and lasts for 8 years. Determine a rational function that gives the...
  50. C

    I am stuck on evaluating the limit of a rational function

    Homework Statement Evaluate the limit: lim_{(x)\rightarrow(2)}\frac{x^2+x-6}{sqrt(x+4)-sqrt(6)} Homework Equations N/A The Attempt at a Solution I've drawn the graph which indicates that the at x=2, y=0, so 0 would seem to be the limit. I could not, however, get the limit...
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