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.

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  1. 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...
  2. 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...
  3. J

    Decomposition of a rational expression

    Homework Statement Decompose the rational expression into a sum of partial fractions: (x+1)/(3(x-2)2) I am familiar with the method of decomposing fractions into a sum of partials fractions (solving for A, B, C, etc.). What is confusing me is the coefficient 3 in the denominator. Do I...
  4. M

    Is there a Proof for the Rational Integral of x/(a^2+x^2)?

    Dear Forum, I've been trying to find a proof for the following: \int \frac{x}{a^2+x^2}dx = \frac{1}{2}\ln|a^2+x^2|+c After many hours I've resorted to asking for help - any ideas anyone? cheers, mazzo
  5. K

    Taylor Expansion for rational function

    Homework Statement Find the taylor expansion of the following formula in the case where r > > d to the first order in \epsilon = \frac{d}{r} \frac{1}{r_{+}} = \frac{1}{\sqrt{r^{2} + (\frac{d}{2})^{2} - rdcos\theta}} Homework Equations (1 + \epsilon)^{m} = 1+m\epsilon, where...
  6. J

    Between any two distinct real numbers there is a rational number

    Homework Statement Let x and y be real numbers with x<y and write an inequality involving a rational number p/q capturing what we need to prove. Multiply everything in your inequality by q, then explain why this means you want q to be large enough so that q(y-x)>1 . Explain how you...
  7. C

    For every rational number, there exists sum of two irrational numbers

    Homework Statement Prove: For every rational number z, there exists irrational numbers x and y such that x + y = z. Homework Equations by definition, a rational number can be represented by ratio of two integers, p/q. The Attempt at a Solution Is there a way to do this by...
  8. M

    Primes in set of rational numbers

    There was a part c and d from a question I couldn't answer. Let R = \{ a/b : a, b \in \mathbb{Z}, b \equiv 1 (\mod 2) \}. a) was find the units, b) was show that R\setminus U(R) is a maximal ideal. Both I was successful. But c) is find all primes, which I believe i only found one...the...
  9. Mentallic

    Rational Coefficients & Non-Rational Roots: A Puzzling Cubic Polynomial

    Homework Statement Not so much a homework problem, but a problem that is annoying me because of its simplicity. Not all cubic polynomials with rational coefficients can be factorized by the rational root theorem (or is this false?). What I am finding hard to comprehend is how a cubic with...
  10. N

    Simplifying rational numbers help

    Express \frac{1+\sqrt{2}}{3-\sqrt{2}} as a+b\sqrt{2} where a and b are rational numbers. I started by \frac{1+\sqrt{2}}{3-\sqrt{2}} * \frac{3+\sqrt{2}}{3+\sqrt{2}} But, I obtain \frac{5}{7}-\frac{4}{7}\sqrt{2} I believe that, here, a and b are rational, but is there a more tidy version? I...
  11. C

    Prove in any interval there exists a rational z

    Homework Statement Prove that in any interval there exists a rational z. Homework Equations The Attempt at a Solution My professor wrote this for me when trying to explain how to prove this: a \notin Q, \epsilon rational [r, s]\in a l([r, s])<\frac{\epsilon}{2}...
  12. D

    Find an equation of a rational function

    Find an equation of a rational function f(x)that satisfies the conditions. We're allowed to use a calculator, by the way. (: Okay, conditions: vertical asymptote: x=-5, x=0 Horizontal asymptote: y=0 x-intercept=7; f(1)=4 On the test i had no idea how to do it, but after seeing...
  13. D

    Arbitrary cycle of digits in rational number?

    Hello all, I have recently been wondering whether there is a way to determine a fraction for which the decimal expansion is a cycle of n numbers? I would like to be able to work this out myself, but I can't wait until I start my mathematics degree. So any help would be greatly appreciated...
  14. N

    Rational approach for specifying a Microscope

    Has anyone worked out a functional Parametric chart that enables one to choose a appropriate Microscope (exclude Scanning electron MicroScopes(MS)) amongst the multitude of choices for a modern academic micro-biological & research lab, hence formulated a general guideline? The MS should feature...
  15. J

    Computing with rational exponets

    Homework Statement Solve \int_-1^1 (x^{4/3} + 4 x^{1/3}) dx Im having difficulties in algebra when solving this problem. Homework Equations The Attempt at a Solution \int_-1^1 (x^4/3 + 4 x^1/3) dx = 3/7 (1)^7/3 + 4 3/4 (1)^4/3 - 3/7 (-1)^7/3 + 4 3/4 (-1)^4/3 = 3/7 +...
  16. A

    Simplfying rational expressions

    Homework Statement 6x (squared) +17x+7 _________________ 2x sqaured + 7x +3 Homework Equations Simplify The Attempt at a Solution 3x+7 is the answer ____ x+3 maybe u can factor 6x by doing 2x (3x) or maybe u don't do that and take 6x (squared +17...
  17. A

    Rational Expressions: Solving Homework with Difficulty

    Homework Statement x (squared) + x - 30 _____________________ x (sqaured) -3x + 2 Find all numbers for which the rational expression is not defined. i know this is really easy stuff, but i have a hard time staying focused in math, and i have to always teach myself with the book, but sometimes...
  18. G

    Rational function helpp needed immedietly

    rational function helpp needed immedietly ! 1. when astraunauts go into space, they feel lighter. this is because weight decreases as a person rises above Earth's gravitational pull according to the formula W(h) = We/(1+h/6400)^2 where We is the person's weight in Newtons, at sea level on...
  19. V

    Limit of rational function to rational power

    Homework Statement Evaluate the limit, WITHOUT using l'Hôpital's rule: \lim_{x \rightarrow -1} \frac{x^{1/3} + 1}{x^{1/5} + 1} Homework Equations The Attempt at a Solution I tried to use the conjugate method which does not produce a useful outcome: \underset{x\to...
  20. K

    Proving Rational Intersection of Sets with Irrational Elements

    Homework Statement Let S={p+q\sqrt{2} : p,q \in Q} and T={p+q\sqrt{3} : p,q \in Q}. Prove that S\capT = Q. Homework Equations See above. The Attempt at a Solution I was thinking possible using S\capT=Q S + T - S\cupT = Q But I have no idea how to combine them? I don't believe it's...
  21. R

    Finite Elements in a Set of Rational Numbers Proof

    Homework Statement This problem is insanely intuitive. Define f : (0,1) \rightarrow \Re by f(x)=\begin{cases} 1/q&\text{if } x \neq 0 \text{, is rational, and }x = p/q \text{in lowest terms}\\ 0&\text{otherwise }\end{cases} Suppose \epsilon > 0. Prove that there are at most a...
  22. J

    Proof: Solve Rational equation for interval

    Prove that if c and d are positive, then the equation \frac{c}{x-2} + \frac{d}{x-4} = 0 has at least one solution in the interval (2,4). I try to use the following theorem: If f is continuous on [a,b] and if f(a) and f(b) are nonzero and have opposite signs, then there is at least one...
  23. S

    Rational and irrational numbers. (semi- )

    Rational and irrational numbers. (semi-urgent) I need to figure this out by tomorrow =/ Homework Statement a. If a is rational and b is irrational, is a+b necessarily irrational? What if a and b are both irrational? b. If a is rational and b is irrational, is ab necessarily irrational...
  24. A

    Integration of rational functions by partial fractions

    The problem is: ((x^3)+x)/(x-1) And i need to break it into partial fractions... I tried long division and got: ((x^2) +x ) But the book gives me the answer of: (x^2)+x+2+(2/(x-1)) Any help would be very much appreciated, thanks.
  25. 3

    Simplifying rational functions with common factors

    Homework Statement Simply these rational functions: [\sqrt{(X^2)+12}-4]/(X-2) (2-\sqrt{(X^2)-5})/(X+3) (X-1)/(\sqrt{X+3}-2) Homework Equations The only example in the book used the technique of multiplying the numerator and denominator by the function p(x) if p(x) is the function...
  26. D

    Difficult Integral of a Rational Function

    Hey guys I'm wondering if someone could hep me solve this integral. I've been working at it for a few days now (as part of a project I'm doing over the summer) and have gotten stuck. I think I need to make some substitution but I can't see what it is to make. -\int\frac{dI}{I(R+BI+CI^2)} I...
  27. M

    How Do You Factor the Numerator of a Rational Function to Find Intercepts?

    Homework Statement Y1=(x3+3x2-4)/(x2)Homework Equations Slant Asymptote at y=x+3. (X-1) and (X+2) appear to be intercepts in the back of the book. How do I factor the numerator to get that?The Attempt at a Solution I know this is simple and there is a method to find the zeros of the numerator...
  28. N

    Rational Functions: Degree of Denominator vs Nominator

    Hi all. I have always wondered: If we e.g. look at functions given by f(x) = \frac{\cos x}{x^2}, \quad g(x) = \frac{\sin x}{x^2}, \quad h(x) = \frac{\exp x}{x^2}, then does the degree of the denominator exceed the degree of the nominator by 1 or by 2?
  29. N

    Complex Analysis: Integrating rational functions

    Homework Statement Hi all. My question has to do with integrating rational functions over the unit circle. My example is taken from here (page 2-3): http://www.maths.mq.edu.au/%7Ewchen/lnicafolder/ica11.pdf We wish to integrate the following \int_0^{2\pi } {\frac{{d\theta }}{{a + \cos...
  30. D

    Finding possible and actual rational roots

    Homework Statement a) 2x^4 - 15x^3 + 23x^2 + 15x - 25 = 0 b) 12x^3 - 20x^2 + 23x - 10 = 0 Homework Equations The Attempt at a Solution I was wondering if you guys could check my answers. For A: Possibles: +/- {1, 5, 25} / {1, 2} Actual: 5, 5/2
  31. Z

    I was wondering what Q/Z, or the rational numbers modulo the integers.

    I was wondering what Q/Z, or the rational numbers modulo the integers. I am struggling to visualize what the cosets may be. Thank you for your time.
  32. J

    What are the x and y intercepts y= x+2/x-7 rational

    what are the x and y intercepts y= x+2/x-7 rational expression is y=-2/7 and x= -2
  33. D

    Annoyingly simple problem - rational functions and limits at infinity

    Hi all This is my first post so please be gentle with me! Limit of this rational function as x approaches infinity? f(x) = (x^3 - 2x)/(2x^2 - 10) I was under the impression that if the degree of the polynomial of the numerator exceed that of the denominator then there could be no...
  34. C

    Linear approximation and rational numbers

    Homework Statement Use linear approximation to approximate sqrt((3.2)^2 + 2(2.1) + (4.3)^2) with a rational number (a ratio of integers). Homework Equations f(x,y) = sqrt(x^2 + 2y + z^2) f(x,y) = (x^2 + 2y + z^2)^1/2 The Attempt at a Solution x = 3 ∆x = 2/10 y = 2 ∆y =...
  35. K

    Simplifying a rational expression

    I've done a calculus problem. The LHS is my answer and the RHS is the answer in the book. \frac{2(x-1)^2}{(x^2-1)^2} = \frac{2}{(x+1)^2} What was done to simplify this equation?
  36. T

    Constructing a Rational Function with Given Asymptotes and Intercepts

    Homework Statement Find an equation of a rational function f that satisfies the given conditions: vertical asymptotes: x=-3, x=1 horizontal asymptote: y=0 x-intercept: -1; f(0)= -2 hole at x=2Homework Equations The Attempt at a Solution Vertical Asymptotes are (x+3) and (x-1). Also (x-2)...
  37. A

    How Do You Compute the Matrix Q for Rational Canonical Form?

    Hi, can someone show me how one would go about finding the matrix Q in Q^(-1) A Q = RationalCanonicalForm(A). Please demonstrate using the example {4, 1, 2, 0} {-4, 0, 1, 5} = A {0, 0, 1, -1} {0, 0, 1, 3} where the characteristic polynomial is (x-2)^4 and the minimal polynomial is...
  38. D

    Is x Necessarily Rational If It Satisfies (ax+b)/(cx+d)=1?

    Suppose a,b,c,d are integers and a DOES NOT equal c. Suppose that x is a real number that satisfies the equation: (ax+b)/(cx+d)=1 Must x be rational? If so, express x as a ratio of two integers. I have no idea how to begin this problem.
  39. W

    Solving Rational Inequalities: (3x+1)/(2x-4) > 0

    Homework Statement 3x+1 2x-4 > 0 The Attempt at a Solution My answer came to be (x< 1/3) U (x>2)
  40. W

    Solving Rational Inequalities: A Simplified Approach

    I wanted help on solving Solving Rational Inequalities. I watched this video and i was wondering if that was the best or easiest way. Our teacher taught us a much harder and weirder way using multiple number lines instead of one. If you do watch the video the man shows examples with open dots...
  41. C

    I with a Complex Rational Expression

    Homework Statement Homework Equations Idk what to do after I multiply the LCD I get confused please help me. The Attempt at a Solution All I know is that the LCD is (x+5)(x+1)(x-2) and when I multiply into every fraction I get this. (X+5)(X-2)(X-2) - (X+5)(X+1)(X+1)...
  42. S

    Multiplying/Dividing Rational Expressions(Playing with AlgebraII Fractions)

    Homework Statement [(4x-4x2)/x2+2x-3] * [(x2+x-6)/4x] Homework Equations The Attempt at a Solution I'm so lazy, though, I do all this stuff mentally, so... Oby-kaby. I individualized all groups into smaller units(I don't know how to say that in algebrish), by dividing by least...
  43. S

    Rational numbers - periodic decimal expansion

    Homework Statement Let n/m be a positive rational number in lowest terms. By examining the long division algorithm, show that the decimal expansion of n/m is eventually periodic, and that the period divides phi(m). For simplicity,you may assume that (m, 10) = 1. Homework Equations...
  44. P

    Rational Functions: Analysis & Agreement

    Hi, I was wondering whether two rational functions f,g whch coincide on the unit circle actually coincide on all of C. I would say yes. Let D be the set of all complex numbers with the poles of both f and g removed (let's assume there are no poles on the unit circle). This is then open...
  45. F

    Problem composing a rational function with itself.

    I'm trying to compose f(x)= x + \frac{1}{x} with itself. e.g. f \circle f I have x + \frac{1}{x} + \frac{1}{x+(1/x)} Now I multiplied \frac{1}{x + (1/x)} * \frac{x}{x} and I got: x + \frac{1}{x} + \frac{x}{x^2+1} This is not the correct answer according to the book. Books is...
  46. B

    Converting rational number to a new base.

    Hello, I found one excercise - convert a rational number 63/64(base - 10) to a number system with a base of 4 using Radix conversion. Searching throught the internet i found this formula (i hope it`s the correct one :) ) - http://img255.imageshack.us/img255/903/races3.jpg Unlike...
  47. P

    *Proof* Sum of Rational and Irrational Numbers

    Homework Statement Prove by contradiction: If a and b are rational numbers and b != 0, and r is an irrational number, then a+br is irrational. In addition, I am to use only properties of integers, the definitions of rational and irrational numbers, and algebra. You guys should also know that...
  48. D

    A rational function which I forget how to integrate

    Hey all! It's been a while since I've done this, how do you integrate a rational function, where the denominator cannot be factored, again? For example, \int \frac{x}{x^{4}-1} dx Thanks, in advance!
  49. U

    Trouble evaluating the integral for a rational function

    Homework Statement i've tried many partial fraction methods but none of y answers are correct in the end, please help me evaluate the integral for f(x)= (10x+2)/(x-5)(x^2 + 1) Homework Equations there are no relevant equations given The Attempt at a Solution A/x-5 + Bx+C/x^2 +1
  50. O

    Integrating Rational Functions: Is It Possible?

    Evaluate the integral: int ((x+1)/(x-1)) dx is this possible? i tried searching for this but there was no answer
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