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. L

    How to Simplify a Rational Function

    Homework Statement u^2 --------------- u^2 - 4 Homework Equations I am told that this is equal to 1+ 4/u^2 - 4The Attempt at a Solution No clue how these two are related. Factor out a u^2/u^2? But that alters the denominator. My next that is partial fraction decomp. Am...
  2. C

    Indefinite integral with a rational function

    EDIT: Problem found. This thread can now be ignored.Homework Statement Find the indefinite integral. Homework Equations ((y^2-1)/y)^2 dy The Attempt at a Solution I've attempted a few things. I first attempted to split the statement inside the outer parentheses into two fractions; (y^2/y...
  3. C

    When the expression is a square of a rational number.

    Homework Statement I am trying to find when the square root of the expression 25+8a^2 is rational, where the number a also needs to be rational. \sqrt{25+8a^2}=b, where a and b are both rational numbers. I am trying to get an expression for a in terms of some other number m, which would always...
  4. L

    Points of Inflection on a rational function

    Homework Statement Find the inflection points and use second derivative test to determine where the function is concave up or down Homework Equations f(x) = (x - 1)/(x2 - 4) The Attempt at a Solution f'(x) = (-x2 + 2x - 4) / (x2 - 4)2 f''(x) = 2x3 - 6x2 +24x -8 / (x2-4)4...
  5. M

    Conjecture Regarding rotation of a set by a sequence of rational angles.

    Conjecture Regarding Rotation of a Set by a Sequence of Angles. Consider the following sequence, where the elements are rational numbers mulriplied by \pi: (\alpha_{i}) = \hspace{2 mm}\pi/4,\hspace{2 mm} 3\pi/8,\hspace{2 mm} \pi/4,\hspace{2 mm} 3\pi/16,\hspace{2 mm} \pi/4,\hspace{2 mm}...
  6. M

    Integrating Rational Functions

    1. ∫\frac{dx}{x3 + 2x} We're suppose to evaluate the integral. Use Partial Fraction Decomposition: \frac{1}{x3 + 2x} = \frac{A}{x} + \frac{Bx + C}{x2 + 2} 1 = A(x2 + 2) + (Bx + C)(x) 1 = Ax2 + 2A + Bx2 + Cx 1 = x2( A + B) + Cx + 2A Solving for A gives \frac{1}{2} Solving for B...
  7. M

    Find Best Books for Rational Mechanics Course

    Hello dears, My faculty is Cvil Engineering. We had a subject "Rational Mechanics". The professor didn't gave textbook. I have my written notes during lectures only. I have many doubts and misunderstandings, so I need good book. Can you help me with this? Actually, I have found some books...
  8. C

    Find the Derivative/Simplification of 2 Rational Expressions

    Homework Statement Find the derivative of: y = [(2x - 5)^4][(8x^2 - 5)^-3] Homework Equations I get: y' = -[(48x(2x - 5)^4) / (8x^2 - 5)^4] + [(8(2x - 5)^3) / (8x^2 - 5)^3] Wolfram gets: http://www.wolframalpha.com/input/?i=derivative+%282x+-+5%29%5E4%288x%5E2+-+5%29%5E-3 The...
  9. N

    Finding the limit of a rational function

    Homework Statement limx→2(x+2)/(x^3+8) 2. I only recently started learning calculus on my own so correct me if I'm wrong. When using direct substitution and the denominator equals 0, the limit is undefined, just like any fraction is when its denominator equals 0. However, it's limit can still...
  10. T

    Rational functions w/ common factors don't simplify?

    Say we have a rational function P(x)=(x^2-3x-4)/(x-4)=[(x+1)(x-4)]/(x-4) I'm a little confused as to why the (x-4) doesn't cancel out. It graphs the same as y=x+1 for x≠4. I feel like I'm missing something from the order of operations.
  11. B

    F-automorphism group of the field of rational functions

    I've been doing some exercises in introductory Galois theory (self-study hence PF is the only avaliable validator :) ) and a side-result of some of them is surprising to me, hence I would like you to set me straight on this one if I'm wrong. Homework Statement Let K(x) be the field of rational...
  12. D

    Solving a Rational exponent(simplify)

    Homework Statement Simplify. Write as positive exponent c(c^5/6)/c^2 Homework Equations None really, just exponent laws. The Attempt at a Solution c(c^5/6) comes out as... c^5/6 Dividing exponents requires subtracting them so... 5/6 - 2/1 Find common denominator... 5/6 -...
  13. M

    Can You Find a Number n with Rational Square Roots for n-7, n, and n+7?

    Hi, I am struggling with this puzzle from a book. Puzzle : Can you find a number n such that, the numbers n-7, n, and n+7 have rational square roots (can be expressed as integers or fractions)? According to the book one of the solutions is n =113569 /14400 This is what I have done so...
  14. Y

    Simplification of a rational polynomial function

    Now if I have a function y=(ab+ac)/a, it can be further factorised, y=(a(b+c))/a. Now if we cancel off the a, we will have only y=b+c that will also give the same y-values as the original form of the function y with respect to the same x-value. This statement implies that cancellation or...
  15. C

    Rational Bezier Curve to Polynomial Bezier Curve Conversion?

    Is there a way to convert a rational bezier curve to a piecewise series of one or more polynomial bezier curves with minimal loss in accuracy, specially cubic ones? I've already tried searching the internet for pre-existing algorithms, but I haven't been able to find any usable results despite...
  16. J

    MHB SE Class 10 Maths - Rational or Irrational Numbers: $\cos(1^0)$ and $\tan(1^0)$

    $\cos(1^0)$ and $\tan(1^0)$ are Rational or Irrational no. Where angle are in Degree help required
  17. F

    MHB Contour integration with rational function and cosh

    Hi All: I am new to the site, so I thought this would be a good time to post an interesting integral I ran across that I am having a time with. It is a miscellaneous problem in Schaum's Outline of Complex Variables, #86 in ch. 7. I have been self-teaching a little CA when I get time and this...
  18. P

    Simplifying the Limit of an Ugly Rational Function: A Student's Guide

    I have been an ugly rational function. I graphed it and put end behaviors and intercepts pods etc... Now after my teacher asked me to take the limit of this function and she has informed me that it is a much more simple way than taking the derivative of the entire peice...
  19. S

    Number systems: construction of rational numbers

    Homework Statement let a and b be rational numbers if a>b>0 show that there exists a natural number n, which can be considered rational as well, st nb>a Homework Equations The Attempt at a Solution I was trying using the peano theorem of natural numbers define set of...
  20. P

    Rational numbers that form a group under addition

    Rational numbers form a group under addition. However, a sequence of rational numbers converges to irrational number. Presumably, group theory does not allow adding an infinite number of rational numbers. This is not indicated in the textbook definition of a group. I might be looking in vain...
  21. S

    MHB Find Horizontal Asymptote of Rational Function F(x)

    F(x)= 6x^2-17x-3/3x+2, find horizontal asymptote
  22. E

    Proving Rational Roots and Irrationality of \sqrt{2}

    Homework Statement Prove that a rational root of a monic polynomial is an integer. Use this to prove that the \sqrt{2} is irrational. Homework Equations The Attempt at a Solution /// I am really not sure where to begin?
  23. H

    Prove or disprove that there is a rational bijective function f : R to (0; 1)

    Homework Statement Prove or disprove that there is a rational bijective function f : R to (0; 1) Homework Equations i found a bijective map from (0,1) to R (y=(2x-1)/(2x^2-2x) The Attempt at a Solution Im just stuck and i was thinking since it has to be a rational function...
  24. M

    The set of squares of rational numbers is inductive

    Homework Statement The set of squares of rational numbers is inductive Homework Equations definition of an inductive set The Attempt at a Solution sorry i know this is probably very easy to most but I am just learning analysis. Okay, so we can see that 1 is in the set because it is...
  25. N

    MHB Rational roots, functions, modeling.

    OKAY I have a trg test make up and due tomroow and I have no clue what I am doing, I've searched online chatrooms and they all want money. so this is basiaclly my last hope the test is over rATIONAL ROOTS ZEROS POLYNOMIALS ETC. please show all work if answering QUESTIONS i NEED HELP ON BELOW...
  26. P

    Proving Modulus of Rational Expression is Equal to 1

    Homework Statement Prove |\frac{e^{2i\theta} -2e^{i\theta} - 1}{e^{2i\theta} + 2e^{i\theta} -1}| = 1 Homework Equations The Attempt at a Solution I feel like this should be fairly simple, anyone have any hints? Also this is just one step in an attempt to solve a much larger problem, so don't...
  27. I

    Putting a matrix into Rational Canonical Form

    I'm trying to put a matrix into RCF, and I keep running into problems. I've checked my work a few times, so I think I must be making a conceptual error. Here's what I've got: $$A=\left( \begin{matrix}2 & -2 & 14 \\ 0 & 3 & -7 \\ 0 & 0 & 2\end{matrix}\right)\quad \text{ so }\quad xI-A=\left(...
  28. Z

    Is Absolute Convergence Required for Evaluating Sums over Rational Numbers?

    it is possible to evaluate sums over the set of Rational so \sum_{q} f(q) with q= \frac{m}{n} and m and n are POSITIVE integers different from 0 ?? in any case for a suitable function is possible to evaluate \sum_{q} f(qx) with f(0)=0 ??
  29. O

    Finding the rational expression of a repeating decimal

    Homework Statement Express the repeating decimal as a series, and find the rational number that it represents 1) 3.2\overline{394} Homework Equations Geometric Series (a/1-r) The Attempt at a Solution I tried putting the value into a series by having n=1 as it goes to infinity Ʃ3.2...
  30. V

    Algebra II Quotients of Rational Expressions

    Homework Statement Simplify. (p4 - q4)/(p + q)2 ÷ 1/(p2 + q2) Answer: (p - q)/(p + q) Homework Equations -- The Attempt at a Solution Transformed it to a multiplication problem. (p4 - q4)/(p + q)2 X (p2 + q2)/1 Difference of the squares in the numerator of the first expression...
  31. C

    Powers w/Rational Exponents: Evaluate (Review My Work)

    Homework Statement Write as a single power, then evaluate: a) (-32)^3/5 x (-32)^-4/5 / (-32)^2/5 b) 4096^3/6 / 4096^2/3 x 4096^5/6 Homework Equations The Attempt at a Solution a) (-32)^3/5 x (-32)^-4/5 / (-32)^2/5 = (-32)^3/5+(-4/5)-2/5 = -32^-3/5 = -1/8 <- not sure...
  32. J

    Integration of Rational Functions by Partial Fractions

    Homework Statement ∫(x3+4)/(x2+4)dx Homework Equations n/a The Attempt at a Solution I know I have to do long division before I can break this one up into partial fractions. So I x3+4 by x2+4 and got x with a remainder of -4x+4 to be written as x+(4-4x/x2+4). Then I rewrote...
  33. L

    Order of error for rational approximation of irrationals

    Hi, I have to approximate an irrational number x by rationals r = p/q. Let ε>0 in ℝ, then, for almost all x exist α and r in (x-ε,x+ε) such that q ≈ c(x) ε^-α, c(x) in ℝ? I know, from Hurwitz theorem (and a conseguence) that α>2, if exists.
  34. V

    Algebra II Simplifying Rational Algebraic Expressions

    Hi everyone and nice to meet you. I'm velox_xox a newbie to PF and a high school correspondence student currently taking Algebra II. Since I'm correspondence, I am basically teaching myself my subjects, which means if I don't understand something it's a big problem. I enjoy math, but I must...
  35. A

    Sequence in Q with p-diatic metric. Show it converges to a rational

    This is the problem I'm trying to slove: Consider the sequence s_n = Sumation (from k=0 to n) p^k (i.e. s_n=p^0+p^1+p^2...+p^n) in Q(rationals) with the p-adic metric (p is prime). Show that s_n converges to a rational number.[/B] Now, I do get some intuition on showing that the...
  36. Biosyn

    Integration of Rational Functions

    Homework Statement ∫(7x^2+22x-54)/((x-2)(x+4)(x-1)) dx Homework Equations Partial functions The Attempt at a Solution ∫(7x^2+22x-54)/((x-2)(x+4)(x-1)) dx = ∫(Ax)/(x2+2x-8) + B/(x-1) = ∫A(x-1)dx + ∫B(x^2+2x-8)dx Or is it...
  37. M

    Integrating rational functions

    Question: When determining the coefficients of the partial fractions for say 5 or more coefficients... Do you find it easiest to set up linear equations and solving? Any advice would be appreciated... Next question.. look in paint doc... why would I3 not be equal to I21??
  38. T

    Every rational number can be written in one way

    Homework Statement Prove that every positive rational number x can be written in ONE way in form x=a1+ a2/2! + a3/3! + ... + ak/k! where a1,a2,...,ak are integers and 0<=a1, 0<=a2<2,... ,0<=ak<k I wrote my solution below. Please check if it is correct and rewrite it for me in a neater way...
  39. F

    Integration of Rational Functions

    Homework Statement Evaluate the integral. (Remember to use ln |u| where appropriate.) ∫ds/s^2(s − 1)^2 Homework Equations The Attempt at a Solution I attempted a solution using the method of partial fractions, but it seems my answer is wrong. Here's what I did... 1=A/s...
  40. F

    Integration of Rational Functions by Partial Fractions

    Homework Statement Evaluate the integral. (Remember to use ln |u| where appropriate.) ∫(x^3 + 36)/(x^2 + 36) Homework Equations The Attempt at a Solution A little bit confused about arriving at the solution for this problem. I get stuck a little ways in. Any help would be...
  41. A

    How to proof that a curve has no rational points

    Hello, I'm trying to do exercise number 20 from chapter 6 of this http://www.people.vcu.edu/~rhammack/BookOfProof/index.html, it asks to show that the curve x2 + y2 - 3 = 0 has no rational points. In the answer it has this tip: first show that a2 + b2 = 3c has no solutions, other than the...
  42. C

    Proving Rational Numbers and Irrational Numbers

    Homework Statement Show that if a\in\mathbb{Q} and t\in\mathbb{I} then a+t\in\mathbb{I} and at\in\mathbb{I} as long as a≠0 The Attempt at a Solution Let a=\frac{x}{y} where x and y are integers. and t is an irrational number If I have a+t . since t cannot be written as...
  43. A

    Sequence that has all rational numbers

    Homework Statement Construct a sequence that has all rational numbers in it Homework Equations None. The Attempt at a Solution Here are my thoughts, though I have no solutions yet. If I construct a sequence Sn= n*sin(n)-1/n, will it work? Thanks guys!
  44. Y

    Calculate the volume with a rational funtion

    Homework Statement Calculate the volume (V) generated by the given function: y^{2}=\frac{x^3}{2a-x} about the line x=2a I suppose you have to use cylindrical shells [because it's difficult to fin y], so: V=2\pi(2a-x)\sqrt{\frac{x^3}{2a-x}}Δx now...
  45. C

    L'Hopital Limit with rational exponent

    Homework Statement I need to find the limit of this using L'Hopital Rule, if it exists. lim_{x\rightarrow0^{+}}(\frac{sinx}{x})^{1/x^{2}} Homework Equations All of our examples used lim f(x)/g(x)=f'(x)/g'(x) The Attempt at a Solution We have not dealt with any limits like this that have...
  46. E

    B^2 = a with a is an integer and b rational => b is an integer

    Homework Statement b^2 = a b is a rational number a is an integer prove that b is an integer. This is self assigned, but I think this is the appropriate place to put my question. Homework Equations see above The Attempt at a Solution Is this legitimate...? Since b is a...
  47. C

    Rational numbers - bounded subset with no least upper bound

    Homework Statement Give an example of a bounded subset of Q which has no least upper bound in Q. Explain why your answer has this property. Homework Equations The Attempt at a Solution [1/8, 1/4, 3/8, 1/2, 5/8, 3/4...infinity] is this correct?
  48. D

    Integrating Log and Rational Functions

    Homework Statement \int_0^{\infty} \ln \left( \frac{e^x+1}{e^x-1} \right) \mbox{d}x \int_0^{\infty} \frac{1}{x^n+1}\ \mbox{d}x\ \forall n >1 Homework Equations - The Attempt at a Solution I've tried IBP and separating the ln into two terms and failed. I've also tried a...
  49. C

    Why Does the Rational Zeros Theorem Not Predict All Zeros?

    When solving 3x^3+4x^2-7x+2 the rational zeros theorem says there can only be a possibility of zeros at plus minus 1, 2, 1/3, and 2/3 but for some reason a zero appears at about -2.4 (or more accurately -1-squareroot of 2). How can this be? It states that if there are any zeros they will...
  50. J

    Is 5^x a Valid Term in a Polynomial?

    y=(5x^4 + 5^x -1)/(2x^3 +1) More specifically is 5^x a valid term in a polynomial?
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