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. karush

    MHB 242t.08.02.09 int of rational expression.

    $\tiny{242t.08.02.09}$ $\textsf{ Evaluate to nearest thousandth}\\$ \begin{align*} \displaystyle I_{8.1.31} &=\int_5^{10} \frac{3x^5}{x^3-5} \, dx =885.576\\ \end{align*} $\textit{ok tried getting to this answer by: }$ $u=x^3-5 \therefore du=3x^2 \, dx$ $\textit{but it didnt seem to go to good}$
  2. L

    MHB Rational function equation problem solving

    Hi I've been tying to figure this out for days. The answer has to be in the form of a rational equation. John wants to start a padlock company. The production cost of one padlock is 12$. John plans to spend 15 000$ on advertising. He decides to sell each padlock for 20$. 1.Which equation gives...
  3. S

    When are Negative Bases Raised to Rational Powers Undefined?

    Homework Statement I'm trying to understand negative bases raised to rational powers, when calculating principle roots for real numbers. I'm not worried about complex solutions numbers at this stage. I just can't find a concise explanation I can understand anywhere. I'm self learning as an...
  4. karush

    MHB 206.07.05.88 Int rational expression

    $\tiny{206.07.05.88}$ \begin{align*} \displaystyle I_{88}&=\int\frac{1}{(x+2)\sqrt{x^2+4x+3}} \, dx \\ &=\int\frac{1}{(x+2)\sqrt{x^2+4x+3+1-1}} \, dx \\ &=\int\frac{1}{(x+2)\sqrt{(x+2)^2-1}} \, dx \\ u&=(x+2) \therefore du=dx \\ I_{88}&=\int\frac{1}{u\sqrt{u^2+1}} du \end{align*} $\textit{so...
  5. Schaus

    Graphing Rational Functions: How to Find Asymptotes and Intercepts

    Homework Statement Sketch the graphs of the following functions and show all asymptotes with a dotted line y = (2x - 6)/ (x2-5x+4) i) Equation of any vertical asymptote(s) ii) State any restrictions or non-permissible value(s) iii) Determine coordinates of any intercept(s) iv) Describe the...
  6. J

    MHB Indefinite integration involving exponential and rational function

    Calculation of $\displaystyle \int e^x \cdot \frac{x^3-x+2}{(x^2+1)^2}dx$
  7. karush

    MHB Integral of a rational function

    $\textsf{evaluate}$ \begin{align} \displaystyle {I}&={\int{\frac{x+2}{x^2+1}dx}}\\ &=\int{\frac{x}{x^{2}{+1}}dx{\ +\ 2}\int{\frac{1}{x^{2}{+1}}}}{\ }dx\\ u&=x^{2}+1 \therefore \frac{1}{2x}du=dx\\ x&=\sqrt{u-1}\\ \end{align} ...? $\textit{calculator answer.?}$ $\dfrac{\ln\left(x^2+1\right)}{2}...
  8. Rectifier

    Antiderivative of a rational function

    I am trying to find primitives to the rational function below but my answer differs from the answer in the book only slightly and now, I am asking for your help to find the error in my solution. This solution is long since I try to include all the steps in the process. The problem $$ \int...
  9. C

    Proving discontinuity for rational numbers (reduced form)

    Hello! This is my first post on these forums. So I was stuck with this question in my Mathematical Analysis exam, and it is as follows: ƒ(x) = 0 if x ∉ ℚ and (p + π) / (q + π) - (p / q) if x = (p / q) ∈ ℚ (reduced form). 1- Prove ƒ is discontinuous at all rational numbers except 1: This is...
  10. Unteroffizier

    Algebra II, Rational Expressions & Square Roots problems

    Problem 1 Simplify/solve: 2*81/2-7*181/2+5*721/2-50 Attempt at solution: a1/2=√a ⇒ 2*√8 - 7*√18 + 5*√72 - 50 = 2√8 - 7√18 + 5√72 - 50 = ? Do not know how to proceed beyond this point. Have experimented with little luck. Problem 2 Simplify/solve: a-1(1+1/a2)-1/2 * (1+a2)1/2 Attempt at...
  11. V

    Finding Equation of Rational Function

    Homework Statement Must find equation meeting these requirements: 1. Hole at x=5 2. Vertical Asymptote at x= 0 3. Horiztonal Asymptote at g(x) = 2 4. Y-int at (-1.5, 0) Homework EquationsThe Attempt at a Solution I know that there must be an (x-5) in both the numerator and denominator due to...
  12. V

    Finding the Time: Solving a Rational Equation for Painting a Wall

    Homework Statement Person 1 can paint 3m2 in x hours, person #2 can paint 7m2 in x + 2 hours however together they can paint 22m2 of the wall in 10 hours. How many hours did person #2 take to paint the wall? Homework EquationsThe Attempt at a Solution Im confused at how to start the question...
  13. L

    Doublecheck reduction of rational expressions

    Homework Statement teacher gave me the correct solution which said that the result from him was as follows ##\frac{1}{a^2b^6}## Homework Equations original problem was as follows ##\frac{ (\frac{1}{a})^2*b^{-3}}{ ab^3 }## The Attempt at a Solution...
  14. Hiero

    Rational roots of 4th degree polynomial with odd coefficents

    Homework Statement A polynomial, P(x), is fourth degree and has all odd-integer coefficients. What is the maximum possible number of rational solutions to P(x)=0? Homework Equations P(x) = k(x-r1)(x-r2)(x-r3)(x-r4) P(x) = 0 when x = {r1, r2, r3, r4} The Attempt at a Solution I expanded the...
  15. M

    I A question about the log of a rational function

    We have the rational function : $$f(x)=\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}\left(\frac{1-ix}{1+ix}\right)^{n/2}\;\;\;,\;\;n\in \mathbb{Z}^{+}$$ It's not hard to prove that ...
  16. karush

    MHB S6.7.r.19 Rational Expression Integral (complete the square?)

    $\Large{S6.7.R.19}$ $$\displaystyle I=\int\frac{x+1}{9{x}^{2}+6x+5}\, dx =\frac{1}{18}\ln\left({9{x}^{2}+6x+5}\right) +\frac{1}{9}\arctan\left[{\frac{1}{2}\left(3x+1\right)}\right]+C $$ $\text{from the given I thought completing the square would be the way to solve this} \\$ $\text{but I...
  17. karush

    MHB LCC 206 {r3} integral rational expression

    $\tiny\text{LCC 206 {r3} integrarl rational expression}$ $$\displaystyle I=\int \frac{3x}{\sqrt{x+4}}\,dx =2\left(x-8\right)\sqrt{x+4}+C \\ \text{u substitution} \\ u=x+4 \ \ \ du=dx \ \ \ x=u-4 \\ \text{then} \\ I=3\int \frac{u-4}{\sqrt{u}} \ du = 3\int {u}^{1/2}du -12\int...
  18. karush

    MHB Steward e6 7r33 rational integral

    $$\displaystyle \int\frac{{x}^{2}}{\left(4-{x}^{2}\right)^{3/2}} \ dx$$ $u=4-{x}^{2} \ \ \ du=-2x dx \ \ \ x=\sqrt{4-u}$ $$\displaystyle \int\frac{4-u}{ \left(u\right)^{3/2}} -2 \sqrt{4-u}\ du$$ Stuck
  19. karush

    MHB -w8.7.28 integral rational expression

    nmh{1000} $\tiny{\text {Whitman 8.7.28 integral rational expression}} $ $$\displaystyle \int\frac{t+1}{{t}^{2}+t-1}\ dt$$ $\text{book answer}$ $$\displaystyle\frac{5+\sqrt{5}}{10} \ln\left({2t+1-\sqrt{5}}\right) +\frac{5-\sqrt{5}}{10} \ln\left({2t+1+\sqrt{5}}\right)+C$$ $\text{expansion}$...
  20. Z

    Problem with Rational Rose Installation

    Hi, I downloaded the file: 7.0.0.4-RATL-RRENT-WIN-all-FP04. Its size is: dir 7* Volume in drive D has no label. Volume Serial Number is B269-AF38 Directory of D:\download 06/04/2016 10:40 PM <DIR> 7.0.0.4-RATL-RRENT-WIN-all-FP04 06/04/2016 09:14 PM 131,195,766...
  21. kaliprasad

    MHB Rational Root of $ax^3+bx+c=0$ is Product of 2 Rational Roots

    if for rational a,b,c $ax^3+bx+c=0$ one root is product of 2 roots then that root is rational
  22. Simon T

    MHB Solving An Equation With Rational Terms

    Hey guys, I am not to good in Math and I am having issues solving this equation. 2x-9/3x + 8 = 3/7x How do you solve this equation? The answer is supposed to be x = 0.3956?
  23. C

    MHB Rational Equation: 9/se^2-4 = 4-5s/s-2

    9/se^{2}-4 = 4-5s/s-2
  24. Noisy Rhysling

    A rational design for a personnel airlock

    In The Martian we see airlocks that an SUV would fit into. Wastes air every time it's opened, even if pumped down. I suggest we make an airlock that clamshells down on a space suit and leaves the minimum amount of room between suit and lock for air to be reclaimed from. You could have banks of...
  25. I

    Rational numbers, supremum (Is my proof correct?)

    Homework Statement Show that the set ##\{x \in \mathbf Q; x^2< 2 \}## has no least upper bound in ##\mathbf Q##; using that if ##r## were one then ##r^2=2##. Do this assuming that the real field haven't been constructed. Homework Equations N/A The Attempt at a Solution Attempt at proof: ##r\in...
  26. P

    B Heard of Another Kind of "Rational" (Fraction) Addition?

    I ran across this years ago. It’s defined by the addition rule: a/b + c/d = (a+c)/(b+d) (Sounds crazy – until you think of x/y as meaning There’s a pile of x things on one side, and one of y things on the other.) Applying it in almost a Pascal's Triangle sort of way generates ... er, a...
  27. woof123

    MHB Rational function transformation

    the question is: Rewrite the rational equation y=(-5x-18)/(x+4) to show how it is a transformation of y=1/x. describe transformations looks like it is shifted 4 to left, then stretched by factor of -5x-18. Is that accurate? would you elaborate beyond that?
  28. Captain1024

    Help Solving Equation Involving A Rational Exponential

    Homework Statement Solve ##y=\mathrm{exp}(\frac{-x\pi}{\sqrt{1-x^2}})## for x when y = 0.1 Homework Equations ##\mathrm{ln}(e^x)=x## The Attempt at a Solution ##\mathrm{ln}(0.1)=\frac{-x\pi}{\sqrt{1-x^2}}## ##(\frac{-\mathrm{ln}(0.1)}{\pi})^2=\frac{x^2}{1-x^2}##
  29. I

    MHB Proof that ax^2 + bx + c has No Rational Zeroes if a,b, and c are Odd

    Well, let's look at how this works. Quadratic equations can have either 1, 2, or no zeroes. If it has no real zeroes, the zeroes it DOES have are complex, so that's obviously not it. Let's imagine ax^2 + bx + c = 0 has one zero, call it \alpha (Cuz it looks pretty). Then that means ax^2 +...
  30. I

    Formula using rational numbers

    I want to convert a recursive real formula to rational number representation, but I get the wrong response. For the real formula: k = 1.9903694533443939 u1 = -12.485780609032208 u2 = -6.273096981091879 u3 = k * u2 -u1 /// 1st iteration u3 = -1.7763568394002505E-15 // approx. zero /// 2nd...
  31. I

    Laurent series of rational function in annulus

    Homework Statement Find the Laurent expansions of ##f(z) = \frac{z+2}{z^2-z-2}## in ##1 < |z|<2## and then in ##2 < |z|< \infty## in powers of ##z## and ##1/z##. Homework Equations Theorem: Let ##f## be a rational function all of whose poles ##z_1,\dots , z_N## in the plane have order one and...
  32. RJLiberator

    Polynomial Existence and Irreducibility over Rational #'s

    Homework Statement Prove that for all n ≥ 1, there exists a polynomial f(x) ∈ ℚ[x] with deg(f(x)) = n such that f(x) is irreducible in ℚ[x]. Homework Equations In mathematics, a rational number is any number that can be expressed as the quotient or fractionp/q of two integers, a numeratorp...
  33. T

    Proof that log2(i) is rational but I think it is wrong

    m and n are integers. log2(i) = m/n 2^(m/n) = i 2^m = i^n 2^0 = i^4 = 1 so that means that log2(i) is rational because there are integers n and m so that log2(i) = m/n , they are m=0 and n=4. But what I do get about this proof is that it seems to imply that log2(i) = 0/4 = 0 while google says...
  34. S

    Has it been proven that all rational numbers repeat ....

    in every radixial representation, except of course for those cases in which the numerator is a factor of some natural-number power of the radix? For the radixial system we know (i.e., because we are bilateral and have arms that have 5 fingers), this would mean that any possible natural number...
  35. B

    The Rational Plane and Dedekind's Axiom

    Homework Statement Here the formulation of Dedekind's axioms that I am using:Suppose that line ℓ is partitioned by the two nonempty sets ##M_0## and ##M_1## (i.e., ##\ell = M_0 \cup M_1##) such that every point between two points of ##M_i## is is also in ##M_i##, for ##i = 0,1##. Then there...
  36. Ravendark

    Integral with sine, cosine, and rational function

    Homework Statement I would like to compute the following integral: I = \int\limits_0^\pi \mathrm{d}\theta \, \frac{\sin^2 \theta}{a^2 + b^2 - 2 \sqrt{ab} \cos \theta} where ##a,b \in \mathbb{R}_+##. 2. The attempt at a solution Substitution ##x = \cos \theta## yields I = \int\limits_{-1}^1...
  37. SlowThinker

    Why we never see a rational charge?

    There is a question that has been puzzling me for quite a few years by now, since the moment I've heard that quarks have charge ##-e/3##, ##2e/3## etc. There seem to be 2 independent answers to the title: 1. There is a Charge Censorship Principle, where you can never observe a charge ##e/3## and...
  38. S

    Is Refusing All Medical Care A Rational Choice For End-Of-Life Planning?

    I am not suicidal. But I am in my 70s and I am trying to make plans and policies appropriate for my age, dealing with end of life. I'm doing this while I'm still healthy. I wear a Medic Alert ID Tag that says "Refuse all Medical Care" PF is visited by civil intelligent people so I would...
  39. D

    I with multiplying rational exponents

    How can the properties of rational exponents be applied to simplify expressions with radicals and rational exponents?
  40. Vinay080

    Who invented differential calculus for rational functions?

    Euler mentions in his preface of the book "Foundations of Differential Calculus" (Translated version of Blanton): I don't understand here, who/who all had invented/discovered the study-of-ultimate ratio (differential calculus) for rational functions long before (Newton and Leibniz), without...
  41. S

    MHB Rational Expressions: Subtract 1/2x-7/x-3

    1/2x-7/x-3 Subtract
  42. G

    Rational exponents (was: Math Discussion)

    Homework Statement (-64)^(3/2) Homework Equations None. The Attempt at a Solution There is no answer that can be reached and it is supposed not be a real number. I was wondering why that is. How is it that there is no "real" answer to this problem?
  43. karush

    MHB What is the Limit of a Rational Expression?

    $$\lim_{{x}\to{\infty}}\frac{\sqrt{3x^2 - 1 }}{x-1}=\sqrt{3}$$ I tried dividing everything by ${x}^{2}$ but not
  44. Mr Davis 97

    Finding the limit of a rational expression

    To find the horizontal asymptotes of a rational function, we find the limit as x goes to infinity. Given the rational function ##\displaystyle\frac{x + 1}{\sqrt{x^2+1}}##, we can find the limit by multiplying the numerator and the denominator by ##\frac{1}{x}##. This gives us ##\frac{1 +...
  45. Mr Davis 97

    Simplifying a rational expression

    Given that we have the expression ##\displaystyle-\frac{1}{(x-2)(x-2)(x-3)}~\cdot~\sqrt{\frac{(x-2)^{2}}{(x-3)(x-1)}} ##, how do we simplify it, step by step? Specifically, I am concerned about the ##\sqrt{(x-2)^{2}}## term. Are we allowed to cancel this with the ##(x-2)## in the denominator?
  46. Math Amateur

    MHB Rational Functions - Polynomials Over a Field - Rotman Proposition 3.70

    I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.5 From Numbers to Polynomials ... I need help with an aspect of the proof of Lemma 3.70 ... The relevant text from Rotman's book is as follows:In...
  47. J

    Do you use two lines (of ruled paper) for a rational functio

    Do you use two lines (of regular ruled paper) for a rational function? FOr example, would you use two lines or one for the equation ## f(x) = (x^2-3) / x+2 ## I can see three ways (in general) of writing this. 1) numerator and denominator each get one line 2) num and den are written smaller...
  48. anemone

    MHB Prove an expression is rational

    The rational numbers $a,\,b,\,c$ (for which $a+bc$, $b+ac$ and $a+b$ are all non-zero) satisfy the equality $\dfrac{1}{a+bc} =\dfrac{1}{a+b}-\dfrac{1}{b+ac}$. Prove that $\sqrt{(c−3)(c+1)}$ is rational.
  49. Titan97

    Finding the number of rational values a function can take

    Homework Statement ##f(x)## is a continuous and differentiable function. ##f(x)## takes values of the form ##^+_-\sqrt{I}## whenever x=a or b, (where ##I## denotes whole numbers) ; otherwise ##f(x)## takes real values. Also, ##|f(a)|\le |f(b)|## and ##f(c)=-1.5##. Graph of ##y=f(x)f'(x)##: The...
  50. Curieuse

    Rational and irrational numbers

    Homework Statement Determine a positive rational number whose square differs from 7 by less than 0.000001 (10^(-6)) Homework Equations - The Attempt at a Solution Let p/q be the required rational number. So, 7> (p/q)^(2) > 7-(0.000001) ⇒ √(7) > p/q > √(7-.000001) ⇒√(7) q> p >...
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