What is Rational: Definition and 625 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. J

    Proof that sqrt(2) is not a rational number

    1. Prove that \sqrt{2} is not a rational number using the rational zero theorem 2. Homework Equations : x2-2 The Attempt at a Solution So, I was thinking to just show that the only rational numbers that have the form \frac{c}{d} are \pm 1, \pm 2, but \sqrt{2} is not any of those...
  2. C

    Finding the domain for a rational function.

    Hi I have the function I have worked out the domain as not equal to 1 using the fact the denominator can't equal 0. Now I am stuck finding the range. How would I find it without graphing. The answer is y<0 , y=>3.
  3. D

    Inequalities of rational expressions

    referring to the photo attached, can anyone expalin and give suitable numerical example please? i couldn't understand the notes.
  4. kaliprasad

    MHB Quadratic equation with rational roots

    form quadratic equation $ax^2 +bx+c=0$ in parametric form such that a,b,c are integers in AP and it has got rational roots
  5. E

    MHB Simplify (Adding and Subtracting Rational Functions)

    5c) Simplify. \frac{2x}{3y} - \frac{x^2}{4y^3} + \frac{3}{5y^4} This is what I did, which is wrong according to the textbook. Could someone point out what I did wrong and how to correct it? Thanks. \frac{(2x)(4y^3)-(x^2)(3y)}{(3y)(4y^3)} + \frac{3}{5y^4} \frac{8xy^3-3x^2y}{(12y^3)} +...
  6. E

    MHB What am I doing wrong? (Simplifying Rational Expressions)

    7a) Simplify and state any restrictions on the variables: \frac{x^2-5xy+4y^2}{x^2+3xy-28y^2} ⋅ \frac{x^2+2xy+y^2}{x^2-y^2} I'm not really sure what a good process would be to simplify this. This is what I tried to do, (below) which is wrong. Could anyone point out what I did wrong and what a...
  7. paulmdrdo1

    MHB What does "not integral" and "not rational" mean here?

    can you explain what it means when they are not "Integral" and not "Rational"? $\frac{4y}{x} = 4yx^{-1 }$ is not integral in x; $3x\sqrt{y}z^3$ is not rational in y.:confused: help please!
  8. V

    What is f(0) if f(x) is a rational number on [0,1]?

    Please see attached. The question is to determine f(0). In my opinion the question wants me to find f(x).. If f(x)=1, then 1. f:[0,1] -> R is continuous (do I need to prove?) 2. Obviously, f(1) = 1 3. f(x) is a rational number for all xε[0,1] so f(0) = 1 Am I correct?
  9. caters

    Irrational^rational = rational

    Is it true that in some cases an irrational number raised to a rational power is rational? If so what kinds of irrational numbers?
  10. J

    MHB Solve Quadratic Equation: -2x^2-14x-49

    Q4.) Quadratic equation: \frac{-2x^2-14x-49}{x^3-7x^2}
  11. mr_sparxx

    How to obtain the rational function integration recursive formula

    I've been dealing with several integrals involving rational functions. I have encountered myself arriving to an integral that requires the application of the following recursive formula: \int\frac{1}{(u^2+α^ 2)^m} \, du= \frac{u}{2 α^ 2 (m-1)(u^2+α^ 2)^{m-1}}+\frac{2m-3}{2 α^2...
  12. srfriggen

    Rational roots theorem to prove irrational

    Homework Statement Use the rational roots theorem to prove 31/2-21/3is irrational. The Attempt at a Solution My teacher strongly hinted to us that this problem had something to do with the fact that complex roots come in conjugate pairs, and all we had to do was, "flip the sign"...
  13. T

    MHB Solve for Rational Term: 2x^2-1 Over x^2+1

    I have a question why is \frac{2x^2-1}{x^2+1} equal to 2-\frac{3}{x^2+1} ??
  14. C

    Write rational function equation given graph

    On The graph, the following information was given: Point of discontinuity( -2,0) and (1,-3) X intercept at ( 2,0) Yintercept (0,-4) This is what I had : (x+2)(x-2)(x-1)/ (x+2)(x-1) But the answer is that u have to put (x+2)^2 in the numerator instead of what I had. But I thought...
  15. hilbert2

    Rational approximation of Heaviside function

    Hi, could someone please help me with this one: I'd need to form a sequence of rational functions ##R_{n}(x)## such that ##\lim_{n \to \infty} R_{n}(x)=\theta(x)##, where ##\theta(x)## is the Heaviside step function. The functions ##R_{n}(x)## should preferably be limited in range, i.e. for some...
  16. D

    Question on the logic of rational roots theorem

    this is not a homework question, but rather I feel like there is a contradiction in the theorem and just want clarity. I know the theorem is correct so I am looking for help in where the mistake is in my logic. take f(x) = x^3 + x^2 - 4x- 7 the rational roots theorem says if there are any...
  17. A

    Rational functions and link with direct substitution property

    Homework Statement Hello, I know the direct substitution property in calculus. But, the definition of a rational function still confuses me. For example, are these rational functions: y=(x^2+2x+1)/(x+1) y=((x^2+2)^(1/2))/(x+1) The denominator of the first one could cancel. So...
  18. M

    Algebra help, Rational Equations

    Homework Statement Solving formulas involving Rational expressions for each excersise, solve for the indicated variable Homework Equations (x-y)/xy = z Solve for x The Attempt at a Solution I started by multiplying both sides by xy, to which I got (x-y) = xyz Then I tried multiplying...
  19. L

    Exploring Indeterminate Limits of Rational Functions at Infinity

    Hi, I am in a first semester Calculus I course in college with an intermediate skill level with precalc and a basic understanding of limits and infinity. I do not understand how to solve this problem I attempted to do so only to find out after completion that ∞/∞ is indeterminate rendering my...
  20. V

    Solving inequalities with rational expressions.

    Homework Statement Identify the solution set of the inequality. Homework Equations 5x + 1 / x- 1 ≥ 7 The Attempt at a Solution I multiplied both sides by x - 1 which gave me 5x+1 ≥ 7x - 7 Then I combined like terms together which gave me 8 ≥ 2x I divided both sides by 2 which gave...
  21. D

    Trying to show that there are no rational solutions for an equation

    Homework Statement I have the equation $$ a^3+2b^3+4c^3-6abc=0 $$ and I want to show that for ##a,b,c\in\mathbb{Q}## and ##a,b,c\not=0## that there are no solutions. The Attempt at a Solution I tried many different approaches but this was the only one that seemed to work. I am not sure...
  22. A

    Transition matrix and rational canonical form

    Homework Statement I want to find the transition matrix for the rational canonical form of the matrix A below. Homework Equations The Attempt at a Solution Let ##A## be the 3x3 matrix ##\begin{bmatrix} 3 & 4 & 0 \\-1 & -3 & -2 \\ 1 & 2 & 1 \end{bmatrix}## The...
  23. K

    Continuity of a Rational Function at a number help

    Homework Statement Find continuity of function f(x)= (x^2-1)/(x-1) at x = 1 Homework Equations limit f(x) as x-> = L The Attempt at a Solution i KNOW it can be easily solved by stating that at x = 1 function becomes infinity,so discontinous it is actually...But as we do in finding domain...
  24. micromass

    Challenge III: Rational Tangles, solved by pwsnafu

    The newest challenge is the following: As an example, we can easily go from ##0## to ##-1/3##. Indeed, we can apply ##T## to ##0## to go to ##1##, we apply ##T## to go to ##2##, we apply ##T## to go to ##3##, and then we apply ##R## to go to ##-1/3##.
  25. Y

    MHB What is the best method for solving rational function integrals?

    Hello, I am trying to solve the next integral, using this method of rational functions. My problem is, that I turn the function into two expressions with the same denominator...when I try to use values of x, I will always make zero of both sides, if you know what I mean \[\int...
  26. MarkFL

    MHB John's Question: Finding Range of Rational Function

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  27. P

    Is Pi a rational number in any other base besides base Pi?

    I'm wondering if Pi is a rational number in any other base besides base Pi. Also is there a formula or function to figure this out? I'm not what the relevance would be if we could find one since the integers might be irrational if we did but I am just curious if indeed Pi is only a rational in...
  28. paulmdrdo1

    MHB Sums and Products of Rational and Irrational Numbers

    Explain why the sum, the difference, and the product of the rational numbers are rational numbers. Is the product of the irrational numbers necessarily irrational? What about the sum? Combining Rational Numbers with Irrational Numbers In general, what can you say about the sum of a rational...
  29. Z

    Power rules for radical roots and rational exponents.

    Hello everyone, I am a bit confused about definitions rules. I can have more questions but for now I want to ask only one question: Let us say I have a number: \sqrt[6]{3x3x3x3x3x3} 3x3x3x3x3x3 is equal to both 27^2 and (-27)^2. But If I write these two expressions separately I can get...
  30. T

    Laurent Series of Rational Functions

    In general I am rather confused by this type of problem. The textbook has a single example and does not show (m)any of its steps so I'm lost. I have a test this coming Thursday and the following is the only question of this type that the prof. has recommended: "23. Use equations (12) and (13)...
  31. mathmaniac1

    MHB Equations with rational degrees

    Is there a way to make these a polynomial? 1)\sqrt{A}+\sqrt{B}+\sqrt{C}=D 2)A1/3 + B1/3 = C Or how else can it be solved?
  32. J

    Differentproof there are more irrational numbers than rational numbers

    you can list and match up all rational numbers with irrational numbers this way.. lets say i have an irrational number 'c'. Rational->Irrational r1->cr1 r2->cr2 . . . rn->crn There exists an irrational number that is not on this matching, (not equal to any of the crx's) this...
  33. S

    Integral of a rational function

    Homework Statement I came across this integral recently while tutoring: ##\displaystyle \frac{1}{5} \int \frac{-x^3+2x^2-3x+4}{x^4-x^3+x^2-x+1}~dx## Homework Equations The Attempt at a Solution I'm not sure how to approach this. At first I suspected partial fraction decomposition...
  34. T

    A proof regarding Rational numbers

    I have found some trouble in trying to prove this question.please help mw with that. Q1) If (a+b)/2 is a rational number can we say that a and b are also rational numbers.? Justify your answer. I have tried the sum in the following way. Assume (a+b)/2=p/q (As it is rational) Lets...
  35. R

    MHB Integration of Rational Functions by Partial Fractions

    1) integral (upper bound:1, lower bound:0) (x^2+1)/(x^3+x^2+4x) dx 2) integral (upper bound:1, lower bound:0) (x^4+x^2+1)/(x^3+x^2+x-3) dx Now I know how to use Partial Fractions,My question is: 1) For the first part ln(x) is not defined at 0 ¼ʃ1/x dx + ¼ʃ(3x-1)/(x²+x+4) dx = ¼ ln|x| +...
  36. Nugso

    Rational Root Theorem Homework: Solving x4 - 4(x3) + 3(x2) -2x +1 = 0

    Homework Statement x4 - 4(x3) + 3(x2) -2x +1 = 0 Homework Equations Rational Root Theorem, q/p The Attempt at a Solution Hello everyone. Today, I've learned the rational root theorem( it's a bit late, isn't it? :( ) and thus wanted to see how it works. According to the...
  37. O

    Rational Numbers That Are Hard To Prove?

    There are lots of examples of numbers where "is it a rational number" has been an open question for a while before being proved as not being rational. Pi and e being famous examples. Some of them are still open, like pi+e, or the Euler-Mascharoni constant, but I think the general consensus is...
  38. J

    QIntegrate rational function: How do I solve (8-16x)/(8x^2-4x+1)dx?

    Homework Statement Integrate (8-16x)/(8x^2-4x+1) dx Q Homework Equations I separate it first 8/(8x^2-4x+1)dx -16x/(8x^2-4x+1)dx Then I have no idea what to do next. The Attempt at a Solution none ;(
  39. S

    Q* (the set of rational cuts) has least upper bound property or not?

    I am struggling to draw this point home: To prove that R has LUB property, we used the following reasoning: First we defined R to be set of cuts (having certain properties) where each cut corresponds to a real number and then we proved any subset A of R has LUB (least upper bound) property...
  40. caffeinemachine

    MHB Show that a rational function under some constraint is actually a polynomial.

    Let $r(x)\in\mathbb Q(x)$ be a rational function over $\mathbb Q$. Assume $r(n)$ is an integer for infinitely many integers $n$. Then show that $r(x)$ is a polynomial in $\mathbb Q[x]$.
  41. MarkFL

    MHB Increase/Decrease of Rational Function | Jillian's Yahoo Answers

    Here is the question: Here is a link to the question: How to find the intervals of increase and decrease? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  42. O

    Radical and its rational exponential form

    Can someone explain how these are equivalent. sqrt((-3)^2) = (-3)^2/2 =sqrt(9) and (-3)^1 3 is not equal to -3 (-3)^2/2 can be expressed as: (-3^2)^1/2 and (-3^1/2)2 (9)^1/2 and (sqrt(-1)sqrt(3))^2...
  43. R

    Why is Pi Irrational for Circles?

    We cannot put the ratio of circumference/diameter in the form p/q. In this case the circumference. Because any number of sides of a regular polygon perimeter to calculate the circumference will not fit to the circumference of a circle. That is the number of sides of a polygon tends to...
  44. C

    Is there an easier way to solve rational equations?

    Hi there, I've been learning about Solving for x in rational expressions with Mono, Bi and tri nomial denominators, and I was curious if there was an easier way to solve them than by factoring. I already understand and fully grasp the concept, but its just annoying to go through all the same...
  45. L

    Question about rational expressions and their domains

    Hi there. I am currently taking "College Math 1" at the local CC and I have encountered something that confuses me regarding rational expressions and their domains. The definition given by the textbook for rational expressions is: "the set of real numbers for which an algebraic expression is...
  46. I

    Complex Analysis - Rational Functions

    Homework Statement I'm studying for my final exam and came across this problem: Let f and g be entire analytic functions and |f(z)|<|g(z)| when |z|>1. Show that f/g is a rational function. The Attempt at a Solution I really have no clue where to go :(
  47. ECHOSIDE

    I'm having trouble factoring this rational expression.

    Homework Statement ((a^3-b^3)/(a^2-2ab+b^2))/((2a^2+2ab+2b^2)/(9a^2-9b^2)) Not using complex number system. Not concerned with domain. Find the quotient and put it in simplest terms. Homework Equations The Attempt at a Solution Too many to transcribe. Apparently I'm missing...
  48. Fernando Revilla

    MHB Jesusluvsponies's question at Yahoo Answers (Real and rational roots)

    Here is the question: Here is a link to the question: Polynomials, please help 10 points!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  49. anemone

    MHB Simplifying Rational Expression

    Simplify \frac{x^2-4x+3+(x+1)\sqrt{x^2-9}}{x^2+4x+3+(x-1)\sqrt{x^2-9}} where x>3.
  50. melese

    MHB No Rational Roots of $x^n+\cdots+1=0$

    (BGR,1989) Prove that for any integer $n>1$ the equation $\displaystyle \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}\cdots+\frac{x^2}{2!}+\frac{x^1}{1!}+1=0$ has no rational roots.
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