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. R

    Prove that a set of positive rational numbers is countable

    Homework Statement Prove that the set of positive rational numbers is is countable by showing that the function K is a 1-1 correspondence between the set of positive rational numbers and the set of positive integers if K(m/n) = p_1^{2a_1}p_2^{2a_2}...p_s^{2a_s}q_1^{2b_1-1}...q_t^{2b_t-1}...
  2. R

    Prove or disprove that there is a rational number x and an i

    Homework Statement Prove or disprove that there is a rational number x and an irrational number y such that x^y is irrational Homework EquationsThe Attempt at a Solution Please guys do not give me an answer. My only question is: what type of proof would you use? It seems like with irrational...
  3. R

    Use proof by contradiction to show there is no rational number such that....

    Homework Statement Use proof by contradiction to show there is no rational number r for which r^3+r+1 = 0 Homework EquationsThe Attempt at a Solution Assume there is a rational number r for which r^3+r+1=0. Then r = (a/b) with a,b ∈ℤ and b ≠ 0, and a/b is in lowest terms Then a/b is a root...
  4. marino

    B Proof That ##\sqrt{x}## Isn't Rational (Unless ##x## is a Perfect Square)

    According to you this theorem is correct? Exercise 1.2 * Proof that ##\sqrt{x}## isn't a rational number if ##x## isn't a perfect square (i.e. if ##x=n^2## for some ##n∈ℕ##). In effect, if ##x=\frac{25}{9}##, so ##x## isn't a perfect square, then ##\sqrt{x}=\sqrt{\frac{25}{9}}=\frac{5}{3}##...
  5. Specter

    Differentiating a a rational function

    Homework Statement Find the first and second derivatives of ##\displaystyle f(x)=\frac {1} {x^2+6}## Homework EquationsThe Attempt at a Solution [/B] ##\displaystyle f(x)=\frac {1} {x^2+6}## ##\displaystyle f(x)=(x^2+6)^{-1}## ##\displaystyle f'(x)=-1(2x)(x^2+6)^{-2}## ##\displaystyle...
  6. V

    I Quantifiers with integers and rational numbers

    Give an example where a proposition with a quantifier is true if the quantifier ranges over the integers, but false if it ranges over rational numbers. I do not know where to go about when answering this, I know that an integer can be a rational number, for example 5 is an integer but can also...
  7. R

    Proving that there is no rational number whose square is two

    Homework Statement The question is to prove that no rational number squared is = 2 Homework EquationsThe Attempt at a Solution I want to understand why for (a/b)^2 = 2, we assume one of the numbers is odd. Is this because, from approximation we know that root 2 is not a whole number, and If...
  8. M

    MHB Calculating The Nth Rational Number

    Hallo If we specify a particular method for mapping the natural numbers to the rationals, could we also specify a "distance" between two consecutive terms in some general way. Also are we able to calculate the nth term in such a progression perhaps incorporating this distance function somehow...
  9. karush

    MHB *aa3.2 Let Q be the group of rational numbers under addition

    aa3.2 Let Q be the group of rational numbers under addition and let $Q^∗$ be the group of nonzero rational numbers under multiplication. In $Q$, list the elements in $\langle\frac{1}{2} \rangle$, In ${Q^∗}$ list elements in $\langle\frac{1}{2}\rangle $ ok just had time to post and clueless
  10. navneet9431

    Need help in solving this question about a rational inequality

    Homework Statement Go through question number 4 Homework Equations The Attempt at a Solution See basically the question is asking us to find the range of the given function x/(x^2+x+1). So,I began solving it this way... I am stuck at this step. I asked my friend for a hint and he told me to...
  11. A

    B A Rational Game: Exploring the Paradox of Aligning Irrational Numbers

    This post is to set forth a little game that attempts to demonstrate something that I find to be intriguing about the real numbers. The game is one that takes place in a theoretical sense only. It starts by assuming we have two pieces of paper. On each is a line segment of length two: [0,2]...
  12. J

    MHB Rational Root Test and Modulo p

    Hi all, I have done the question in two methods. The first method is done by rational root test and the second method is by modulo p (theorem is as attached). It seems that my answers for both methods do not tally. 1. Where have I done wrong in the attached for the methods? Which is the...
  13. M

    MHB What are the steps for finding asymptotes of rational functions?

    Hello everyone. Time to get back to math. I have forgotten how to find asymptotes of rational functions. I think there are three types of asymptotes. Can someone show me how to find asymptotes of rational functions? What exactly is an asymptote?
  14. BWV

    I Rational powers of irrational numbers

    √2 is irrational but √22 is rational Is there any way to know if given some irrational number α, if αn is rational for some n? Or can it be proven that ∏n or en are irrational for all n?
  15. R

    Does x = -1 Make the Function FoG Undefined Even If Part of It Equals Zero?

    Homework Statement Say you have a fn FoG = x+2/x+1 + x+1/x+2 if x = -1, the first one is undefined. But the second one would end up as 0, a real number I'm trying to understand, would x = -1 make the whole term FoG undefined? because only x+2/x+1 would be undefined, x+1/x+2 would be =...
  16. J

    MHB Integral of Rational Exponential

    Hi, I'm new to this forum. This semester I took Calculus I and just took the final yesterday. There were a few questions that were unexpected that I didn't know how to handle. This integral has got me stumped.\int_{0}^{1} e^{x}/(1 + e^{2x}) \,dx The techniques I know at this point include u...
  17. Delta2

    I Rational sequence converging to irrational

    In the textbook I have (its a textbook for calculus from my undergrad studies, written by Greek authors) some times it uses the lemma that "for any irrational number there exists a sequence of rational numbers that converges to it", and it doesn't have a proof for it, just saying that it is a...
  18. M

    MHB Graph Rational Function By Hand

    Graph $f(x) = \frac{2}{(x - 3)}$ on the xy-plane by building a table of values. 1. How many values of x must I use to graph this function?2. Must I use the same amount of negative values of x as positive values of x to form an even number of points in the form (x, y)?3. Is graphing by hand an...
  19. M

    MHB Find Inverse of Rational Function

    Find the inverse of f(x) = 2/(x - 3). Let y = f(x) y = 2/(x - 3) Replace y for x. x = 2/(y - 3) x(y - 3) = 2 Solve for y. xy - 3x = 2 xy = 2 + 3x y = (2 + 3x)/x Replace y with f^-1 (x). f^-1(x) = (2 + 3x)/x 1. Is f^-1(x) the inverse of f(x)? 2. What does f(x) and f^-1(x) look like...
  20. M

    MHB What Determines the Range of a Rational Function Like f(x) = 2/(x - 3)?

    Find the range of f(x) = 2/(x - 3). 1. What exactly are we looking for when we say RANGE of a rational function? 2. Is the domain of the inverse the range of the given function? 3. What is the easiest way to find the range? Graphing?
  21. M

    MHB What is the domain of the rational function f(x) = 2/(x - 3)?

    Find the domain of f(x) = 2/(x - 3). 1. Are we looking for the domain of f or f(x)? 2. Solution Set x - 3 = 0 and solve for x. x - 3 = 0 x - 3 + 3 = 3 x = 3 Let D = domain D = ALL REAL NUMBERS except for x = 3. Yes? P.S. Does x = 3 mean there is a hole at the point (3, 0) for this...
  22. P

    Rational expressions and domains

    Homework Statement Okay, I have two examples that are confusing me. I am not sure where all the numbers that must be excluded from the denominators so that we're not dividing by zero are coming from. a) x2 + 6x +5 / x2 - 25 b) x-7 / x-1 multiplied by x2-1 / 3x-21 Homework Equations None...
  23. E

    MHB How to Find the Zeros Using the Rational Roots Theorem?

    I can't find the zeros to 4x^5-10x^4-14x^3+49x^2-28x+4 I found my positive zeros, 2, 1/2 using synthetic division and possible zeros. But from there I'm stuck.
  24. D

    Looking for Intuitive Reasoning Behind Rational Exponents

    Solving these seem fairly simple so far. But I don't know why this works. I asked my instructor and she couldn't give me an intuitive reason as to why. Homework Statement ##\sqrt[3]y\cdot\sqrt[5]y^2## Homework Equations N/A The Attempt at a Solution $$\sqrt[3]y\cdot\sqrt[5]y^2$$ $$y^\frac...
  25. B

    Simple demonstration with real, rational and integers

    Homework Statement Let ##\alpha \in \mathbb{R}## and ##n \in \mathbb{N}##. Show that exists a number ##m \in \mathbb{Z}## such that ##\alpha - \frac {m}{n} \leq \frac{1}{2n}## (1).The Attempt at a Solution If I take ##\alpha= [\alpha] +(\alpha)## with ##[\alpha]=m## (=the integer part) and...
  26. lfdahl

    MHB Prove that the radius of the incircle of △ is rational

    Let $\bigtriangleup$ be an isosceles triangle for which the length of a side and the length of the base are rational. Prove that the radius of the incircle of $\bigtriangleup $ is rational if and only if the two right triangles formed by the altitude to the base are similar to a right triangle...
  27. lfdahl

    MHB Rational trigonometric expression show tan^218°⋅tan^254°∈Q

    Show, that $$\tan^2 18^{\circ} \cdot \tan^254^{\circ} \in \Bbb{Q}.$$
  28. Zafa Pi

    I Can a tetrahedron have all dihedral angles rational?

    At each edge of a tetrahedron the 2 common faces form a dihedral angle. Can each of these 6 angles be rational multiples of pi?
  29. Math Amateur

    MHB Some Properties of the Rational Numbers .... Bloch Exercise 1.5.9 (3)

    I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Section 1.5: Constructing the Rational Numbers ... I need help with Exercise 1.5.9 (3) ...Exercise 1.5.9 reads as follows: We are at the point in Bloch's book where he has just...
  30. Math Amateur

    MHB Sum of a rational number and an irrational number ....

    I am trying without success to provide a rigorous proof for the following exercise: Show that the sum of a rational number and an irrational number is irrational.Can someone please help me with a rigorous solution ...I am working from the following books: Ethan D. Bloch: The Real Numbers and...
  31. Math Amateur

    Sum of a rational number and an irrational number ....

    Homework Statement I am trying without success to provide a rigorous proof for the following exercise: Show that the sum of a rational number and an irrational number is irrational. Homework Equations I am working from the following books: Ethan D. Bloch: The Real Numbers and Real Analysis...
  32. kupid

    MHB How to calculate LCM for rational equations ?

    To calculate an LCM for a rational function, follow these steps: 1. Factor all denominator polynomials completely. 2. Make a list that contains one copy of each factor, all multiplied together. 3. The power of each factor in that list should be the highest power that factor is raised to in any...
  33. awholenumber

    Factoring rational equations -- looking for worked out practice problems

    Where can i find some worked out practice problems ?
  34. kupid

    MHB Practice problems of Rational Equations ?

    Does anyone know where i can find some practice problems of Rational Equations ?
  35. M

    MHB Range of Rational Functions....3

    Find the range of y = sqrt{2x - 4}. I need the steps. According to the textbook, graphing the function leads to finding the range. This may be true for others but not for me. I am not clear on the range idea.
  36. M

    MHB Range of Rational Functions....2

    Find the range of y = (x + 2)/(x - 2). I need the steps. According to the textbook, graphing the function leads to finding the range. This may be true for others but not for me. I am not clear on the range idea.
  37. M

    MHB Range of Rational Functions....1

    Finding the range of functions can be complicated. Rational functions can be complicated. I have a terribly hard time finding the range of functions. Find the range of y = 1/(x + 4). How is this done? I want the steps.
  38. M

    MHB Domain of Rational Function: What is the domain of the given rational function?

    Find the domain of the rational function. y = (6x^2 + 11x + 4)/(3x + 4) Solution: Let denominator = 0 and solve for x. 3x + 4 = 0 3x = -4 x = -4/3 This means the domain is any real number except for x = -4/3. When x = -4/3, the denominator becomes 0, which is undefined. Correct?
  39. SSequence

    B Subsets of Rational Numbers and Well-Ordered Sets

    This isn't original or anything, but I was thinking about how would one go about formalizing (in a general sense) an informal wikipedia picture such as this: https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Omega-exp-omega-labeled.svg/487px-Omega-exp-omega-labeled.svg.png For example...
  40. M

    MHB Exploring Rational and Irrational Numbers with Examples of Arithmetic Operations

    Give an example of irrational numbers a and b such that the indicated expression is (a) rational; (b) irrational. 1. a + b 2. a/b Must I replace a and b with numbers that create a rational and irrational number?
  41. M

    MHB Finding the Range of Rational Inequalities: An Algebraic Approach

    How do I find the range of [(4 - 4x^2)/(x^2 + 1)^2] > 0 algebraically? Do I set the numerator to 0 and solve for x? Do I set the denominator to 0 and solve for x?
  42. B

    MHB Function - one linear, one rational - is the following True or False

    The function ƒ(x) is a linear function and g(x) is a rational function. These functions have the following values: ƒ(3) = 7 g(3) = 5.6 ƒ(4) = 5 g(4) = 6.7 There is a solution to the equation ƒ(x) = g(x) between x = 3 and x = 4 that must be closer to 3 than 4. TRUE or FALSE?
  43. B

    Integer and Rational Number Subtleties in an Algebra Problem

    Homework Statement Let ##S = \{\frac{1}{n} + \mathbb{Z} ~|~ n \in \mathbb{N} \}##. I am trying to show that ##f : \mathbb{N} \rightarrow S## defined by ##f(n) = \frac{1}{n} + \mathbb{Z}## is a bijection. Surjectivity is trivial, but injectivity is a little more involved. Homework EquationsThe...
  44. doktorwho

    Integral that is reduced to a rational function integral

    Homework Statement Suggest an integral that is reduced to a rational function integral when this substitution is used: ##a)## ##t=\sin x## ##b)## ##t=\sqrt[6] {x+5}## ##c## ##\sqrt{1-9x^2}=-1+xt## Homework Equations 3. The Attempt at a Solution [/B] I found this to be a very interesting...
  45. J

    MHB Algebraic Rational Expressions

    I am attempting to find the solution to the following question. Simplify and state the restrictions on the variables\frac{5a^5b^6}{10a^2b^3}\div\frac{2a^4b^2}{20a^3b^5} Not really understanding how to find the restrictions with these set variables.
  46. Arman777

    How Do You Simplify the Integral of a Rational Function?

    Homework Statement ##∫\frac {dx}{(x^2-1)^2}## Homework Equations The Attempt at a Solution I tried to divide ##\frac {1} {(x^2-1)^2}## as ##\frac {Ax+B} {(x^2-1)} +\frac {Cx^3+Dx^2+Ex+F} {(x^2-1)^2}## but this looks so complex..I don't know how to do ? Maybe I can...
  47. L

    Find limit for rational function

    Homework Statement f [/B] f(x)= x^2 +4 find the limit as x approaches 1, there is something wrong with the latex code but I don't know what. Limit $$\lim_{x\to 1} \frac{{f(x)}^4-{f(1)}^4}{x-1}$$ Homework Equations -methods for finding limits -factorising polynomials -possibly polynomial long...
  48. Mr Davis 97

    Is the group of positive rational numbers under * cyclic?

    Homework Statement Is the group of positive rational numbers under multiplication a cyclic group. Homework EquationsThe Attempt at a Solution So a group is cyclic if and only if there exists a element in G that generates all of the elements in G. So the set of positive rational numbers would...
  49. karush

    MHB Q5|8.5.17 int of rational expression...

    $\tiny{Q5|8.5.17}$ $\textsf{Evaluate}$ \begin{align*}\displaystyle I&=\int_0^{9}\frac{x^3 \, dx}{x^2+18x+81} \color{red}{=243\ln2-162} \end{align*} OK even before I try to get this answer trying to see what road to take u substituion long division and remainder partial fractions (the...
  50. 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*} $\textit{ok tried getting to this answer by: }$ $u=x^3-5 \therefore du=3x^2 \, dx$ $\textit{but it didnt seem to go to good}$
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