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. Agent Smith

    B A Pi Question: Why do we use the awkward approximation 22/7 ?

    As a high school student we were told to use ##\frac{22}{7}## as a rational approximation for ##\pi##. However, to the same level of accuracy, ##\frac{314}{100} = \frac{157}{50}## is also ##\pi## and since there's a ##100## and a ##5## in the denominator many calculations would've been far...
  2. M

    Proof by induction for rational function

    For this problem, My solution is ##P(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0## where n is a member of the natural numbers Base case (n = 1): ##P(x) = a_0x^0 = a_0## Thus ##\lim_{x \to \infty} \frac{P(x)}{e^x} = \lim_{x \to \infty} \frac{a_0}{e^x} = a_0 \lim_{x \to \infty}...
  3. M

    Proving limit of rational function

    For this problem, The solution is, However, I'm confused how ##0 < | x - 1|< 1## (Putting a bound on ##| x- 1|##) implies that ##1 < |x+1| < 3##. Does someone please know how? My proof is, ##0 < | x - 1|< 1## ##|2| < | x - 1| + |2| < |2| + 1## ##2 < |x - 1| + |2| < 3## Then take absolute...
  4. M

    Proving convergence of rational sequence

    For this problem, The solution is, However, does someone please know why this did not use ##2n ≤ 2n^2 + 2n + 1## which would give ##\frac{3n - 1}{2n^2 + 2n + 1} ≤ \frac{3n}{2n} = \frac{3}{2}##? In general, after solving many problems, it seems that when proving the convergence of a rational...
  5. M

    Proving rational function converges from first principles

    For this problem, I am confused how they get $$| x - 4 | > \frac{1}{2}$$ from. I have tried deriving that expression from two different methods. Here is the first method: $$-1\frac{1}{2} < x - 4 < -\frac{1}{2}$$ $$1\frac{1}{2} > -(x - 4) > \frac{1}{2}$$ $$|1\frac{1}{2}| > |-(x - 4)| >...
  6. Vanadium 50

    I Quality of rational approximations

    22/7 is a very good approximation for π. Sqrt(2) doesn’t do that well until 99/70 and e doesn’t do that well until 193/71. 355/113 is even better. Is there some reason for this? Perhaps geometrical? Why do the ratios of small integers work better for π than other numbers? Or is it just...
  7. M

    Limit of a rational function with a constant c

    For this problem, Did they get ## x## approaches one is equivalent to ##t## approaches zero because ##t ∝ (x)^{1/3} + 1##? Many thanks!
  8. Euge

    POTW Definite Integral of a Rational Function

    Evaluate the definite integral $$\int_0^\infty \frac{x^2 + 1}{x^4 + 1}\, dx$$
  9. brotherbobby

    Proving that the inverse of a rational number exists

    Problem statement : I cope and paste the problem as it appears in the text below. Attempt : Not being a math student, I try and prove the above statement using an "intuitive" way. Let us have a rational number ##b = \frac{n}{m}##. Multiplying with ##a## from the right, we see ##ab =...
  10. Eclair_de_XII

    B Can the continuity of functions be defined in the field of rational numbers?

    I argue not. Let ##f:\mathbb{Q}\rightarrow\mathbb{R}## be defined s.t. ##f(r)=r^2##. Consider an increasing sequence of points, to be denoted as ##r_n##, that converges to ##\sqrt2##. It should be clear that ##\sqrt2\equiv\sup\{r_n\}_{n\in\mathbb{N}}##. Continuity defined in terms of sequences...
  11. M

    If ## a ## is a positive integer and ## \sqrt[n]{a} ## is rational?

    Proof: Suppose ## a ## is a positive integer and ## \sqrt[n]{a} ## is rational. Then we have ## \sqrt[n]{a}=\frac{b}{c} ## for some ## b,c\in\mathbb{Z} ## such that ## gcd(b, c)=1 ## where ## c\neq 0 ##. Thus ## \sqrt[n]{a}=\frac{b}{c} ## ## (\sqrt[n]{a})^{n}=(\frac{b}{c})^n ##...
  12. B

    Need help in manipulating rational absolute value inequalities

    How does one manipulate rational absolute inequalities? For example, I want to transform the absolute value inequality ##|x-3|<1## to ##\frac{|x+3|}{5x^2}<A \ ##, for some number ##\text{A}##, to find an upper and lower bound on the latter term using the constraint in the first term, and not...
  13. B

    Rational epsilon-delta limit proof questions

    Summary:: Good afternoon. I have more questions about the details of epsilon-delta proofs. Below is a simple, rational limit proof example with questions at the end. The scratch work and proof are a bit pedantic but I don't follow proofs very well which omit a lot of details, including scratch...
  14. chwala

    Show that ##(b-c)x^2+(c-a)x+a-b=0## has rational roots

    If we have a quadratic equation, ##px^2+qx+d## ,then the condition that the roots are rational is satisfied if our discriminant has the form ## q^2-4pd≥0## (also being a perfect square). Therefore we shall have, ##(c-a)^2-4(b-c)((a-b)≥0## ##(c-a)^2-4(ab-b^2-ac+bc)≥0##...
  15. wrobel

    I A curve that does not meet rational points

    This is just to recall a nice fact: Any two points ##A,B\in\mathbb{R}^n\backslash\mathbb{Q}^n,\quad n>1## can be connected with a ##C^\infty##-smooth curve that does not intersect ##\mathbb{Q}^n##. The proof is surprisingly simple: see the attachment
  16. C

    I Finding a Rational Function with data (Pade approximation)

    Dear Everybody, I need some help understanding how to use pade approximations with a given data points (See the attachment for the data). Here is the basic derivation of pade approximation read the Derivation of Pade Approximate. I am confused on how to find a f(x) to the data or is there a...
  17. R

    B Prove that s/t is rational where s and t are rational

    Is my proof correct? The steps from hypothesis to conclusion are in order below: 1) given rational numbers s,t with t != 0 2) take s = p/1 and t = q/1 where p,q are integers 3) (p/1)/(q/1) = p/q is rational 4) therefore by substitution s/t is rational
  18. K

    A Exploring Bertrand's Theorem: Why is β the Same Rational Number for All Radii?

    Wikipedia on Bertrands theorem, when discussing the deviations from a circular orbit says: >..."The next step is to consider the equation for ##u## under small perturbations ##{\displaystyle \eta \equiv u-u_{0}}## from perfectly circular orbits" (Here ##u## is related to the radial distance...
  19. Trysse

    I Five points in space with rational distances that are not co-linear

    Hi there, experts on three-D space! while thinking about (physical) space, I have come up with the following (geometry) question: Is it possible to define five points (A, B, C, D, E) in Euclidian space, so that all distances (AB, AC, AD, AE, BC, BD, BE, CD, CE, DE) can be expressed in rational...
  20. M

    Can anyone please check/verify this proof about rational numbers?

    Show sqrt(3), sqrt(5), sqrt(7), sqrt(24), and sqrt(31) are not rational numbers.
  21. chwala

    Determine which functions are rational

    Ok in my thinking, i would say that it depends on ##x##, if ##x## belongs to the integer class, then the rational functions would be ##i ## and ##iii##...but from my reading of rational functions, i came up with this finding: I would appreciate your input on this.
  22. karush

    MHB 1.1.21 simplify rational expression

    simplify $\left(\dfrac{a^2b^3-2a^{-3}b^3}{2a}\right)^2=$ OK this could get confusing quickly but I don't think we want to square it first
  23. brotherbobby

    Prove a rational fraction is equal to another

    Problem Statement : If ##\dfrac{x}{b+c-a}=\dfrac{y}{c+a-b}=\dfrac{z}{a+b-c},## prove that ##\boxed{\boldsymbol{\dfrac{x+y+z}{a+b+c}=\dfrac{x(y+z)+y(z+x)+z(x+y)}{2(ax+by+cz)}}}## Attempt : Let the fractions (ratios) ##\dfrac{x}{b+c-a}=\dfrac{y}{c+a-b}=\dfrac{z}{a+b-c} = \boldsymbol{k}##...
  24. MidgetDwarf

    Is it possible to make graphs of subsets of Rational Numbers in Mathem

    Is it possible to make subsets of rational numbers in Mathematica using the plot command, or any other command? Ie., say I want to graph the set of rational numbers from 0 to 1.
  25. N

    MHB What is the Limit of (5x)/(100 - x) as x Approaches 100 from the Left?

    Find the limit of (5x)/(100 - x) as x tends to 100 from the left side. The side condition given: 0 <= x < 100 To create a table, I must select values of x slightly less than 100. I did that and ended up with negative infinity as the answer. The textbook answer is positive infinity. Can you...
  26. N

    MHB How to Find the Limit of (1 - x)/[(3 - x)^2] as x Approaches 3?

    Find the limit of (1 - x)/[(3 - x)^2] as x---> 3. I could not find the limit using algebra. So, I decided to graph the given function. I can see from the graph on paper that the limit is negative infinity. How is this done without graphing?
  27. N

    MHB Is the Limit of 1/(x^2 - 9) as x Approaches -3 from the Left Positive Infinity?

    Find the limit of 1/(x^2 - 9) as x tends to -3 from the left side. Approaching -3 from the left means that the values of x must be slightly less than -3. I created a table for x and f(x). x...(-4.5)...(-4)...(-3.5) f(x)... 0.088...0.142...…...0.3076 I can see that f(x) is getting larger and...
  28. N

    MHB Is the Limit of 5/(x^2 - 4) as x Approaches 2 from the Right Positive Infinity?

    Find the limit of 5/(x^2 - 4) as x tends to 2 from the right side. Approaching 2 from the right means that the values of x must be slightly larger than 2. I created a table for x and f(x). x...2.1...2.01...2.001 f(x)...12...124.68...1249.68 I can see that f(x) is getting larger and larger...
  29. N

    MHB What is the Limit of (3x)/(x-2) as x Approaches 2 from the Left?

    Find the limit of (3x)/(x - 2) as x tends to 2 from the left side. Approaching 2 from the left means that the values of x must be slightly less than 2. I created a table for x and f(x). x...0...0.5...1...1.5 f(x)...0...-1...-3...-9 I can see that f(x) is getting smaller and smaller and...
  30. N

    MHB Horizontal Asymptote of Rational Function

    Given f(x) = [sqrt{2x^2 - x + 10}]/(2x - 3), find the horizontal asymptote. Top degree does not = bottom degree. Top degree is not less than bottom degree. If top degree > bottom degree, the horizontal asymptote DNE. The problem for me is that 2x^2 lies within the radical. I can rewrite...
  31. potatocake

    How to prove rational sequence converges to irrational number

    I attempted to solve it $$ x = \frac {1}{4x} + 1 $$ $$⇒ x^2 -x -\frac{1}{4} = 0 $$ $$⇒ x = \frac{1±\sqrt2}{2} $$ However, I don't know the next step for the proof. Do I need a closed-form of xn+1or do I just need to set the limit of xn and use inequality to solve it? If I have to use...
  32. brotherbobby

    Quadratic equation with no rational roots

    Given : Equation ##x^2+(2m+1)x+(2n+1) = 0## where ##m \in \mathbb{Z}, n \in \mathbb{Z}##, i.e. both ##m,n## are integers. To prove : If ##\alpha,\beta## be its two roots, then they are not rational numbers. Attempt : The discriminant of the equation ##\mathscr{D} = (2m+1)^2 - 4(2n+1) =...
  33. anemone

    MHB Rational Number: Proving $x+\dfrac{1}{x}$ is Rational

    Let $x$ be a non-zero number such that $x^4+\dfrac{1}{x^4}$ and $x^5+\dfrac{1}{x^5}$ are both rational numbers. Prove that $x+\dfrac{1}{x}$ is a rational number.
  34. greg_rack

    Issue calculating the derivative of a rational function

    First, I calculated the derivative of $$D(\sqrt{ax})=\frac{a}{2\sqrt{ax}}$$ Then, by applying the due theorems, I calculated the deriv of the whole function as follows: $$ f'(x)=\frac{\frac{a}{2\sqrt{ax}}(\sqrt{ax}-1)-\sqrt{ax}(\frac{a}{2\sqrt{ax}})}{(\sqrt{ax}-1)^2}=...
  35. A

    MHB Rational Inequalities: Solve & Understand | Math

  36. bigchaka

    Sum of rational and irrational is irrational

    Summary:: i get a proof that sum of rational and irrational is rational which is wrong(obviously) let a be irrational and q is rational. prove that a+q is irrational. i already searched in the web for the correct proof but i can't seem to understand why my proof is false. my proof: as you...
  37. M

    How Do You Formulate and Solve Rational Inequalities?

    My attempt so far: I put all the terms to become smaller than zero: so ##x<-4## becomes ##x-4<0## ##-1\leq x\leq 3## becomes ##-1-x\leq 0## and ##x-3 \leq 0## ##x>6## becomes ##x-6>0## which is the same as ##-x+6<0## (i think)... I am now stuck on making it a rational inequality... anyone...
  38. M

    Rational motion combined with 2 springs

    I first calculated the speed of two blocks using angular speed, then find the centripetal force of them, but I don't know how to proceed my calculation, what value should I plug into Hooke's law?
  39. F

    I Asymptotes of Rational Functions....

    Hello, I know that functions can have or not asymptotes. Polynomials have none. In the case of a rational functions, if the numerator degree > denominator degree by one unit, the rational function will have a) one slant asymptote and b) NO horizontal asymptotes, c) possibly several vertical...
  40. chwala

    Proving rational surd inequalities

    my attempt, i am not good in this kind of questions ...i need guidance.
  41. S

    Question about asymptotes of rational function

    I tried graphing the function in the calculator, and the graph seems to have a horizontal asymptote at y=0, not at y=1. Why is this so? Thanks for helping out.
  42. coreanphysicsstudent

    Rational power of the imaginary unit

    but the real answer is −1, (1±i√3)/2 What's wrong with my solution? please help me through.
  43. R

    I Inverse Laplace transform of a rational function

    I struggle to find an appropriate inverse Laplace transform of the following $$F(p)= 2^n a^n \frac{p^{n-1}}{(p+a)^{2n}}, \quad a>0.$$ WolframAlpha gives as an answer $$f(t)= 2^n a^n t^n \frac{_1F_1 (2n;n+1;-at)}{\Gamma(n+1)}, \quad (_1F_1 - \text{confluent hypergeometric function})$$ which...
  44. Leonardo Machado

    A Rational Chebyshev Collocation Method For Damped Harmonic Oscilator

    Hello everyone. I'm currently trying to solve the damped harmonic oscillator with a pseudospectral method using a Rational Chebyshev basis $$ \frac{d^2x}{dt^2}+3\frac{dx}{dt}+x=0, \\ x(t)=\sum_{n=0}^N TL_n(t), \\ x(0)=3, \\ \frac{dx}{dt}=0. $$ I'm using for reference the book "Chebyshev and...
  45. e2m2a

    I The Fundamental Theorem of Arithmetic and Rational Numbers

    The fundamental theorem of arithmetic applies to prime factorizations of whole numbers. Can this theorem also correctly be invoked for all rational numbers? For example, if we take the number 3.25, it can be expressed as 13/4. This can be expressed as 13/2 x 1/2. This cannot be broken...
  46. S

    Algebra Rational exponents in the real number system?

    Are there rigorous texts that treat the topic of raising real numbers to rational powers without treating it a special case of using complex numbers? I'm not trying to avoid the complex numbers for my own personal use! My goal is to determine whether students who have not studied complex...
  47. S

    I Rational functions in one indeterminate - useful concept?

    The examples of "formal" power series and polynomials in one indeterminate are familiar and useful in algebra. However, I don't recall the example of rational functions (ratios of polynomials) in one indeterminate being used for anything. Is that concept useful? - or trivial? -or equivalent...
  48. Ackbach

    MHB Solving Rational ODE's of the form (ax+by+c) dx+(ex+fy+g) dy=0.

    There are essentially three cases of the rational ODE $(ax+by+c)\,dx+(ex+fy+g)\,dy=0,$ since there are two straight line expressions multiplying the differentials. We will think of this geometrically, then translate to the algebraic approach. The tricky part to these problems is keeping track of...
  49. Math Amateur

    MHB Understanding Garling's Corollary 3.2.7 on Real Numbers and Rational Sequences

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ... I am focused on Chapter 3: Convergent Sequences I need some help to fully understand the proof of Corollary 3.2.7 ...Garling's statement and proof of...
  50. N

    MHB Is 2√(7)+4 an Irrational Number? Exploring the Proof of Its Irrationality

    See picture for question and answer.
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