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. K

    Shoudn't the fraction -2/-1 be less than one?

    Because technically, the numerator is smaller than the denominator as −2<−1 I know it's an extremely stupid question. I mean I know that I can just multiply −1 to the numerator and the denominator and I'll get 2/1 which is greater than one. But what exactly is happening here? A number that...
  2. karush

    MHB Can You Verify the Integral of Secant Squared Over (1+Tangent)^3?

    $$\int\frac{\sec^2 \left({t}\right)}{\left(1+\tan\left({t}\right)\right)^3}dt$$ $$u=1+\tan\left({t}\right)\ du=\sec^2\left({t}\right) dt$$ So far ok?
  3. karush

    MHB How to Expand Rational Expressions for Integration

    $$\int\frac{x^4}{4+x^2}dx$$ this was homework for a section of expanding rational expressions $$\frac{x^4}{4+x^2}=x^2+\frac{16}{x^2 +4}-4$$ I don't see how W|F got this expansion?
  4. K

    MHB Proving Ideal Property of f(x)=0 for Every Rational x in $\mathcal{F}(\mathcal{R})$

    I am asked: Prove that each of the following is an ideal of $\mathcal{F}(\mathcal{R})$: a. The set of all f such that f(x)=0 for every rational x b. The set of all f such that f(0)=0 My question is how do I know what the multiplicative operation is within the ring? Is multiplication the...
  5. C

    How to Solve a Rational Equation with Unknown Vertical Stretch/Compression?

    Homework Statement Homework Equations y = f(x) y=k(x+4)(x)(x-6) y=1/f(x) y= 1/ (k(x+4)(x)(x-6)) The Attempt at a Solution I'm more looking for clarification on how people would approach this. There is no explicit point given to deduce the value of k to determine the vertical stretch or...
  6. Shahab Mirza

    What Angles Reveal Second Order Bright Fringes in Diffraction?

    Hi, 1. Homework Statement Q : A diffraction grating with 10000 lines per CM is illuminated by yellow light of wavelength 589 nm, At what angles is the 2nd order bright fringes seen ?Homework Equations From my textbook , I got this equation , d sin theta = m (λ) The Attempt at a Solution Ok so...
  7. C

    Rational functions: combine and simplify terms

    Homework Statement (4a/a+4)+(a+2/2a) Homework Equations Just combine and then factor out The Attempt at a Solution It's actually fairly simple, but I'm having difficulty at the end. /multiply each term by opposite denominator 4a(2a)/a+4(2a) + a+2(a+4)/2a(a+4) /combine 4a(2a)+(a+2)(a+4) /...
  8. Jaco Viljoen

    Without solving the equation show it has 2 rational roots

    Homework Statement Without solving the equation 3x^2-8x-3=0 show it has 2 different rational roots.[/B]Homework Equations ax^2+bx+c=0 The Attempt at a Solution I would appreciate if someone would check my work, and advise if I have done the right or wrong thing? Thank you, Jaco [/B]...
  9. J

    What does it mean to take positive rational numbers to whole

    What does it mean to take positive rational numbers to whole-number powers?
  10. X

    Calculus Michael Spivak 3th Edition Pag 42 Rational Function

    Pag 42 says: Seems inconsistent. in (9): p = x +x²+x.sin²x q = x.sin x + x.sin²x But by definition p and q are not polynomial functions. It is a mistake in the book?
  11. lep11

    Left-handed limit of a rational function

    Homework Statement What is Lim (1+x2)/(4-x) as x approaches 4 from the left? Prove using the definition. Homework EquationsThe Attempt at a Solution Well x≠4. Function approaches positive infinity as x approaches 4 from the left side. Let m>0 and 0<x<4. Then (1+x2)/(4-x) > x2/(4-x) > x/(4-x) >...
  12. Fallen Angel

    MHB Proving $\sin\left(10^{\circ}\right)$ is Rational or Irrational

    Is $\sin\left(10^{\circ}\right)$ rational or not? Prove it.
  13. Mr Davis 97

    Justification for cancellation in rational functions

    For example, say we have ##\frac{x^4(x - 1)}{x^2}##. The function is undefined at 0, but if we cancel the x's, we get a new function that is defined at 0. So, in this case, we have ##x^2(x - 1)##, then ##x^2(x - 1)(1)##, and since ##\frac{x^2}{x^2} = 1##, we then have ##\frac{x^4(x - 1)}{x^2}##...
  14. J

    What does it mean to take positive rational numbers to whole

    How do you do it
  15. S

    Finding the range of a rational function

    Homework Statement A curve is given by the parametric equations ##x=t^2 +3## ##y=t(t^2+3)## Find dy/dx in terms of t and show that (dy/dx)^2 >=9 Homework Equations Parametric derivatives The Attempt at a Solution Using the chain rule, I arrived at...
  16. Math Amateur

    MHB Every interval (a,b) contains both rational and irrational numbers

    I am reading Chapter 1:"Real Numbers" of Charles Chapman Pugh's book "Real Mathematical Analysis. I need help with the proof of Theorem 7 on pages 19-20. Theorem 7 (Chapter 1) reads as follows: In the above proof, Pugh writes: " ... ... The fact that a \lt b implies the set B \ A contains...
  17. A

    MHB Evaluating a rational function with contour integration

    Hello, I am looking to evaluate: $$I = \int_{0}^{1} \frac{x^4(1-x)^4}{1+x^2} dx$$ I will use a rectangular contour. The image looked weird here so the upload of the image is here: http://i.stack.imgur.com/W4BfA.jpg $R$ is more like the radius of the small semi circle, we have to let $R \to...
  18. A

    MHB Complex Contour integration of rational function

    Hello, Evaluate: $$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ We know that because $f(x)$ is even:$$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx = \frac{1}{2} \cdot \int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ Consider a complex function, with $z = x + iy$ $$f(z) =...
  19. P

    De Moivre's Theorem for Rational Exponents

    ##cosθ + isinθ = cos(θ + 2kπ) + isin(θ + 2kπ)## for ##k ∈ ℤ## ##[cosθ + isinθ]^n = [cos(θ + 2kπ) + isin(θ + 2kπ)]^n## ##cos(nθ) + isin(nθ) = cos(nθ + 2nkπ) + isin(nθ + 2nkπ)## ##cos(nθ + 2mπ) + isin(nθ + 2mπ) = cos(nθ + 2nkπ) + isin(nθ + 2nkπ)## for ##m ∈ ℤ## Now consider the special case ##n =...
  20. J

    What is the easiest method for adding and subtracting rational numbers

    What is the easiest method for adding and subtracting rational numbers
  21. B

    Gaps In The System of Rational Numbers

    I have a question which comes from Rudin's Principles of Mathematical Analysis; specifically, from the introduction. In example 1.1, the author clearly shows that no rational numbers satisfy the equation ##p^2 = 2##. So, I am trying to imagine myself in a scenario in which I am in a time before...
  22. B

    Pre-Calc. Question: Graphing Rational Functions

    When you have a rational function, such as: 3x-5/x-1 After attaining things like the x and y intercepts and asymptotes, how do you know how many "pieces" of the graph there are? With linear functions/equations, you know it's a single line. Even quadratic graphs are a single piece - albeit...
  23. P

    MHB Rational Expression word problem

    Nancy's lawn care specializes in residential lawn cutting and fertilizing. Nancy the owner has tracked her income and expenses. She has determined that her profit can be represented by the expression $5A, where A is the area of the lawn in square metres. The time in hours that it takes her to...
  24. J

    MHB Range of t in rational expression

    Evaluation of range of real values of $t$ for which $\displaystyle 2\sin t = \frac{1-2x+5x^2}{3x^2-2x-1}\;,$ Where $\displaystyle t\in \left[-\frac{\pi}{2}\;,\frac{\pi}{2}\right]$
  25. Y

    MHB Limits of Rational Functions: Dividing by Highest Power?

    Hello all I have a general question. When I look for a limit of a rational function, there is this rule of dividing each term by the highest power. I wanted to ask if I should divide by the highest power, or the highest power in the denominator, and why ? I have seen different answers in...
  26. G

    Are all rational numbers normal?

    This number is rational and normal, right? http://www.wolframalpha.com/input/?i=0.01234567890123456789... edit - You'll have to edit in the ".." because the forum doesn't recognize it as part of the link.
  27. nuuskur

    Antiderivative of a rational function

    Homework Statement Consider the integral: \int\frac{2x^3 -4x^2 +8x +7}{(x-1)^2 (x^2 +4x +8)}{\rm{d}}x Homework EquationsThe Attempt at a Solution The degree of the denominator is 4 and the numerator's is 3, hence I thought I would try partial fractions: \frac{A}{x-1} +\frac{B}{(x-1)^2}...
  28. evinda

    MHB At which p-adic fields does the equation have no rational solution?

    Hello! (Wave)I have to check if the equation $3x^2+5y^2-7z^2=0$ has a non-trivial solution in $\mathbb{Q}$. If it has, I have to find at least one. If it doesn't have, I have to find at which p-adic fields it has no rational solution.Theorem: We suppose that $a,b,c \in \mathbb{Z}...
  29. evinda

    MHB For which primes, does the equation have a rational solution?

    Hi! (Smile) I have to find for which primes $p$, the equation $x^2+y^2=3z^2$ has a rational point in $\mathbb{Q}_p$. According to my notes: Obviously, $\forall p \in \mathbb{P}, p \nmid 2 \cdot 3$, there is a rational solution in $\mathbb{Q}_p$. But,why is it obvious that the equation has a...
  30. T

    [Limit] Algebraic Manipulation of Rational Function

    My https://www.amazon.com/dp/0073532320/?tag=pfamazon01-20 gives a rule of thumb to divide by the highest power in the denominator for the following problem to demonstrate a slant (oblique) asymptote: \lim_{x\to\infty} \frac{4x^3+5}{-6x^2-7x} = \lim_{x\to\infty}...
  31. F

    Prove limit x approaches 0 of a rational function = ratio of derivatives

    1. The problem statement, all variables and given/known dat If f and g are differentiable functions with f(O) = g(0) = 0 and g'(O) not equal 0, show that lim f(x) = f'(0) x->0 g(x) g'(0) The Attempt at a Solution I know that lim as x→a f(a) = f(a) if function is continuous. since its...
  32. PcumP_Ravenclaw

    Square root of 2 divided by 0 is rational?

    Dear All, Please help me understand how ## \sqrt{2} ## divide by 0 is rational as stated in the excerpt from alan F beardon's book?
  33. J

    Partial Fraction Decomposition for Integrating Rational Functions

    Im trying to find the integral of ( sec(t)^2 ) / ( (tan(t)^3) + (tan(t)^2) ). I've managed to get the integral into the form 1 / (u^3 + u^2) where u = tan(t), however I am having difficulty proceeeding from there. Could someone take a look at the working out I have attached and let me...
  34. kaliprasad

    MHB Is the sum of two cube roots of irrational numbers rational?

    prove that $\sqrt[3]{45+29\sqrt{2}}+ \sqrt[3]{45-29\sqrt{2}} $ is rational
  35. H

    Adv. Functions: Rate of Change in Rational Functions

    Homework Statement mtan for f(x) = 5/√ 3x ... at x=1 Homework Equations msec = y2-y1 / x2-x1 The Attempt at a Solution The two points I got from the equation: (1, 5/√ 3) and (1+h, 5/√ 3+h) msec = f(1+h) - f(1) / h = (5/√ 3+h - 5/√ 3) / h ... multiply top and bottom by denominators (√ 3+h)...
  36. A

    Understanding limits of rational functions at infinity

    Is there a way to distinguish between rational functions that have the same limit at both ends and those that don't? I think I might have answered my own question, but let's say I evaluate a rational function, and it turns out to be a coefficient ratio with no variables (3/2). Does that mean...
  37. O

    MHB List all values of x for which each rational expression is undefined.

    List all values of x for which each rational expression is undefined: \frac{x^2-9}{x^2-3x-10} Answer is 3, -3 q1) Are these -, and positive answer interchangeable ones because it is a rational expression so when I see rational that's going to signal it's going to need 2 answers (either...
  38. P

    MHB Rational Number equations help

    I have a question in math I need help with please: Show that the product of a rational number and it's inverse is equal to 1, with one exception. what is the exception? can anyone help please?
  39. johann1301

    Is ln(√e^π)/π a rational number?

    is ln(√eπ)/π a rational number? where π =3.14...
  40. H

    MHB Simplifying a rational expression

    Hello, I am having difficulty solving my math problems. Simplify the expression: (12+r-r^2)/(r^3 +3r^2) The answer is (4-r)/r^2 I know that i can expand 12+r-r^2 as (-r+4)(r+3) But i cannot figure out the rest. Please help me. Thanks
  41. Dethrone

    MHB Piecewise function - rational and irrational

    $$g(x)=\begin{cases}x^2, & \text{ if x is rational} \\[3pt] 0, & \text{ if x is irrational} \\ \end{cases}$$ a) Prove that $\lim_{{x}\to{0}}g(x)=0$ b) Prove also that $\lim_{{x}\to{1}}g(x) \text{ D.N.E}$ I've never seen a piecewise function defined that way...hints?
  42. M

    Solve Rational Inequality: Find Integer Roots [-2, 3]

    Homework Statement Find all integer roots that satisfy (3x + 1)/(x - 4) < 1. The Attempt at a Solution I would do this: Make it an equation and find x such that (3x + 1)/(x - 4) = 1. 3x + 1 = x - 4 2x = -5 x = -5/2 Then check if the inequality is valid for values smaller than x and for...
  43. A

    MHB Simplifying rational expressions

    If its not problem for you to check these last two. I got no more of these. If you have some kind a book with tasks like this on internet I would love to print it out so I can have some fun. 1ST answer 20/3 2ND answer -20/3
  44. H

    The factorial of a rational number, the gamma function not used

    My first question is: is this formula (at the bottom) a known formula? In this subject i haven't explained how i build up the formula. So far i think it is equal to the gamma function of Euler with \Gamma\left(\frac{m_1}{m_2}+1\right)= \frac{m_1}{m_2}\ ! with m_1 , m_2 \in...
  45. kaliprasad

    MHB Rational: $(p^2+1)(q^2+1)(r^2+1)$ is Square

    Show that for $p,q,r$ if $pq+qr+rp=1$ then $(p^2+1)(q^2+1)(r^2+1)$ is square of rational number
  46. V

    Rational ratio of frequencies leads to isolating integral of motion

    Hello All, Padmanabhan's discussion of dynamics mentions that in general the two dimensional harmonic oscillator fills the surface of a two torus. He further notes that there will be an extra isolating integral of motion provided that the ratio of frequencies is a rational number. This last...
  47. loops496

    Proof of rational density using Dedekind cuts

    The Problem Let x and y be real numbers such that y<x, using the Dedekind cut construction of reals prove that there is always a rational q such that y<q<x What I've done Since I can associate a cut to every real number, let x^∗ be the cut associated to x and y^∗ the one associated...
  48. L

    Rational function that approximates e^x

    Is there a rational function,not series, that approximates e^x ? for example (x+1)/(x+3)
  49. D

    MHB What Does It Mean for Expressions to Be Non-Integral and Non-Rational?

    can you explain what it means when they are not "Integral" and not "Rational"? $\frac{4y}{x} = 4yx^{-1 }$ is not integral in x $3x\sqrt{y}z^3$ not rational in y
  50. N

    Lambert W function with rational polynomial

    Hi all, During my research i ran into the following general type of equation: \exp(ax+b)=\frac{cx+d}{ex+f} does anyone have an idea how to go about solving this equation? thx in advance
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