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. 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...
2. 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...

34. 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...
35. 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...
36. 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?
37. 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...
38. Proving rational surd inequalities

my attempt, i am not good in this kind of questions ...i need guidance.
39. 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.

41. 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...
42. 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...
43. 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...
44. 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...
45. 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...
46. 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...
47. 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...
48. MHB Is 2√(7)+4 an Irrational Number? Exploring the Proof of Its Irrationality

See picture for question and answer.
49. MHB Determine if (27/4)/(6.75) is a whole number, natural number, integer, rational or irrational.

Let Z = set of real numbers Determine if (27/4)/(6.75) is a whole number, natural number, integer, rational or irrational. I will divide as step 1. 27/4 = 6.75 So, 6.75 divided by 6.75 = 1. Step 2, define 1. The number 1 is whole or natural. It is also an integer and definitely a rational...
50. I Error(?) in proof that the rational numbers are denumerable

If someone can straighten out my logic or concur with the presence of a mistake in the proof (even though the conclusion is correct, of course), I would be much obliged. I’m looking at the proof of the corollary near the middle of the page (image of page attached below). I simply don’t find...