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.
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...
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...
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)| >...
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...
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 =...
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...
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 ##...
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...
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...
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##...
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
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...
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
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...
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...
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.
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}##...
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.
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...
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?
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...
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...
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...
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...
I attempted to solve it
$$ x = \frac {1}{4x} + 1 $$
$$⇒ x^2 -x -\frac{1}{4} = 0 $$
$$⇒ x = \frac{1±\sqrt2}{2} $$
However, I don't know the next step for the proof.
Do I need a closed-form of xn+1or do I just need to set the limit of xn and use inequality to solve it?
If I have to use...
Given : Equation ##x^2+(2m+1)x+(2n+1) = 0## where ##m \in \mathbb{Z}, n \in \mathbb{Z}##, i.e. both ##m,n## are integers.
To prove : If ##\alpha,\beta## be its two roots, then they are not rational numbers.
Attempt : The discriminant of the equation ##\mathscr{D} = (2m+1)^2 - 4(2n+1) =...
Let $x$ be a non-zero number such that $x^4+\dfrac{1}{x^4}$ and $x^5+\dfrac{1}{x^5}$ are both rational numbers. Prove that $x+\dfrac{1}{x}$ is a rational number.
First, I calculated the derivative of
$$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}=...
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...
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...
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?
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...
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.
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...
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...
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...
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...
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...
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...
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...
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...
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...