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.
Homework Statement
Here's the question: int(1/(x(x^2+3)*sqrt(1-x^2)))dx
Homework Equations
According to the textbook, the answer should be: (1/3)*ln((1-sqrt(1-x^2))/x)+(1/12)*ln((2+sqrt(1-x^2))/(x^2+3))+C
The Attempt at a Solution
1) let t = sqrt(1-x^2), so dt = (-x)/sqrt(1-x^2)...
Homework Statement
I have a picture of a rational function and I need help finding out the equation of the function.
Homework Equations
Vertical Asymptote/ (Bx+C)
The Attempt at a Solution
I know I have to find X and Y intercepts, which is what I did. But how do I find the...
prove that the square root of 2 plus the square root of 3 is not rational?
does always the sum of two not rational numbers is a not rational number?
i know the proof 2 = a^2/b^2
i separately proved that square root of 2 and square root of 3 are irrational
how two prove that the sum...
My textbook says the following:
For a closed interval J_n = [a_n, b_n]
"A nested sequence of rational intervals give rise to a separation of all rational numbers into three classes (A so-called Dedekind Cut). The first class consists of the rational numbers r lying to the left of the...
Consider the set of rational numbers, under the usual metric d(x,y)=|x-y|
I am pretty sure that this space is totally disconnected, but I can't convince myself that the set {x} U {y} is a disconnected set.
It seems obvious, but I can't find two non-empty disjoint open sets U,V such that U...
Hello.
Between a rational and a irrational is there a rational? and a irrational? and vice-versa?
I know that between 2 rationals there is a rational and a irrational and that between 2 irrationals there is a rational and a irrational, but i cannot figure this out... please help.
Thanks.
Hello!
I've been solving a few of these problems but I'm stuck on this one, trying to simplify one of the steps.
Homework Statement
Find dy/dx of: y = x(x^2 +1)^1/2
Attempt at a solution
y1 = x (1/2)(x^2 + 1)^-1/2 * (2x) + (x^2 +1)^1/2 * 1
I get the the equation above but I have...
Homework Statement
prove that it is possible that an irrational number raised to another irrational, can be rational.
you are given root2 to root2 to root2
Homework Equations
The Attempt at a Solution
i have shown that root2 to root2 to root2 is rational, but would appreciate a...
Cross multiply ...
and then multiply by the common denominator...
how do you do it
do you do it in your head..by cross multiplying
or the other way
It seems like cross multiplying is less formal ...and I don't know if its 'right' to take the shortcut --since you have to do this...
1. Let a and b be rational numbers. Prove or provide counterexample that
A) a+b is a rational number.
B) Is ab necessarily a rational number?
2. How can you prove that the sume of two rational number is rational? Well I am not really good at math
3. This is what I've tryed to...
please... i need a help in integrating the partial fractions
i can't proceed to the integration part if i don't understand the patter in finding the constant...
that is...
if the given is:
ʃ ( (x^5+1) / ((x^3)(x+1)) )dx
then;
ʃ ( x-2 + ( 4x^3+1 ) / ( x^4 + 2x^3) )
ʃ ( x-2 + (...
In my calc class we are reviewing rational and polynomial functions before we start with the actual calculus part of the course.
In my book we had 3 problems that we had to do for homework and none of my classmates could understand why the book answered them a certain way.
Question:
State...
I've programmed an algorithm to numerically compute the logarithm of numbers in phinary base easily. I could avoid float multiplications if I can find a pair of rational numbers x and y such that
x^{\sqrt{5}}=y
Is it possible?
Probably not, but I cannot prove it :(
Homework Statement
∫1/ x^3-1 dx, ok how would i do this
Homework Equations
∫dx/ x^2+a^2= 1/a tan^-1 (x/a) +c
i tried to simplify x^3-1 = (x+1)(x-1)(x+1)
It can easily be shown that the recurring decimal x = 1.123123... is rational, as follows:
10^{3}x-x = 1123.123...-1.123123...=1122 => x = \frac{1122}{999} \in Q
Show that the recurring decimals 0.3712437127... and 0.9999999...are rational numbers.
3. The Attempt at a Solution...
1. x/20 = (3/8)-(4/5)
2. solve
3. My attempt as far as I can tell there is no LCM so
3/8 becomes 15/40 4/5 becomes 32/40
(15/40)-(32/40)= 17/40 which equals 8.5/20 which means x=8.5
For some reason I don't think I got the right answer?
I've been working through "A Course of Pure Mathematics" and there is one problem I'm really stuck on. I'm wondering if anyone could help me out. To avoid typing it all out, I here's a link...
Hi, this is a question to the members with some knowledge in algebraic geometry:
1. what are rational curves with self-intersection -2? How do they look like?
2. do you know why these correspond to the vertices of some of the Dynkin diagrams?
3. just something that's bothering me...
I've been thinkng about this one for a while. Is i rational or irrational. i is an imaginary number, so logically, it would be irrational. But \frac{-1}{i} = i so it has a fractional equivilant. But then, it doesn't have a real number decimal equivilant...
So, what is it? Is i rational or...
and, consequently, infinitely many.
I am new to proofs so could you please check if this proof is correct?
Let x be an irrational number in the interval In = [an, bn], where an and bn are both rational numbers, in the form p/q.
Let z be the distance between x and an, So:
x - an = x...
Please tell me one of the bases for the infinite dimenional vector space - R (the set of all real numbers) over Q (the set of all rational numbers). The vector addition, field addition and multiplication carry the usual meaning.
Homework Statement
I have volunteered to help a friend's son with his Algebra 2 (thinking - no problem, I've had Calc 1-3, differential equation, complex variables, probability / stats and so on.
So I start to help and the first questions:
Why aren't these rational expressions...
Homework Statement
Suppose there are two polynomials over a field, f and g, and that gcd(f,g)=1. Consider the rational functions a(x)/f(x) and b(x)/g(x), where deg(a)<deg(f) and deg(b)<deg(g). Show that if a(x)/f(x)=b(x)/g(x) is only true if a(x)=b(x)=0.
Homework Equations
None
The Attempt at...
I want to integrate:
1/[(x + 1)*(x^2 + x +1)] dx
Now the quadratic has complex routes, and we have not done any integration with that yet, so I broke it up into its partial fractions.
A/(x +1) + (Bx + C)/(x^2 + x +1)
But I cannot seem to find the numbers A B C. mamybe I am just...
Homework Statement
Find asymptotes for f(x) = (x^2 -1) / (x - 1). (if exist)
Homework Equations
g(x) is a (horizontal or oblique) asymptote if lim |f(x) - g(x)| = 0
(here, lim is to be limit as x goes to infinity. don't know how to type it)
or
if q(x)/p(x) = g(x) + r(x)/p(x)...
1) Prove that the acute angle whose cosine is 1/10 cannot be trisected with straightedge and compass.
...
I worked it out and at the end found out that , if I can prove that the cubic polynomial 40x3 - 30x -1 has no rational roots, then I am done.
Now, is there any way to prove (e.g...
Homework Statement
Is there a non-empty perfect set that contains no rational number?
Homework Equations
None
The Attempt at a Solution
I thought the answer was no, but my professor said that there is. My reasoning is as follows (please let me know if I'm wrong here):
If p is an...
Homework Statement
Show that the square root of 3 is not rational
Homework Equations
The Attempt at a Solution
A number is irrational if χ is not ε. Q=p/q: p, q ε z and q is not=0, z=integers
If p/q: p, q is not ε or q=0, then square 3 is rational. If p=square root of 3 and q...
I'm a new comer, even in Math. I need hands for this (simple, may be for most people) question:
Can a subset of Rational Number Set be open (and closed)? If does, how can it be? If not,why?
thks!
Ka Yan:smile:
Homework Statement
\frac{5(y-2)}{y+1} x \frac{y+1}{10}
Homework Equations
The Attempt at a Solution
Does this equal 5(y-2)(y+1)/10(y+1) ? Or are there no brackets on that first y+1 ?
First Question
If: f(x) = (x^2+1)/x
Then: f(x) = x + (1/x)
From my understanding, x would be the oblique/slant asymptote. Why is that?
Second Question
Why and how can horizontal asymptotes be crossed?
Homework Statement
Factor the following over the set of rational numbers. Simplify if possible.
cos³ x-1
I do not know how to deal with the cubic cosine. Help is greatly appreciated.
Homework Statement
Determine all positive rational solutions of x^y=y^x.Homework Equations
The Attempt at a Solution
Obviously, x=y will always work. I think that is the only solution. If I can show that x^y must be rational, I think it will be easy because then both x and y must have the same...
i'm trapped with a problem: \int\frac{dx}{x\sqrt{2-x-x^2}}.
i think this problem could be solved by subtitutions: \ x+\frac{1}{2}=\frac{3}{2}sint and \ u=tan\frac{t}{2}.
and finally we would get an expression in \ u: \frac{\sqrt{2}}{4} log\left|\frac{2\sqrt{2}+u-3}{2\sqrt{2}-u+3}\right|
(am...
[SOLVED] functions on rational numbers
Homework Statement
Find all functions from Q to Q which satisfy the following two conditions:
i)f(1)=2
ii)f(xy)=f(x)f(y)-f(x+y)+1 for all x,y in Q
Homework Equations
The Attempt at a Solution
I can show by integers that if x is an...
Homework Statement
Queuing Theory (study of lines for stores) says that for a drive through window at a Macdonalds, the function
f(x)= 9/(x(x-9))
represents the average time in hours a customer will wait in line. X=average number of people an hour.
How long will a customer have to...
Hello, this question is for anyone who is kind enough to shed some light.
I am not actually taking a physics class currently, but a philosophy of science course. One of the guest lecturers we've had this semester spoke on QM; EPR and Bells. My question is basically this, I am not doubting...
Homework Statement
Find two constants for 'a' and 'b' such that the verticle asymptote will be \pm \frac{3}{5}
y=\frac{ax^2+7}{9-bx^2}
I rearranged so that it becomes -bx^2+8 in the denominator since i know that there are two roots that are \pm it must be a square and since 3 is the...
I have trouble with Graphing rational functions, please help me,test is tomorrow
I do not know how to use horizontal , vertical , oblique asymptotes to graph a rathional functions.
like y=2x+3+3/x+1;
y=x^2-4/x-4
thank you very much
if two rational numbers added together is still rational then wouldn't an infinite sume of rational numbers that converge also be rational and if that is the case then an irrational number is therefore rational which makes no sense though. i don't see where the flaw in this lies because it is...
This isn't really a question about homework specifically, it's more just that I don't understand part of my chapter...I am just starting Principles of Mathematical Analysis by Ruben...
Here is what I don't understand:
It is proving that p^2 = 2 is not satisfied by any rational p. And it...
Homework Statement
Proved that the set of all rational numbers of the form 3^m *6^n are integers , is a group under multiplicationHomework Equations
No equations for this particular proof
The Attempt at a Solution
Assume that all rational numbers are in the form 3^m *6^n . Therefor 3^m*6^n =...
Hi,
here is the question, if A is a closed set that contains every rational number r: [0,1], show that [0,1] is a subset of A.
But, how could A be closed? If A is closed, R^n-A is open, so any point in R^n-A would have a open sphere around it and this open sphere wouldn't intersect A...