Rational Definition and 616 Threads

I am trying to show that some given simple affine algebraic varieties are rational (i.e. birationally equivalent to some A^k). Are there any tricks or even nice algorithms for finding the birational maps and their inverses? Examples are the curve x^2 + y^2 = 1 and the surface x^2 + y^2 + z^2 =...

Homework Statement The rate at which people enter an amusement park on a given day is modeled by the function E defined by: E(t) = 15600/(t² - 24t + 160). The rate at which people leave the same amusement park on the same day is modeled by the function L defined by: L(t) = 9890/(t² - 38t...

I'm stuck on how to do this problem My attempt: the answer to the second one is 3x + 2/(x+2)(x-2) Can someone point out my mistake?

can a non-factorable function has rational real roots?

Hi. I need help computing a range. The question is : Find the domain and range of y=\frac{|x+2|}{x}. The domain is obvious, x can't be 0, (-inf,0,) U (0,inf). But how do I find the range?? Can someone help me out? I have tried messing around with the definition of absolute value... if...

I know the set of positives rationals form a group under multiplication and that the negative irrationals do not form a group under multiplication because there is no identity or inverse. My question is does the set of all rational numbers under multiplication form a group.

Can anyone point me to a text or link that summarizes the rules when evaluating/simplifying an expression of the form (a^n)^(1/m) for a < 0. (a^n)^(1/m) yields different answers for a^(n/m) and (a^(1/m))^n . Ex: (-8)^(2/6) = (-8)^(1/3) = -2 (-8)^(2/6) = ((-8)^2)^(1/6) = 2...

Is the number 1, or any other whole/negative number a rational number?

My teacher gave us as excercices this: I'm pretty certain you have to prove it by contradiction, but I don't get how to represent to periodic decimal expension in a proof? Any hint is welcome, thanks in advance.

I don't get it... For which value of x is each rational expression not defined? a) 3/x are you supposed to solve for x? or?

Since pie is the ratio of the circumference of the circle to its diameter, isn't it possible that there exist a fraction for all nonrepeating going on forever decimal values?

4 The height, x metres, of a diver above a swimming pool at time t seconds after he has bounced from the diving board can be modeled by the function x(t)= 3- 3t (3t^2/2) a How long, in seconds, after he has bounced from the diving board does the diver reach his maximum height? b What is the...

Hello, Recently I have been trying to reason why certain rational functions such as (-2(2x^2+x-31))/((x-3)*(x+4)*(x+1)) have varying near horizontal asymptote slopes. I know that the direction of the horizontal asymptote can be varied by altering the parent function x+1/x, but several...

how to calculate the calculus of this rational function: \int \frac{dx}{{x+x^6}} could anyone get me through the solution?!

In a motorcycle race, one lap of the course is 650m. At the start of the race, Genna sets off 4 seconds after Tom does, but she drives her motorcycle 5m/s faster and finishes the lap 2.5 seconds sooner than he does. a) Find the speed at which each of them is driving. b) Find the tim etaken...

Hi I wanted to know what is the expansion of (1+x)^n when n is a rational number and |x|<1 ... Please let me know as soon as possible.. Thanks for your help Sincerely Sparsh

This is a question I have to do for tomorrow \frac{1+4x+4x^2}{x}\times\frac{3x^4}{12x^2-3}. I just factor the tops and then cancel where I can. This is what I came up with: \frac{3x^4}{12x^3-3x} But the answer key says: \frac{x^3(1+2x)}{2x-1} What have I done wrong?

Trigonometry made more "rational" I read an article the other day on "rational trigonometry" and decided to go exploring... and I feel as if a lightbulb has turned on. This seems like such a wonderful new technique to doing otherwise cumbersome problems, and I agree that it does feel like a...

I thought I knew what I was doing (because I've done this before) but I can't see what's happening in this one example...I understood what was going on right up until the last step. If you are dividing two rational expressions, flip the second expression and multiply the two rational...

Is e^pi rational? I seem to have heard from one of my tacher that research was going. How far we have gone?

I want to find a nice and elegant proof of the Rational Root theorem, but I get stuck. I read some stuff on the Internet, but I have not found a complete proof of the theorem. Here's my try: Say we have a polynomial: F(x) = \sum ^{n}_{r = 0} a_{n}x^{n} = a_{n}x^{n} + a_{n - 1}x^{n-1} +...

The question I have been given is essencially this: Give a brief description, with some explanation, of which rational numbers between 0 and 1 have finite decimal expansions? [An example is 3/40 = 0.075, however 2/3 = 0.66666666666...] I am truly :confused: Please help. I have looked...

Im trying to either prove or disprove that if a and b are rational numbers, then a^b is also rational. I tried doing it with a contradiction, but i can't seem to correctly arrive at a solution. this is how i started the problem defn of rational number: a,b = {m/n: m,n are all nonzero...

Sligtly more complex than the average one, I'd assume. How would I go about proving that the limit of of a rational expression consisting of two polynomials of the same degree goes to one and the limit of one where the degree of the bottom is greater than the degree of the top goes to zero. I'd...

okay i need help on rational functions I am not sure if that what they are called but it all should be the same.What i need to find is the holes zeros vertical asymtotes horizontal asymtotes and the slant asymtotes can someone please tell me how to find each on of those and the...

Right now in math class we are learning rational expressions. Since I am in an advanced math class, it seems like we learn a new lesson everyday. So if you don't understand something, you pretty much need to teach yourself. I don't really understand rational expressions, so can someone tell...

Dear Experts. Do you know some literature on representation of rational numbers by formulas? TIA.

Need help with proving: Show that (a,b) + (c,d) = (a+c, b+d) is not well-defined in the rational numbers. [Note: (a,b) + (c,d) = (ad+bc, bd) is well-defined because (a,b) is related to (c,d) when ad = bc.)]

what is a spline what is a b spline what is rational b spline what is a uniform rational b spline what is non uniform rational b spline(NURBS)

I was wondering if someone could check these questions to make sure that I am doing these correctly. I have attached a few questions in word document as I could not type it properly on here.

Given that {\left\{ {\left( {a,b,x} \right)|0 < x < \left( {b - a} \right),\left( {a,b,x} \right) \in \mathbb{Q}^3} \right\}} , Can I list all the rational numbers in \left[ {a,b} \right] as the sequence represented by a + \mathop {\lim }\limits_{x \to 0} x\left\{ n...

prove that between any two real numbers there is a number of the form \frac{k}{2^n} where k is an integer and n is a natural number.

I need some closure on the following, Question: Prove that if a linear system of equations with only rational coefficients and constants has a solution then it has at least one all-rational solution. Must it have infinitely many? My Solution: If the RHS of the equations is a rational...

I am asked to list the positive rational numbers in one list so that the pattern of the order is clear and so that all the positive rational numbers would eventually appear on the list. Then I have to explain the pattern and why every positive rational number will eventually be on the list...

I've been grinding my brain at this problem because I am trying to figure out if the product of two rational numbers is always, never, or sometimes rational. a rational number would either have to terminate, or be infinitely periodic, so i would say that the product of two rational numbers is...

Can someone prove that there exists x and y which are elements of the reals such that x and y are irrational but x+y is rational? Certainly, there are an infinite number of examples (pi/4 + -pi/4 for example) to show this, but how would you prove the general case?

Integrals of Rational Functions... The integral of:... (x-1)/x^4+6x^3+9x^2 , dx...i factored out the bottom getting: x^2(x+3)(x+3)...so, my new integral is: (x-1)/x^2(x+3)^2... now when i muiltlpy both sides by (x-1)/x^2(x+3)^2...i get... x-1= A(x+3)^2 + Bx^2(x+3) + C x^2...for A i got...

heres a problem i stumbled: find three rational numbers a,b,x such that: x^2 + 5 = a^2 x^2 - 5 = b^2

Or something like that... I need definition,, explanation and examples. I have an exam in Rings and Fields on Sunday, and he used that term during the course- I have no idea what it is. I'd appreciate any help. Thanks in advance!

How do I convert a piece wise function like this: y[n]= {1 -N<=n<=N {0 otherwise to something like this: p0+p1e^(-jw)+...+pMe^(-jwM) ------------------------------ d0+d1e^(-jw)+...+dNe^(-jwN) Basically, what formula do I need to use to calculate the coeffients? [Edit] I'm...

Ok I have to find the Lowest Common Denominator for 3 ration expressions. I don't think the numerators are important so Ill just leave them out. The denominators are x^2 - 4 and x -2 I got the LCD as x-2. correct? When finding the LCD in expressions like this you just have to factor...

Suppose 2 cuadratic functions: ax^2+bx+c, dx^2+ex+f. Suppose that the first one is upside with its minimum above the x line reference, and the second one is downside with its maximum above the x reference, and suppose that the two functions intersect at two points that pass through straight line...

I need to show that a rational + irrational number is irrational. I am trying to do a proof by contradiction. So far I have: Suppose a rational, b irrational. Then a = p/q for p, q in Z. Then a + b = p/q + b = (p + qb) / q But I don't know where to go from here because I still have a...

How do I solve this integral?? (it's in the attachment) (my-efforts.gif just explains a failed attempt)

Hello, I doubt it, but is there any number system (base) where π or e run out of digits?

state the vertical asymptote(s), x-intercept(s), y-intercept(s), domain and range of f(x)=(1)/(4x^2)-1 ok I factored the denominator and got (2x+1) (2x-1) I solved for x, so the x-intercepts are x=-1/2, and x=1/2 for only (4x^2)-1 the reciprocal function has no x-intercepts. Sub in x=0 to...

Ok, so before every says that I am just looking for answers, I would like to say my teacher has been on "stress leave" for the past 5 weeks and we have been tought by a very bad "math" teacher Im having a hard time with Rational Equations, I guess because I have no notes on it :cry: So I am...

Allright, I have a math test next week and there's some things i want to clear up before i take it. I can't use a calculator btw. 1) We've learned a lot about testing symmetry (y-axis, x-axis, origin). It seems on most problems we just test y and origin symm. When do you know you shoudl test...

When is x rational and irrational? Also When is r positive and negative? :confused:

What's actually the definition of a^{m/n} where m and n are integers and a is any real number? Suppose I define it as the n-th square root of a^m. Wouldn't it be inconsistent with other stuffs? What stuffs? For example, a^1 is supposed to be a. But 1 = 2/2 and, using my earlier definition...