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Don't I get to sleep at all?
Listen to MathIsFun. You aren't defining set A correctly. Set A is _NOT_ [0, 255]. Set A is all of the possible combinations between two sets which are [0, 255]. Set A contains 65,536 elements.
What are you creating nurbs in and for?
The only way I would know is to linearly search each nurb. But I'll ask some friends I have, see if they can offer any advice.
rickyoswaldiow, that only works for 3.
f(x, y) = x + y
f(x, y) = 15, what are x and y?
You used excel to find the number of permutations for each sum, from 300 to 2997? How, and how long did that take?
And is it permutations or combinations? Order doesn't matter, does it?
You are correct, f(x, y) is different than f(x) and f(y), but so is the explanation.
Remember, I never named the sets I was talking about in my explanation. This is because it applies to every set and every function between those sets.
Let A = {(x, y) | x ∈ [0,255], y ∈ [0,255]}, the domain. A is a set of ordered points, just like those in your function. How many elements are in A? 255*255.
Let B = {z | z ∈ [0, 256]}, the range.
Now apply my previous post.
Good question, by the way.
Move it, but don't delete this thread, and place a "Thread moved to here: " link for this thread.
But then it said some functions can't be integrated, like e^-2x dx. But the fundamental theorem of calculus garentees e^-2x dx has an antiderivative. Seems like a self defeating statement. If there is no function that can be differentiated to get e^-2x dx, then how can it have an antiderivative?
If you mean e^x^2, then you're going to love this one. There is an antiderivative of e^x^2. But we can't integrate it. There is a function of it's antiderivative, but it was a function that was unknown until we were investigating the antiderivative of e^x^2. Here is an example you will understand:
That is the defintion of ln(x). The same thing happens when you take the integral of e^x^2. The function is defined by that integral.
I believe I was partially wrong in the above. I think the farther it is from the line, the more pull a single point has on it. Although I'm sure you can implement it both ways.
As for finding the intersection, there isn't any formula you can use. That's because there is no formula for a cubic nurb, besides knowing where the control points are.
irspow, there is nothing wrong with arguing. In fact, that's normally how progress is made. If you disagree, then please, disagree with me. I can be just as wrong as anyone. The way I was taught is to take only the positive unless otherwise stated. In my experience, it has worked out. If you can find an example where it doesn't (an answer is missed), then post it here.
But seriously, never take what anyone says at face value unless you agree with it.
I believe the standard is that the √x is always positive. If you want both roots, you simply put ±√x. If you take the root in the middle of a problem, you must put the ± in:
x² = y
±√x² = ±√y
±|x| = ±√y
x = ±√y
What he means by cubic nurb is different than a cubic polynomial.
A nurb is a straight line with control points. The control points work like gravity. The closer it is to the line, the more it pulls it. So placing a control point above a line will make it arc up, almost like -x^2. Placing one above and below the line gives you a cubic nurb, it looks like a cubic polynomial.
You can move the control points to where ever you want, and it thus makes it easy to draw specific curves. Very useful for modeling surfaces of things like cars.
Sure is. Like I said in that other post, x^2 isn't a 1-1 function. Since it's not 1-1, you can't figure out what x was from y.
Ricky, isn't the problem really that there are so many instances in which we use the identity √x² = x and it winds up working for our purposes anyway.
If √x² = x works, then so will √x² = |x|. Only one of them is correct however...
Technically, doesn't √x^2 = x and-x?
I think you meant the piece-wise function:
√x^2 = x if x > 0
√x^2 = -x if x ≤ 0
In which case, you'd be right.
Ah, you're right. I completely forgot about roots.
You can always simplify, but in this case, you just simplified wrong.
√x² ≠ x, this is never valid, whether dealing with limits or not.
√x² = |x|
Scratch all of that. What you are doing is dividing xπ/yπ, in which both pi's cancle out. The only time there are no pi's (starting to get a bit hungry), is when x=0 or y=0. If x=0, xπ/yπ = 0. If y = 0, you don't have a real value. So every number is rational.
Thanks irspow. That is certainly one of my options. I'm not quite sure what to do yet. At least I have a few years to decide.
People who use degrees are just afriad of Pi.
Heh, just kidding of course. Degrees works just as well.
sin(x)/sin(y) is only rational when x = 180*n or x = 360*n*y, where n is an integer.
These are fairly in depth things. I will try to explain it the best I can, but I won't be surprised if you don't understand. These topics take quite a while to fully grasp. If you want to look up more information for yourself, this is known as Set Theory and more specifically, the Pigeon Hole Principle. If you have any questions, just ask.
You have sets A and B. A is the domain, B is the range. Also keep in mind that |A| means the number of elements in set A. 1-1 means:
For all x∈A and all y∈A, if f(x)=f(y), then x=y.
What this basically means is that every domain value goes to one range value, and no two different domain values go to the same range value. Your function must be 1-1 to work, otherwise, you wouldn't be able to tell the first two numbers from the third.
An example of a non 1-1 function is f(x) = x^2. (-3)^2 = 3^2, but -3 ≠ 3. An example of a 1-1 function is f(x) = x. Because if f(x) = f(x), x = x, which is always true for any value x.
If a function is 1-1, than the number of values in the range has to be greater or equal to the number of values in the domain. |B| ≥ |A|
This is no problem. But you wish your function tho be onto:
For all b∈B, there exists an a∈A such that f(a) = b.
What this says is that for every single value in the range, there is some value in the domain which goes to it.
An example of a non onto function is (again) f(x) = x^2. There is no value in this function which goes to -1. f(x) = -1 just isn't possible. On the other hand, f(x) = x is onto. You can find any real value in the range.
If a function is onto, than the number of values in the domain must be greater than or equal to the number of values in the range. |A| ≥ |B|.
Put 1-1 and onto together, and you get |A| ≥ |B| and |B| ≥ |A|. This means that |A| = |B|. However, for you, your domain is 256*256 and your range is 256. 256*256 ≠ 256. Therefore, it is impossible to make a function that is 1-1 and onto.
No soorejmg, that's not what I said. It will _NOT_ exist. There is _NO_ such function.
Is there any method by doing some combination of different AND OR operations or any such operations on the two numbers and later then doing some other steps to obtain the two numbers from the single number??
soorejmg, no function exists, has ever existed, or will ever exist.
How about this? I don't know if this is for a computer program or not, but I think that it can be adapted for any such situation.
a + (b/1000) = c
Any answer will produce a.b
Sorry irspow, it can't be. There are a few reasons. The first, is that there is a limitation. a, b and c can only be from [0, 255]. That's because a, b, and c can only be 8 bits each. A float (decimal value number) is 8 bits as well, but it can only store just as many numbers as an 8 bit integer can.
Further more, 1000 can not be stored exactly in binary. So you will get rounding errors. It will be close, but not exact.
Oh, is there C++ code that mimmicks trigonomic functions? I will need them if I am going to use the computer to do this one.
#include <math.h>
http://www.cplusplus.com/ref/cmath/
R = [2sin((90n - 180)/n) / sin(360/n)] - 1
sin(x)/sin(y) is only rational when x = 0 or x = 2Pi*n*y, where n is an integer. It should be easy to solve from there.
C = A + B×256
Nicely done. I'm assuming you, MathsIsFun, know why this works, but for an explanation on why it works, all you need to know is bit shifting. Bit shifting is the way all multiplication works. If you shift all the bits in a number:
01000100
10001000
By one place, you multiply it by 2. If you do so by 2, you multiply it by 4. 3, 8. 4, 16. And so on. There are 8 bits in a char (which is [0, 255]) and 2^8 = 256. So multiplying it by 256 shifts it 8 bits. So for example:
00000000 01000100 (space is there just for readability)
Becomes:
01000100 00000000
Adding another 8 bit (again, [0, 255]) value:
01000100 00000000
+00000000 01100100
-------------------------
01000100 01100100
And thus, it is just like putting each number side by side.
Dnt u have any idea like in puzzles some tricks are there to find out number by special operations all....any idea like that in this case?
I can prove through the use of the pigeon hole principle, that it is not possible if the function must be onto (valid for every value of C). Once you prove something impossible, there are no such tricks.
Assuming that you are programming this...
unless |X| means absolute value which I am not sure it will be recognized on a computer.
Absolute value is extremely easy to do on a computer. All it requires is to switch the very first bit of a signed number and then subtract that from the highest possible signed value. Most languages support an abs() function, but it also works if you just do: x = (unsigned int) x or something similar.
But if you're on a computer, the above isn't needed.
if (X < 0) return -1; return 1;
x³ / ((x²)^9)^(1/6)
That will work but I don't know how you would do it in computer language.
Pretty easy actually:
(x*x*x) / pow((pow(x, 18)), 1/6)
If your language doesn't support a power function, you can write it yourself, although doing non integral powers is a bit tricky. But, as I said, you don't need to do all this.
This is a really good example on why Numerology is baloney. Nice work guys.