Does not exist. Or, if you want to try and think about it in some graphical sense, you could think of it as being both +/- infinity, and so is not well defined, as so does not exist.
None of this has anything to do with limits. The answer is also not infinity, so stop saying that people. Infinity is not a number.
Let me explain something about solutions to linear equations. There are three possible sizes for a set of solutions to an equation: Either no solutions, one solution, or infinitely many solutions. When there is exactly one solution, we say the relation is well-defined; when there are no solutions or infinitely many solutions, the relation is undefined.
Let me give you an example for each:
2*x=6 <- There is only one solution: x=3.
0*x=6 <- There are no solutions, because no matter what value for x you put, this equation will never be true.
0*x=0 <- There are infinitely many solutions, because any number for x makes this equation true.
The reason we say the first solution is well-defined is because we can just write it: x=3. x has exactly one possible value.
The reason we say the last two are undefined is because we can't just write x=something; there isn't one something. You have to express the answer in terms of sets. The solution set of x=0/0 is everything, or {x|xeR}, whereas the solution set of x/0 for x!=0 is the empty set.
This is why we say you can't divide by zero, because you don't get just one answer.
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Anonymous2006-05-16 16:13
division with 0 is completely different than multiplication with it because it has no multiplicative inverse. your use of equal signs is flawed, especially in your statement x=0/0 where x is not a set, but an element of it, implying that there is some set such that all x in R is a solution to 0/0 implies that 0/0 = x1 = x2. x/0 is not well defined for any x.
also, there are also equations with other finite amounts of solutions. as such, your statement that there are only three possibilities for the size of the set of solutions leads me to believe that you may have skipped over algebra 1 in your math studies. e.g.: quadratic, cube roots.
theoretically, there could exist a polynomial equation of infinite degree with an infinite number of solutions, such that not all x in R solves it.
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Anonymous2006-05-16 19:48
>>10, he said linear equations. Quadratic, cubic != linear.
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Anonymous2006-05-16 22:14
y kant u red
"Let me explain something about solutions to linear equations."
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Anonymous2006-05-16 22:27
Division is a method of solving those equations, but that isn't its definition. Goddamn. Go learn about functions.
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Anonymous2006-05-17 0:04
>>10
I quite clearly specified LINEAR equations. Division by zero is linear. My equal signs are correct everywhere except x=0/0, I meant to write 0*x=0 again.
Accusing people of skipping their math classes does not help your argument.
>>10
You can't just use linear functions to prove such a general statement. That is so basic and flawed reasoning. As >>10 said, multiplying with zero is totally different with dividing by 0. If you learnt functions and limits, you would know that by dividing any polynomial by zero, you would get a vertical asymptote. Thus it has an infinite range which would be undefined for a particular x value(s). It has EVERYTHING to do with limits.
>If you learnt functions and limits, you would know that by dividing any polynomial by zero, you would get a vertical asymptote.
What the hell is it with people accusing me of not having learned math?
>>16
My statement was hardly meant to be any sort of proof, but yes, you can use linear functions to prove this statement because division is defined in terms of multiplication.
Division by zero has NOTHING to do with limits. For example the limit of x=0/a as a->0 is zero. Is zero the answer? Of course not. Limits tells you absolutely nothing about the actual value at a point; the entire purpose of limits is to evaluate how the function converges to that point, and division by zero has nothing to do with convergence.
>>18
we agree with you, we're just pressing for more information so its more justified.
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Anonymous2006-05-17 20:39
Inverse of multiplication, yes. Solution set to an equation, no.
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Anonymous2006-05-17 21:02
unless you define this division with a limit it is meaningless (in the usual sense of division which is the inverse of multiplication on the real line). if you do, the answer depends on what it is you're dividing... you can get anything.
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Anonym 2006-05-18 1:13
I think it has to do something with Za Warudo....
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Anonymous2006-05-18 6:32
>>18
On the contrary. They are using limits to describe how it is impossible to divide by 0. This is because it will appear to converge to a certain x-value, but then will suddenly become a vertical asymptote. Thus, there is an infinite limit. That is +-infinity. The very fact that it does not converge to a specific y-value, proves that it is impossibe to divide by zero.
Also, if you take an inverse of a function, you will effectively change its range and domain. Thereby your explaination with linear function is extremely inadequate. Consider dividing higher polynomials or rational functions by 0 (something which is easily described by limits).
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Anonymous2006-05-18 7:38
>>24
"The very fact that it does not converge to a specific y-value, proves that it is impossibe to divide by zero."
Wrong. Limits have nothing to do with the actual value at a point. For example, take the piecewise function x = {1 if x is rational, 0 otherwise}. The limit does not exist ANYWHERE, and yet the function is defined EVERYWHERE. Convergence has nothing to do with whether or not the function is defined.
There are infinitely many examples of functions which are defined at points where the limit does not exist. Take the unit step function for example. Just like ordinary division, the limit does not exist at zero; it does not converge to a specific y-value. Yet it's still defined there. Its definition at zero has nothing to do with limits.
"Also, if you take an inverse of a function, you will effectively change its range and domain. Thereby your explaination with linear function is extremely inadequate."
How so? Alright, let me explain it with domain and range then. Multiplication of x by zero has the trivial range {0}, meaning the entire domain is in the kernel. This means it's not injective, hence it's not invertible. What did any of that have to do with limits? And what do higher polynomials or rational functions have to do with this?
>>25 Converge means goes towards some specific number, which is just another way of describing what a limit is. If something converges to a specific y-value, it tends to that value as the distance decreases. So don't go accusing other people of being wrong then crapping on about irrelevant stuff.
>>25
"The limit does not exist ANYWHERE, and yet the function is defined EVERYWHERE."
What the hell? If a function is defined everywhere, then as x heads towards a particular number, the limit will easily evaluated.
ie. lim(x->1) (3x+5) = 8
"Take the unit step function for example."
Er. The reason why a limit does not exist is because the right hand limit, doesn't equal the left hand limit. This is the same case with asymptotes. When dividing by 0, the left hand limit equals +infinity, whilst the right hand limit equals -infinity.
"Also, if you take an inverse of a function, you will effectively change its range and domain. Thereby your explaination with linear function is extremely inadequate."
True. When you take an inverse of a function, it's range and domain swap. Generally range becomes domain, vice versa.
Please read up on the Wikipedia link you suggested us.
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Anonymous2006-05-18 14:14
>>30
just because a function is defined everywhere doesn't mean it has a limit in any given place.
e.g. something like: f(x) = {x when x!=1; 923 when x=1}
i think youre thinking of continuity
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Anonymous2006-05-18 15:16
>>30
Wrong. >>25 gave an good example of a function where the limit does not exist but the function is well defined.
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Anonymous2006-05-21 11:52
>What the hell? If a function is defined everywhere, then as x heads towards a particular number, the limit will easily evaluated.
ie. lim(x->1) (3x+5) = 8
Okay genius, if you think limits exist everywhere for the piecewise function f(x) = {1 if x is rational, 0 otherwise}, then tell me: what's lim{x->5} f(x)?
"When dividing by 0, the left hand limit equals +infinity, whilst the right hand limit equals -infinity."
Again, this has nothing to do with it. There are certainly examples of functions which have limits that tend to infinity but are defined. Heck you can just define anything piecewise, for example take g(x)={0 when x=pi*n-pi/2, integer n; tan(x) otherwise}. The limit of this function does not exist at any pi*n-pi/2, because the left and right hand limits go to + and - infinity, just like division by zero. However it IS defined at the point.
"Please read up on the Wikipedia link you suggested us."
What the hell? The Wikipedia link I suggested says exactly what I've been saying. Limits have absolutely nothing to do with why it's undefined, and you can't use limits to remove the singularity.
I can't explain it any more clearly than this. Limits have nothing to do with division by zero.
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Anonymous2006-05-21 12:01
>>30
As >>31 said, the reason lim(x->1) (3x+5) = 8 because the function is continuous at x=1; this means the limit at 1 matches the value at 1. If you were asked to prove this on a test on elementary function theory, writing
>>37 literally meaning "how many of this item is in zero of this item"
No, there is no literal meaning. "/" is a well defined mathematical operator. The only thing undefined is what you get when you divide by zero.
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Anonymous2006-05-26 18:58
So you have a pie weight A, and you have B people.
If you evenly distribute the pie by weight, how much does each person get? A/B. In general, A/B < A, however if you have fractional people, A/B > A
Dividing by 0 is like having 0 people.
So, here we have a pie of size A, and we have 0 people.
If you evenly distribute the pie by weight, how much does each person get?
Lemma: If a tree falls in the forest and no one is there to hear it, it makes a detectable sound.
A falling object will inevitably have an impact, at which point kinetic energy will be transfered into different types of energy. A minute amount of this energy is involved in an exchange of momentum, resulting in movement of the earth. This movement will create extremely low frequency sound around the planet. This sound exists, and is therefore potentially detectable, with sufficient tools.
Similarly, we have a pie that is cut, but no one is around to eat the pieces. The pieces exist, and may be larger than the whole, but they are edible given sufficient means.
Contradiction.
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Anonymous2006-05-26 20:13
Wrong.
sound (n.) 1. the particular auditory effect produced by a given cause; "the sound of rain on the roof"; "the beautiful sound of music"
Sound is defined by perception. So the answer is no, the tree makes no sound. Yes, the air vibrates and all that jazz, but the fact that no ears interpret the vibration means there is no sound. Don't blame me, blame the dictionary.
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Anonymous2006-05-27 1:55
wrong, you.
your definition is ill-suited.
sound is just a kinetic disturbance that propagates through matter as a wave.
so the answer is yes, everything makes a sound. the question is regarding the existence of something (sound) that no one is around to hear. it exists, and is perceivable, but you might not hear it.
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Anonymous2006-05-27 12:09
>>44 Yes. Sound does not require itself to be heard by ears to exist.
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Anonymous2006-05-27 13:03
DEPENDS ON THE DEFINITION:
THE SOUND AS IN "SOUND VS. NOISE" IS MEANINGLESS WITHOUT AN OBSERVER TO TELL THE DIFFERENCE WHEREAS THE SOUND AS IN "VIBRATION IN MEDIA OF FREQUENCIES FROM 20Hz to 20kHz" IS INDEED INDEPENDENT OF PERCEPTION.
I FEEL LIKE I'M EXPLANING THINGS TO NIGGERS
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Anonymous2006-05-27 14:18
that's because youre a dumbass, and you think you understand what you're talking about.
it's definitely not restricted to 20hz to 20khz, thats just the range that humans can hear. other animals can hear outside of that range even, does that mean what they're hearing isnt sound? i feel like im explaining this to someone who's existence is defined by selective reading of dictionary.com
noise still has about nothing to do with this conversation. i seriously have no clue why you would bring any of that up.
and i'm rather curious as to what you would call a 19Hz wave travelling through the air? what do you think about sonar? echolocation?
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Anonymous2006-05-28 7:29
infrasound you fucking bixnood!
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Anonymous2006-05-28 13:00
alright, so now tell me what your definition of infrasound is.
hint: since you don't think it's sound, try not to use the word sound in your definition.
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Anonymous2006-05-28 13:25
you defined it in your last post shithead
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Anonymous2006-05-28 22:17
so your definition of infrasound is 'a 19hz wave travelling through the air'? i'm asking for your interpretation of the definition since you clearly don't understand what sound is or what you're talking about.
you're putting up the weakest argument ive heard since i told your mom to suck my cock.
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Anonymous2006-05-29 2:13
lol. sound is 20Hz-20kHz just like light is 380-780nm, and not the entire EM spectrum.
there are no interpretations, looking it on the internet. you lose. get over it
"Sound is a disturbance of mechanical energy that propagates through matter as a wave. Sound is characterized by the properties of sound waves which are frequency, wavelength, period, amplitude and velocity or speed."
"Sound is perceived through the sense of hearing. Humans and many animals use their ears to hear sound, but loud sounds and low frequency sounds can be perceived by other parts of the body through the sense of touch."
"The range of frequencies that humans can hear is approximately between 20 Hz and 20,000 Hz. This range is by definition the audible spectrum, but some people (particularly women) can hear above 20,000 Hz."
"The human ear is sensitive to sound waves in the frequency range from about 20 to 20,000 Hz, which is called the audible range. You may have heard the term sound range, but sound waves can also be above audible range (ultrasonic) and below audible range (infrasonic). "
just google sound, go ahead. it doesn't matter how many times you say it's only 20-20k hz, it's not.
youre right about light though, since its fucking defined as the visible part of the spectrum, unlike sound, which is not defined in terms of its audibility.
if you don't get it by now, you're just stubborn.
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Anonymous2006-05-29 9:42
You're all wrong. Both yes and no are correct, for the following reason:
sound1 Audio pronunciation of "sound" ( P ) Pronunciation Key (sound)
n.
1.
1. Vibrations transmitted through an elastic solid or a liquid or gas, with frequencies in the approximate range of 20 to 20,000 hertz, capable of being detected by human organs of hearing.
2. Transmitted vibrations of any frequency.
3. The sensation stimulated in the organs of hearing by such vibrations in the air or other medium.
Sound has multiple definitions. 2 says it's the vibration, 3 says it's the sensation, 1 says it's vibration capable of being heard. When a tree falls and no one hears it, 2 is satisfied but 3 is not. It's a matter of semantics; it just boils down to which 'sound' are you referring to when you ask the question.