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x/0=infinity ... Experimentally verified by myself.





ocalhoun
I'm taking electronics classes at the moment, and I've noticed something very interesting about a special case in one of the equations.

Capacitive reactance is the measure of how much a capacitor resists the flow of power. Capacitors allow high frequency to pass easily, but tend to block low frequencies. DC power is the lowest frequency possible (0), so it is blocked completely (infinite capacitive reactance).

The formula for capacitive reactance is: X=1/(2(pi)fC)
f is for frequency and C is for capacitance
Let's suppose we have a .05 farad capacitor and DC current (0 frequency)
That would make the equation: X=1/(2(pi) * 0 * .05)
Anything multiplied by 0 = 0 so: X=1/0
Normally we would be stuck at this point, but we already know the answer: X must equal infinity.
So, 1/0=infinity

I know there are problems with it being reversible, but I'm working on that. The best I've been able to work out as a solution to that so far is X/0=(infinity * 0 * infinity)/X
Peterssidan
you can't use infinity as a number! 1/0 is not equal to infinity. 1/0 is undefined. if we have 1/a and let a be smaller and smaller 1/a will go towards in the infinity.
Afaceinthematrix
Well 1/0 is undefined because the value would be too infinitely large to describe in numbers. But infinity isn't a number so I wouldn't say that 1/0 is infinity.
Indi
Other people have already pointed out the "infinity is a number" fallacy, so i won't repeat it. (i will offer the standard alternative i usually do: infinity is not a number, it is a direction.)

Instead, i'm gonna snipe at the carelessness of this "experimentally verified" claim.

Even assuming that one were to actually set up this "experiment" - which doesn't really need to be run, because it is a principle that many devices already use in practice, so we can consider it a fait accompli - it "verifies" nothing.

For starters, there is demented logic of trying to verify a mathematical concept by applying that mathematical concept. In simple terms, it is like saying: "We don't know the ratio of a circle's diameter to its circumference, so we'll check it experimentally. So i make a circle, and measure, and since i already know the answer is π, and that's i seem to get, that means it's proven!" See the problem? ^_^

But the real problem is the impossibility of such an endeavour. In real life, meters have limits. In real life, you would suppose you have a capacitor of 0.050 ± 0.002. When you attach your DC ammeter next to your capacitor and find a current of 0 A, what you are really finding is something more like 0.000 ± 0.002 µA. Small, but not certainly zero. Which means that if you use your voltage drop (let's say 5.000 ± 0.001 V) to find your reactance, you aren't really dividing by 0.

But, more importantly, in real life, there is no such thing a truly 0 Hz (true DC) signal. It is a theoretical idea only, and impossibly to get in reality. Even the most expensive battery drains slowly as it works. And even if you were to somehow set up a system that did not drain, it is impossible to maintain a perfect potential difference when charges are moving - quantum chromodynamics tells us this with certainty... but if you take a moment and realize that charges are little packets of negative charge, and imagine trying to set up a situation where the charges on both sides of the circuit are constant while these little packets keep jumping, you will see that there will be tiny fluctuations that cannot be corrected faster than the speed of light times the length of the circuit.

Be careful with your claims. ^_^ "Experimentally verified" is a bold claim.
Arnie
Is that formula correct in extreme situations such as this one? If yes: are your instruments correct in such situations?
Afaceinthematrix
I was thinking about this today, and I thought of another "proof" for you. If you're going to try and prove that x/0 is infinity (even though infinity isn't a number; I prefer to say that x/0 is too infinitely large to be defined by numbers, which is why it's undefined), then try doing this instead.

Integral between pi/2 and p/4 of sec^2(x)...

When you take the antiderivative of sec^2(x), you'll get tan(x). After plugging in the endpoints, you'll see that you'll get (undefined - 1)... Graphing this out, you'll see that sec^2(x) has a vertical assympote at pi/2... so obviously the area under the curve will be infinity...
Arnie
sec x is such an American function. At my university we just call it 1/cos x. Anyway, sec˛ x is the derivative of tan x, probably the only reason why it's defined...

In the example above: cos(0.5pi) = 0 so sec(0.5pi) divides by zero. This means that at exactly that point of the graph the function is undefined, but when approaching it you approach infinity.

Whether the area will be infinity is another question, because although the y component in the area near the asymptote is extremely large, the x component is extremely small.
powers1983
You've proved the theory using, um, theory. There is always going to be a small leakage current through the capacitor and so it's not fully blocked. But if you're doing electronics then that sounds like you're more likely to be taught the engineers way of thinking which is to approximate, simplify, linearise, simplify a bit more and then it works easy (but add in a factor of safety just in case) rather than the scientist/mathematician thinking which requires no such simplifications but rather needs to expand on it in order to fully explain it. So an engineer is quite happy to accept that there is a current of practically zero which means a practically infinite reactance- engineers don't need to prove it, just use it (unless they're at uni in which case all they do is prove things).
Afaceinthematrix
Arnie wrote:
sec x is such an American function. At my university we just call it 1/cos x. Anyway, sec˛ x is the derivative of tan x, probably the only reason why it's defined...

In the example above: cos(0.5pi) = 0 so sec(0.5pi) divides by zero. This means that at exactly that point of the graph the function is undefined, but when approaching it you approach infinity.

Whether the area will be infinity is another question, because although the y component in the area near the asymptote is extremely large, the x component is extremely small.


I'm sorry; we use secx at my school (which is in America) and then when I talk to other people or right problems out that other people see, I try to use 1/cosx (because that's universal; many places use secx, but many places - like your country - do not). But sometimes I forget when I go back and forth and simply use secx out of habit. I was also unsure for a while rather it would be infinity or not, but after thinking about it for a while and going over with several professors/colleagues, I came to the conclusion that it must be infinity. After studying the graph and seeing the reaction of multiple computer programs, it seems like it will be infinity despite the small x component.
nivinjoy
x/0=infinity ????? Question Question Then i think you will have to correct everyone out there...!!!

Laughing Cool Laughing
czc587
ocalhoun wrote:

Capacitive reactance is the measure of how much a capacitor resists the flow of power. Capacitors allow high frequency to pass easily, but tend to block low frequencies. DC power is the lowest frequency possible (0), so it is blocked completely (infinite capacitive reactance).


You lost me here... I thought capacitors are DC devices and only allow (vs block) DC current.



PS are you a 13 year old and reviewing some NASA publications also Smile ?
http://www.freerepublic.com/focus/f-news/2002114/posts
Bikerman
No - that's a rather oversimplistic view of the capacitor (well, actually plain wrong - the simple view is the other way around - capacitors block DC).
Wiki has a reasonable article on the matter
http://en.wikipedia.org/wiki/Capacitor
czc587
hmmm ... I've built circuits for years, they block current in that they store similar to a battery, only under certain circumstances do they actually block DC. Look on any circut board, those guys are not there to block DC. For ac circuits I use them to bring down the spikes or curves in the voltage or signal.
Bikerman
czc587 wrote:
hmmm ... I've built circuits for years, they block current in that they store similar to a battery, only under certain circumstances do they actually block DC. Look on any circut board, those guys are not there to block DC. For ac circuits I use them to bring down the spikes or curves in the voltage or signal.

That'll be why I said it was a simplistic view then Smile
I've also built circuits for years, and frequently built both low pass and high pass filters, both of which use...yep, capacitors.
The fact that capacitors block DC (in ALL circumstances) can be easily seen from the basic current derivative of a capacitor.

(where i(t) is current at time t, V voltage, C capacitance).
Note that for direct current dv/dt=0 so the current passed (regardless of capacitance) is 0.

In DC circuits their primary use is for temporary storage of current, or power shunts. Their main use is in AC circuitry for filters and tuning circuits, and in mixed circuits they act as couplers/decouplers between DC and AC.
ptolomeo
1/0=Infinity then ------- 0*Infinity=1 -----------. I didnt know this notable equation!!!
Davidgr1200
I always like the following argument:
1/2 means you take one thing and divide it into 2 parts
1/0 means you take one thing and divide it into no parts - what do you have then? When you phrase it like that you realise how meaningless the question is.
nilsmo


You're looking at the left side of the graph and seeing that the red line at x=zero must be infinity high.

Well, now look at the "right side" of zero. You get negative infinity.. Hm... that's why we don't say that 1/0 is infinity: it might be, for example, negative infinity.
KeithK
I'm going to give you the benefit of the doubt here. Simply put, we have two parts to any fraction. The numerator and the denominator. Simply put, the numerator is the amount of something you have. The denominator is like the limit.

For example: You have 20 apples. If you have five people to feed and each of those people can eat four apples a day then you have the equation. 20 / (4 * 5) would equal 20 / 20, or just one apple. The limit is equal to the amount, then it will equal one. You have one days supply.

Now, lets change that a bit and say you have 10 apples. 10/(4*5) = 10/20 = 1/2. You now only have half a days supply. Because your limit exceeds your amount, you have less than one.

Now if you had zero apples it'd just be 0/20. You'd have nothing to meet the limit. You have no supply of apples. Certainly not enough to meet anyone's expectations.

Now lets say you have 20 apples again, only no one eats apples. There is no limit because there is no expectation. You have 20 apples to feed 0 people. You now have an infinite supply of apples. (Assuming they don't rot. But who cares if they aren't getting eaten Razz)

So, by that logic 20/0 = infinite. In reality I could say any number over zero is infinite. Simply because Infinite = an amount without a limit. I could say 272399/0 is infinite.

This is not to be confused with the concept of infinity, which is the state of being infinite.

Which would lead us to the conclusion that: infinity = y/0.

However there is a problem with this. The concept of the opposite, which is nothing, or Zero. Contrary to popular belief, it does not stand for simply 0. It has to be zero of something. Zero apples, Zero bolts, Zero motivation, Zero interest. As you can see without some limiting factor, zero would remain undefined. Zero needs a limit.

Which would lead us to the conclusion that: zero = 0/x.

I could tell you that I have 0/4, and you would say I have how much? that's right. I have Zero. We shorthand those fractions as simply 0 when we use zero as a whole number, but you could plug any number into the denominator and reach the same conclusion. This is because the amount simply doesn't exist, while the limit does.

To recap, we now have these two equations for Inifinity and Zero.

Infinity = y/0 and Zero is equal to 0/x.

however, both the State of Zero and the State of Infinity can't exist at the same time, can they? After all, if y = 0 in the infinity equation, then wouldn't it now be equal to the Zero equation. How can there be no limit to nothing? The same could be said of if x in our Zero equation equaled 0. Can we have no limit without an amount? This is where my mind begins to reel back and shy away, because we're talking about a different concept completely.

For simplicity sake, lets say: infinity = y/0 where y does not equal 0. zero = 0/x where x does not equal 0. And for that lack of amount that has no limitations, lets call that concept Oblivion and say that Oblivion = 0/0.
asnani04
1/0 is not infinity. Infinity is not a number, it is just a limit of some expressions. For example, as a person has already pointed out, 1/a, where a tends to zero, is a very large number, and lim 1/a where a tends to zero is taken as infinity. But, at a=0, the the quantity 1/a is undefined.

Hope I have been of some help. Smile
kelseymh
asnani04 wrote:
1/0 is not infinity. Infinity is not a number, it is just a limit of some expressions. For example, as a person has already pointed out, 1/a, where a tends to zero, is a very large number, and lim 1/a where a tends to zero is taken as infinity. But, at a=0, the the quantity 1/a is undefined.

Hope I have been of some help. Smile


You haven't. First, "infinity" is not a limit (only finite numbers can be limits), it is the identification of an expression which does not have a finite limit. Second, infinities can in fact be treated as numbers.

Please review Cantor's analysis of cardinalities for a proper discussion of the issue. If you don't have enough mathematical background to get through Cantor, don't worry about it; he's pretty dense. But that also means you don't have enough background to be knowledgeable on the subject.
Dennise
asnani04 gets close.

Infinity is a very useful mathematical construct and is not A number, but as has been mentioned before, is an arbitrarily large number.

Consider any large number you wish and add as many zeros to that number as you wish. I will come along and just add 1 to that number, making my number larger and call that infinity. We could have an infinity contest and do this ad nauseum.

Now consider the capacitor experiment. If one were to use a cheap capacitor and a good ammeter, one would actually measure some very small DC leakage current indicating the reactance indeed is NOT infinity but a large number. Conversely, use a high quality capacitor and a cheap ammeter and one would measure zero DC current and conclude the the reactance is infinity when in fact it is only a larger number than the first 'infinity'. One could try this with many combinations of poor and high quality capacitors and ammeters and measure different but tiny DC currents. Infinity is arbitrary and depends on how and where it is used.
Radar
In the physical world, in one sense, and as far as we can tell, you don't really have infinite anything. You simply have numbers too large to comprehend.

In the theoretical mathematical sense, yeah, okay, infinity, but we shy away from actually examining it as much as possible.
capricornis
Division by 0 doesn't give you infinity or any kind of a number.. In strictly mathematical terms, the result of division by 0 is 'undefined' i.e. it doesn't have any meaning or concept whatsoever. It is there only to round-up the axiomatic basis of any mathematics (and you could have many different 'mathematics' depending which 'axioms' you accept as the basis of your system and deem them 'self-evident' that is, not needing a proof, but that's a different conversation altogether).
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