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Nothing multiplied by infinity equals a finite number


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A finite number divided by nothing (zero) is infinity.

A finite number divided by infinity is nothing (zero).

 

I'm afraid not - anyhting divided into 0 equal parts is nothing.

 

A finite number divided by infinity is nothing (zero).

 

Not true again you just get an infinatly small number. The only number by which you can divide something to get zero equal parts is zero.

 

Therefore' date=' nothing multiplied by infinite equals a finite number.

[/quote']

 

The statement is false because the other two are also false :)

Zero times anyhting at all is always zero. Think about it: how can you divide infinity into zero equal parts? The only way is with a zero as the answer.

 

Its not always true though, mostly division by zero is classed as undefined.

 

A few read for you:

 

http://mathforum.org/dr.math/faq/faq.divideby0.html

 

 

Cheers,

 

Ryan Jones

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well,

 

suppose you have two functions, f and g which are functions of x. Now also suppose that as x goes to zero, f approaches infinity and g approaches zero.

 

Now if I define h = f/g, then what is the value of h at zero? Its infinity over zero. Luckiliy calculus gives us a way of figuring out sticky situations like this using L'Hopital's rule. We can then figure out if in the limit x->0, what does h approach. Is it infinity, is it zero or some finite value.

 

http://mathworld.wolfram.com/LHospitalsRule.html

 

the rule also works for situations involving inf/inf, 0/inf, 0/0 etc

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L'Hopital's rule can only be applied to the cases of 0/0 and inf/inf, other similar forms (such as 0*inf etc) have to be altered to one of the first two forms!

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A finite number divided by nothing (zero) is infinity.

A finite number divided by infinity is nothing (zero).

 

Therefore' date=' nothing multiplied by infinite equals a finite number.[/quote']

I think you're missing some limits in there. A finite number © divided by 0 isn't going to give a logical answer. Dividing by 0 in calculus doesn't make sense (I'm not sure how it's handled in maths higher than calculus). What you are thinking of is a finite number © divided by a variable that's limit is zero goes to infinity.

 

Likewise, the second one should read "a finite number divided by a varialbe that goes to infinity tends to zero."

 

These aren't stict equations like you are usual them. They're limits, or a general idea of what is happening in the equation. They can't just be multiplied around so easily.

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Thanks RyanJ...

 

Your welcome - a good try too :)

 

Just remember that anything divided by zero is a big no-no in mathematics, its either 0 or undefined depending on the situation.

 

If you could divide something by infinity you;d get an infinatly small value and you culd say it approches zero but it never quite gets there even at infinity :)

 

Have a read arround the Dr.Math site - you may find some other thing to spark your thinking from :)

 

Cheers,

 

Ryan Jones

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The reason it approaches zero but doesn't actually reach it is because infinity is not actually a quantity that can be manipulated as if it were finite. "4/infinity" is just as nonsensical as "4/happiness." You can, however, take the limit of 4/x as x approaches infinity, since you aren't saying that x is infinitely large, only indefinitely large, meaning it is larger than any given finite value.

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A finite number divided by nothing (zero) is infinity.

Correct.

 

A finite number divided by infinity is nothing (zero).

Correct

 

These remarks are correct, up to some imprecise formulation. But I can accept them.

 

A more formal saying is:

 

Given a number N/Z, for non-zero finite N, and Z approaching zero, the quotient approaches infinity.

Given a number N/I, for finite N, and I approaching infinity, this quotient approaches 0.

 

Therefore, nothing multiplied by infinite equals a finite number.

This is not correct in the general case. You need more information about the nature of the limit.

 

Suppose you have a function f(z)=sin(z) and g(z)=1/z. Now if you look at the product p(z)=f(z)g(z) and you let z approach the value 0, then you'll see that this product approaches a finite number, being 1. In fact, the funtion p(z) can be analytically extended to 0, by defining p(0) = 1. This function is infinitely differentiable at 0 without any anomality.

 

Now suppose we have another function h(z)=1/(z*z). Now if we take the product q(z)=f(z)h(z), and you let z approach the value 0, then you'll see that this product approaches infinity. This function has a first order singularity in 0.

 

Now we take the mad function m(z) = exp(1/z) (here exp() is the e-power of its argument).

Now, if z is approaching 0 from above, then m(z) goes to infinitiy. If z is approaching 0 from below, then m(z) goes to 0. So, now you have a really weird situation. This function has a so called GESP (General Essential Singular Point) in 0, which can be regarded as an ordinary singularity of infinite degree.

 

Just remember that anything divided by zero is a big no-no in mathematics, its either 0 or undefined depending on the situation.

As you can see, this statement is too coarse. There are many subtleties in division by 0. E.g. sin(z)/z is perfectly valid in 0, cos(z)/z has a first order singularity in 0 and exp(1/z) has a GESP in 0.

 

When the field of complex numbers is extended with one single point, called "infinity", then any arithmetic with singularities of final degree becomes perfectly legal. E.g. the function (1-z)/z has two zeros in that case, one zero being 1 and the other being infinite. The functoin (1-z)/(z*z) has three zeros, one zero being one and two coinciding zeros at infinity. The function sin(z)/z has zeros in pi, 2pi, 3pi, ... and -pi, -2pi, -3pi.... The function exp(z) has a GESP at infinity and the function exp(1/z) has a GESP in 0.

 

Have a look at this:

 

http://en.wikipedia.org/wiki/Riemann_sphere

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A finite number divided by nothing (zero) is infinity.

Correct.

...

These remarks are correct' date=' up to some imprecise formulation. But I can accept them. [/quote']

Perhaps in the chemistry-forum, the formulation isn't that important but in mathematics, a finite number divided by 0 is not defined.

 

I don't think it's a good idea to confirm his wrong idea about this, certainly not when there was already pointed out that this is wrong and that you have to talk about a limit in order to be mathematically correct.

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nothing * infinity still equals nothing.

 

0*(any number finite or not) = 0

That's not correct. 0*(a finite number) is 0 but 0*inf is an indeterminate form, just as 0/0, inf/inf, inf-inf, ...

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Perhaps in the chemistry-forum' date=' the formulation isn't that important but in mathematics, a finite number divided by 0 is not defined.

[/quote']

Yes, you're right. In my response I already mentioned the sloppy formulation and gave a more formal formulation, but I agree with you that in mathematics formulations must be precise, in order to avoid subtle pitfalls, especially in more complex situations.

 

So, concluding: N/0 is not defined, but one can say that N/Z, with N finite and non-zero, approaches infinity when Z approaches 0.

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Correct, with that nuance that it could be negative infinity as well of course -depending on the sign of the numerator and the way of approaching 0.

Yes, but that is a whole other story. Taking into account direction of approach opens up a whole bucket of new bizarre things.

 

Related to this, there is a limit, which is finite and yet undefined:

 

(A+B*i)/(A-B*i), with i*i = -1.

 

Now choose (A, B), such that A+B*i equals eps*exp(i*phi) and let eps go to 0 from above.

 

Now you see that the value of this limit depends on the direction from which 0 is approached. You can make it equal to 1, to -1, to i or even sqrt(2)+i*sqrt(2), just as you like. Choose a good direction and you get any value you like on the unit circle in the complex plane.

 

So, the actual value of this limit is undefined and this even in a more bizarre way shows that 0/0 cannot be expressed correctly, not even in a sloppy way.

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That's true but I don't see the relevance of your complex example. Also in the reals, a limit isn't defined if it depends on the way you approach it (meaning: it has to be the same, coming from the left or from the right). In that way, the limit for x going to 0 of 1/x isn't defined because depending on coming from the left or right, you get - or + infinity.

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That's true but I don't see the relevance of your complex example. Also in the reals, a limit isn't defined if it depends on the way you approach it (meaning: it has to be the same, coming from the left or from the right). In that way, the limit for x going to 0 of 1/x isn't defined because depending on coming from the left or right, you get - or + infinity.

If you look at the real numbers, yes, but when you extend to the complex plane plus {inf}, the Riemann-sphere, then the pole in 0 of 1/x has a meaning, while something like z/z* (z* means conjugate of z) is ill-defined in zero, one cannot speak of a zero or a pole. An entity like z/z (or sin(z)/z from my previous examples) can be defined in 0 without problem. What I want to point out is that the whole issue of division by 0 and the corresponding limits is a much more subtle matter than most people think of.

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