The peculiar results of an infinite universe

[Brendan]

Why does the equation seem to work out so well? - Because You make a mistake by dividing by infinity.

I don't know how to write the infinity symbol here, so I use "i" instead. Let's say I have a number "a" which is neither zero nor infinity. Your assumption is that:

a / i = 0 (for all (a != 0 and a != i))

This is wrong. It is just as wrong as

a / 0 = i

would be. Tempting, but wrong. There is a trivial proof to it. Take the equation

a / i = 0

and multiply it with i:

a = 0 * i

0 and i are constants, but a is a variable. Remember? a stands for every number that is neither 0 nor i! So a could be 1, 2, 3 or -2.9873 . How could two constants mulitplied with each other result in a variable? So now: What's the result of zero mutliplied with infinity?

Changing a / i = 0 to a = 0 * i is like doing this:

a / i = 0

a * i / i = 0 * i

a * 1 = 0 * i

a = 0 * i

Isn't i / i an indeterminate form? So it shouldn't be treated like the number one. Zero times infinity is also an indeterminate form.

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Better ways of writing the equations a / i = 0 and a / 0 = i would be in terms of limits like this:

lim x->i a/x = 0
lim x->0 a/x = i

When I was in precalculus half a decade ago, the teacher had us write the = symbol in equations like the two above as -> so we'd understand that the limit is approached by the function as the variable, like the x in the examples, approached a number or infinity.

[Fightspit]

For me, the equation:

Code: Select all

a / i = 0

gives:

Code: Select all

a = 0    for all i different of 0

That's all !

Edit:
I edited about i different of 0.

[Crowley]

Changing a / i = 0 to a = 0 * i is like doing this:

a / i = 0

a * i / i = 0 * i

a * 1 = 0 * i

a = 0 * i

Isn't i / i an indeterminate form? So it shouldn't be treated like the number one. Zero times infinity is also an indeterminate form.

That's exactly what I think! Moreover, I am sure that "a / i" is also indeterminate.

Better ways of writing the equations a / i = 0 and a / 0 = i would be in terms of limits like this:

lim x->i a/x = 0
lim x->0 a/x = i

That's definetly better. But it has the backdraw that You don't have a handy value for x=i then, so You cannot base any imagination about an infinite universe on that then.

When I was in precalculus half a decade ago, the teacher had us write the = symbol in equations like the two above as -> so we'd understand that the limit is approached by the function as the variable, like the x in the examples, approached a number or infinity.

Hmm. As You write of it... I vaguely remember that there was something similar in my math class.

[Crowley]

For me, the equation:

Code: Select all

a / i = 0

gives:

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a = 0    for all i

That's all !

What do You mean with "for all i"? In this part of the discussion, "i" is short for "infinity"! (I don't see an easy way for writing the mathematical infinity symbol in this forum, that's why I chose "i".)

[Fightspit]

Of course, if we know a ( a = 4 for exemple), "i" must be infinity to be near of 0.

[Crowley]

Of course, if we know a ( a = 4 for exemple), "i" must be infinity to be near of 0.

"i" is not a variable, the mathematical constant for infinity (???) is meant! And the question is what is

a / ???

for all a != 0 and a!= ??? ?

My arguments point out that You cannot compute this value, it's just not defined. Defining it, for example as zero, would lead to contradictions.

[jebus]

Well I will try to explain a few things about infiniity (also the use of i for infinity drives me crazy as i = sqrt(-1) )

0/0, ???/??? , 0*???, ???-???, 1^??? are all indeterminite forms.

Now concerning the case of a/b=0, this means that a is 0 and b is finite, or a is finite and b is infinite.