Little math theory question

[Tomoyo]

What is infinity/(infinity^2)? It should be 1/infinity, zero, but I think it can also be 1.

Or can infinity not be squared?

 

[Yom]

It is 0, in practice. It is actually infinitesimal. For instance, if you want to find the limit of X/(X^2) as X approaches infinite, the answer is 0.

 

[BasketCase]

Whenever you add an infinity into an equation, the results usually turn out to be meaningless.

1/infinity is not truly zero--it's just extremely close to zero. 1/0 is not infinity, because you're not permitted to divide by zero in the first place. You cannot make 1 out of ANY number of zeroes, therefore dividing by zero is simply not permitted even if you scribble 1/0 with your pencil. It's meaningless.

Now, the mathematician Georg Cantor (yes, his first name is apparently spelled that way) did prove that there are multiple orders of infinity, but I don't think infinity^2 is one of them. The first-order infinity is the number of integers, and the second-order infinity is the number of real numbers (after that it starts getting really wierd, and I didn't get any further in the book because I'd developed a MONSTER of a headache).

 

[luiz]

For practical purposes, L'Hôpital's theorem grants us that the answer is zero.

 

[Dida]

The limit appoaching zero is equal to zero.
So 1/infinity is not 'extremely close to zero' but it is equal to zero.

 

[Perfection]

For practical purposes, L'Hôpital's theorem grants us that the answer is zero.

No, L'Hopital's says it's an indeterminate form

Limx->inf e^x/(lnx)^2 is infinite even though e^x is infinite and lnx is infinite. You can't take those and plug them in infinity/infinity^2.

 

[newfangle]

Now, the mathematician Georg Cantor (yes, his first name is apparently spelled that way) did prove that there are multiple orders of infinity, but I don't think infinity^2 is one of them. The first-order infinity is the number of integers, and the second-order infinity is the number of real numbers (after that it starts getting really wierd, and I didn't get any further in the book because I'd developed a MONSTER of a headache).

The number of integers is usual called "eloph-naught" (an X looking thing with a zero subscript).
The number of reals can then be represented as 2^eloph-naught, which is a helluva lot bigger.

 

[Perfection]

The number of integers is usual called "eloph-naught" (an X looking thing with a zero subscript).
The number of reals can then be represented as 2^eloph-naught, which is a helluva lot bigger.

How does that work, I'm familiar with Cantor, but I don't get the 2^eloph-naught thingy

 

[newfangle]

I don't know the derivation by heart, sorry.

Though I'm fairly sure its pretty meaningless.

 

[Perfection]

I don't know the derivation by heart, sorry.

Though I'm fairly sure its pretty meaningless.

Yeah, I thought it was just like eloph-one

 

[newfangle]

Here's some info:
To talk about cardinal numbers, including transfinite cardinal
numbers, you need to talk about counting the numbers of elements in
sets.

If a set A is finite, there is a nonnegative integer, denoted #A,
which is the number of elements in A. That is one of the finite
cardinal numbers.

To do arithmetic with cardinal numbers, you use facts about finite
sets and the number of elements in them, such as the following:

If A and B can be put into one-to-one correspondence, then #A = #B,
and conversely.
If A is contained in B, then #A <= #B.
If A is disjoint from B and C is their union, then #C = #A + #B.
If A and B are sets, and C = A x B is the set of all ordered pairs
of elements, the first from A and the second from B, then
#C = (#A)*(#B).
If C is the set of all subsets of A, then #C = 2^(#A).

Try some of these with small sets to satisfy yourself that they are
so.

If the set is infinite, the corresponding cardinal number is not one
of the finite cardinal numbers, so it is called a transfinite (or
infinite) cardinal number.

The first transfinite number is called aleph-sub-zero (or aleph-
naught, or aleph_0). (Aleph is the first letter of the Hebrew
alphabet.) It is the cardinal number of the set of positive integers.
Sets having this cardinal number are called countably infinite sets,
or countable sets, because they can be put into one-to-one
correspondence with the positive integers, or counting numbers.

The above rules for computing with finite cardinal numbers were
extended by Cantor to apply to transfinite cardinal numbers. In this
way he could talk about doing arithmetic with transfinite cardinal
numbers. He discovered and proved that, if n is any finite cardinal
number,

aleph_0 + n = n + aleph_0 = aleph_0,
aleph_0 + aleph_0 = aleph_0,
aleph_0*n = n*aleph_0 = aleph_0 (n > 0),
aleph_0*aleph_0 = aleph_0,
aleph_0^n = aleph_0 (n > 0).

This is a pretty strange arithmetic! Note that subtraction and
division are not definable operations in this arithmetic. The
Associative Laws of Addition and Multiplication hold, and the
Commutative Laws of Addition and Multiplication do, too, and the
Distributive Law, but there is no zero, no unity, no negatives,
and no reciprocals.

Cantor also proved that 2^aleph_0 > aleph_0. This means that the
set of all subsets of the integers cannot be put in one-to-one
correspondence with the integers, that this set is really a different
size of infinite set, truly larger. Cantor called 2^aleph_0 = C,
which stands for "continuum". C is the cardinality of the set
of real numbers. Once again, Cantor showed that

n^aleph_0 = C (n > 1),
aleph_0^aleph_0 = C,
C + n = n + C = C,
C + aleph_0 = aleph_0 + C = C,
C + C = C,,
C*n = n*C = C (n > 0),
C*aleph_0 = aleph_0*C = C,
C*C = C,
C^n = C (n > 0),
C^aleph_0 = C,

but 2^C > C. This trick of exponentiating one cardinal number to get
a larger one works for all transfinite cardinal numbers.

In this way, Cantor showed that there are infinitely many different
transfinite cardinal numbers. In fact the set of transfinite cardinal
numbers itself has a cardinal number, which is transfinite! Let's not
think about that - it gets very confusing!

But are there any transfinite cardinal numbers between aleph_0 and C?
The name aleph_1 has been given to the smallest transfinite cardinal
number larger than aleph_0. Then aleph_0 < aleph_1 <= C. The above
question can be rephrased as, "Is aleph_1 = C?" Cantor guessed that
this is so, but was unable to prove this. It was one of the great
unsolved problems of mathematics, called "The Continuum Hypothesis".
Finally it was shown that this was not provable from the usual axioms
of set theory! It is usually assumed as an additional axiom.

I hope that this is helpful. If you need more help, write again.

-Doctor Rob, The Math Forum

 

[Bluemofia]

1/infinity is not truly zero--it's just extremely close to zero. 1/0 is not infinity, because you're not permitted to divide by zero in the first place. You cannot make 1 out of ANY number of zeroes, therefore dividing by zero is simply not permitted even if you scribble 1/0 with your pencil. It's meaningless.

Something to add on:

1.) a = b
(multiply both sides by a)

2.) a^2 = b*a
(subtract b^2 to both sides)

3.) a^2 - b^2 = b*a - b^2
(factor it out)

4.) (a - b) * (a + b) = b * (a - b)
(cancel out (a-b))

5.) a + b = b
(replace b with a because of step 1.)

6.) 2a = a
(divide by a)

7.) 2 = 1

Thus if you divide by 0, then all of math is destroyed.

 

[newfangle]

Thats a stupid reason why dividing by zero is bad.

Here is the actual reason.

Lemma: In any algebraic ring, 0r=0 for all r in the ring.
Proof: 0r = (0+0)r = 0r + 0r by distributive axiom. Now subtract 0r from both sides and we get 0r=0.

Theorem: If 0 is in a ring R, 0 has an inverse if and only if the ring is trivial (ie, R={0})
Proof: Assume 0 has an inverse, call it 'a'. So a0=1. But a0=0 from the above lemma as well. So 0=1 in R. This only happens when R={0}, as desired.

So what does this all mean?

If you can divide by zero, you are dealing with a very trivial structure.

 

[luiz]

No, L'Hopital's says it's an indeterminate form

Limx->inf e^x/(lnx)^2 is infinite even though e^x is infinite and lnx is infinite. You can't take those and plug them in infinity/infinity^2.

Hence the for practical purposes...

 

[wit>trope]

What is infinity/(infinity^2)? It should be 1/infinity, zero, but I think it can also be 1.

Or can infinity not be squared?

hmm it seems to me that a/b=c just in case c*b=a

So if aleph_0/aleph_0^2 is to equal 0 then aleph_0^2 times 0 must equal aleph_0 but it doesn't.

And if aleph_0/aleph_0^2 is to equal 1 then aleph_0^2 times 1 must equal aleph_0 WHICH IT DOES

So your second intuition is right! aleph_0/aleph_0^2 is equal to 1.

This can be generalized for for all equations of the form:

aleph_N/aleph_N^2=1

 

[BasketCase]

Heh. Uhh, newfangle: I think Bluemofia was just messing with our heads. He posted a variant of a famous mathematical fallacy. One that had me totally stumped for years until I just plugged in some numbers and started actually doing the equations--then I discovered why the fallacy doesn't actually prove 2 = 1.

1.) a = b
(multiply both sides by a)

2.) a^2 = b*a
(subtract b^2 to both sides)

3.) a^2 - b^2 = b*a - b^2
(factor it out)

4.) (a - b) * (a + b) = b * (a - b)
(cancel out (a-b))

5.) a + b = b
(replace b with a because of step 1.)

6.) 2a = a
(divide by a)

7.) 2 = 1

Note in step 4 that both sides of the equation are divided by (a-b). Then note in step 1 that a = b. That means (a-b) = zero. BINGO! When step 4 cancels out an (a-b) from each side of the equation, what step 4 is really doing is dividing by zero! That's where the error is.

 

[punkbass2000]

I thought that was his point...

 

[WillJ]

I thought that was his point...

No, I think his point was that 2 equaling 1 isn't a good enough reason to be unwilling to divide by 0.

Edit: Although it's certainly a good enough reason for me!

 

[Damnyankee]

Here's some info:
To talk about cardinal numbers, including transfinite cardinal
numbers, you need to talk about counting the numbers of elements in
sets.

If a set A is finite, there is a nonnegative integer, denoted #A,
which is the number of elements in A. That is one of the finite
cardinal numbers.

To do arithmetic with cardinal numbers, you use facts about finite
sets and the number of elements in them, such as the following:

If A and B can be put into one-to-one correspondence, then #A = #B,
and conversely.
If A is contained in B, then #A <= #B.
If A is disjoint from B and C is their union, then #C = #A + #B.
If A and B are sets, and C = A x B is the set of all ordered pairs
of elements, the first from A and the second from B, then
#C = (#A)*(#B).
If C is the set of all subsets of A, then #C = 2^(#A).

Try some of these with small sets to satisfy yourself that they are
so.

If the set is infinite, the corresponding cardinal number is not one
of the finite cardinal numbers, so it is called a transfinite (or
infinite) cardinal number.

The first transfinite number is called aleph-sub-zero (or aleph-
naught, or aleph_0). (Aleph is the first letter of the Hebrew
alphabet.) It is the cardinal number of the set of positive integers.
Sets having this cardinal number are called countably infinite sets,
or countable sets, because they can be put into one-to-one
correspondence with the positive integers, or counting numbers.

The above rules for computing with finite cardinal numbers were
extended by Cantor to apply to transfinite cardinal numbers. In this
way he could talk about doing arithmetic with transfinite cardinal
numbers. He discovered and proved that, if n is any finite cardinal
number,

aleph_0 + n = n + aleph_0 = aleph_0,
aleph_0 + aleph_0 = aleph_0,
aleph_0*n = n*aleph_0 = aleph_0 (n > 0),
aleph_0*aleph_0 = aleph_0,
aleph_0^n = aleph_0 (n > 0).

This is a pretty strange arithmetic! Note that subtraction and
division are not definable operations in this arithmetic. The
Associative Laws of Addition and Multiplication hold, and the
Commutative Laws of Addition and Multiplication do, too, and the
Distributive Law, but there is no zero, no unity, no negatives,
and no reciprocals.

Cantor also proved that 2^aleph_0 > aleph_0. This means that the
set of all subsets of the integers cannot be put in one-to-one
correspondence with the integers, that this set is really a different
size of infinite set, truly larger. Cantor called 2^aleph_0 = C,
which stands for "continuum". C is the cardinality of the set
of real numbers. Once again, Cantor showed that

n^aleph_0 = C (n > 1),
aleph_0^aleph_0 = C,
C + n = n + C = C,
C + aleph_0 = aleph_0 + C = C,
C + C = C,,
C*n = n*C = C (n > 0),
C*aleph_0 = aleph_0*C = C,
C*C = C,
C^n = C (n > 0),
C^aleph_0 = C,

but 2^C > C. This trick of exponentiating one cardinal number to get
a larger one works for all transfinite cardinal numbers.

In this way, Cantor showed that there are infinitely many different
transfinite cardinal numbers. In fact the set of transfinite cardinal
numbers itself has a cardinal number, which is transfinite! Let's not
think about that - it gets very confusing!

But are there any transfinite cardinal numbers between aleph_0 and C?
The name aleph_1 has been given to the smallest transfinite cardinal
number larger than aleph_0. Then aleph_0 < aleph_1 <= C. The above
question can be rephrased as, "Is aleph_1 = C?" Cantor guessed that
this is so, but was unable to prove this. It was one of the great
unsolved problems of mathematics, called "The Continuum Hypothesis".
Finally it was shown that this was not provable from the usual axioms
of set theory! It is usually assumed as an additional axiom.

I hope that this is helpful. If you need more help, write again.

-Doctor Rob, The Math Forum

::head explodes::

 

[newfangle]

Heh. Uhh, newfangle: I think Bluemofia was just messing with our heads. He posted a variant of a famous mathematical fallacy. One that had me totally stumped for years until I just plugged in some numbers and started actually doing the equations--then I discovered why the fallacy doesn't actually prove 2 = 1.



Note in step 4 that both sides of the equation are divided by (a-b). Then note in step 1 that a = b. That means (a-b) = zero. BINGO! When step 4 cancels out an (a-b) from each side of the equation, what step 4 is really doing is dividing by zero! That's where the error is.

Yes yes, I know he's dividing by zero.

But I was merely pointing out that there's a much better way to show why trying to divide by zero is a bad idea. Why? Because if you could, we'd only have one real number, ie, zero.