If 4−4=0, and 5−5=0, then what is ∞−∞? (Opera)

[qazz12]

Why do some claim ∞−∞≠0?

[hhwdavid]

"I would like to express this answer in a simple but hopefully convincing way.
Firstly, infinity is not a number.
The number of numbers in the set 1, 2, 3, 4, 5, 6…… (carrying on for ever) is what we would say is infinite.
(Note: FINITE means definite. INfinite means NOT definite.)
I actually do not like the word “infinity”, I prefer to say something is non finite or infinite.
If we MUST use the word “infinity” then the number of numbers in the set of counting numbers 1, 2, 3, 4, 5, 6, 7, 8, …… = infinity.
but the number of numbers in the set of even numbers 2, 4, 6. 8, 10, 12, …. also = infinity.
If we think about subtracting these we would get:
(1,2,3,4,5,6,7,8,9,….) - (2,4,6,8,10,….) = (1,3,5,7,9,….)
counting numbers - even numbers = odd numbers
I hope you agree that the odd numbers are also infinitely big.
So in a very crude way, this says infinity - infinity = infinity."

[qazz12]

Your claim is strong.

If (2,4,6,8,10,….) + (1,3,5,7,9,….) = (1,2,3,4,5,6,7,8,9,….)

and (2,4,6,8,10,….) + (1,3,5,7,9,….) is quasi equal to (2,4,6,8,10,….) + (2,4,6,8,10,….)

then

(2,4,6,8,10,….) + (1,3,5,7,9,….) = (2,4,6,8,10,….) + (2,4,6,8,10,….)

(1,2,3,4,5,6,7,8,9,….) = 2 (2,4,6,8,10,….)

(1,2,3,4,5,6,7,8,9,….) - 2 (2,4,6,8,10,….) = 2 (2,4,6,8,10,….) - 2 (2,4,6,8,10,….)

∞−∞ tends to 0

But your point is clarified.

[turf3]

Depends on how you're defining "infinity".


For example, if you subtract the number of integers (infinite) from the number of irrational numbers (infinite) you're left with an infinite number of irrational numbers.


Questions like this basically boil down to "can you map one infinite set of numbers onto another infinite set of numbers, or are there some in one set that can't be mapped onto the other set?"


Vite: number theory.

[turf3]

Originally Posted by hhwdavid

"I would like to express this answer in a simple but hopefully convincing way.
Firstly, infinity is not a number.
The number of numbers in the set 1, 2, 3, 4, 5, 6…… (carrying on for ever) is what we would say is infinite.
(Note: FINITE means definite. INfinite means NOT definite.)
I actually do not like the word “infinity”, I prefer to say something is non finite or infinite.
If we MUST use the word “infinity” then the number of numbers in the set of counting numbers 1, 2, 3, 4, 5, 6, 7, 8, …… = infinity.
but the number of numbers in the set of even numbers 2, 4, 6. 8, 10, 12, …. also = infinity.
If we think about subtracting these we would get:
(1,2,3,4,5,6,7,8,9,….) - (2,4,6,8,10,….) = (1,3,5,7,9,….)
counting numbers - even numbers = odd numbers
I hope you agree that the odd numbers are also infinitely big.
So in a very crude way, this says infinity - infinity = infinity."

You can map the even numbers onto the natural numbers - very easily. 2X.


So the "quantity" of numbers in the set of even numbers is equal to the "quantity" of natural numbers.


Similarly, all rational numbers can be mapped onto the set of integers.


On the other hand, irrational numbers can't be mapped onto integers, because you can't get an irrational number by dividing one integer into another.


Do a little research into the concept of Aleph-nought, Aleph-one, etc.

[guidoLaMoto]

Eg- x/0 is not 0, it's undefined because the operation "division by 0" is undefined.

Infinity is undefined, ergo infinity - infinity is undefined.

[stanley-88888888]

^ also why is 5 ÷ 0 = ∞
but 0 ÷ 0 = undefined ?

[rjshae]

You can have different types of infinities. For example: the count of all integer ratios vs. the count of all irrational numbers. What's the difference between those two "numbers"? It's an undefined operation.

[rjshae]

Originally Posted by stanley-88888888

^ also why is 5 ÷ 0 = ∞
but 0 ÷ 0 = undefined ?

The first can be handled through limit theory. For example, what is 5/x as x goes to zero? It clearly shoots off to infinity. On the other hand, for 0/x you'd always get zero, but if you take x/x as x goes to zero, you always get one. Ergo it's taken as undefined because there isn't a unique limit.

[Thoreau424]

Since infinity is no specific number, subtracting a non-specific number from another of the same is unlikely to result in 0 / zero. It's as theoretically impossible for the two numbers to be identical, and trying to define infinity (as an actual value) in the first place.