Maths geniuses, enlighten me on this one...?

Question

Infinity is a number that is so large to the extent that it is undefined, right? Ok, sure hope you can help me with these questions:

What is Infinity - infinity? Is it zero or undefined? If it's zero, how can you subtract something with an undefined value from something which is already undefined? If undefined is the answer, how can have a minuend and a subtrahend with the same values and not get a zero?
How about 0/0 - 0/0?If we can subtract 2 undefined values (Infinities), surely we have the answer for this?

How about Infinity / Infinity? Is it one, or undefined? Is the answer equal to (0/0)/(0/0), or (0^0) / (0^0)?

What are the answers to 0 ^ Infinity, and Infinity ^ Infinity?

Thanks!

Answer 1

Infinity is NOT a number; it is simply never ending.

To answer your question about infinity minus infinity, here is probably the simplest way to explain this: infinity minus infinity has three possible answers: 0, infiity, and some constant I will provide an example for each answer using one anology:

Imagine you have a bag of infinite marbles. What would happen if you removed ALL the marbles in the bag, you would have none in the bag (thus infinity minus infinity is zero).

What happens if you remove all the even numbered marbles? Since there is an infinite number of even marbles, as well as an infinite number of odd numbered marbles, you have infinity minus infinity equaling infinity (all the infinite even marbles were removed, but there are still an infinite number of odd marbles in the bag).

What happens if you want to have all but 3 marbles in the bag? You have to remove the all of them but 3. Thus infinity minus infinity equals a constant.

To answer your question about (0/0) - (0/0) This is simply undefined: you CANNOT devide by 0. An undefined value does not necessarily mean that it is infinite; there is a difference.

Infinity divided by invinity will provide you the same answers as the subtraction problem. Zero raised to any power will always be 1 and infinity raised to the infinite power will more than likely be infinity.


I hope this answers your question as to the concept of infinity.

Answer 2

Under the standard rules for arithmetic on integers, rational numbers, real numbers and complex numbers, division by zero is undefined. This is just a set rule that is compatible and does not give rise to a number of paradoxes, which would if say, n/0 = infinity.
For example assuming that 0 / 0 = 1 generates the absurdity that 2 = 1.

Also, it is known that (lim b to 0) a/b = +infinity
and, (lim b to 0) a/-b= -infinity
But CANNOT say a/0 = infinity
as we cannot determine whether it is + or - infinity.

We cannot determine 0/0 it is undefined,
it can be seen from (lim x to 0) f(x)/g(x)
in which both f(x) and g(x) approach 0 as x approaches 0, may converge to any value or may not converge at all. See L'Hopital's Rule.

It is useful to think of a/0 as being infinityty, provided a is not zero. This infinity can be either positive, negative or unsigned, depending on context.

In physics these are used:
infinity/infinity = 1
0^infinity = 0 (since 0 *0 =0)
infinity ^infinity = infinity (cannot 'blow up' more than infinity)

Answer 3

Infinity is not a number in the traditional sense, and defies such manipulations. The minuend and subtrahend do not have the same value, as you assert, since infinity is not a value.

You cannot subtract an undefined "number" from another undefined "number", nor can you proceed with other standard mathematical operations because of the nature of infinity.

Answer 4

OK, you're confusing Mathematics with physics, here.

In pure maths, there is no such 'number' as infinity. Mathematicians never use it to describe a quantity, as 'quantity' implies measurement, and 'infinity' is immeasurable.

Infinity is a concept invented by physicists. Usually to excuse the fact that they simply don't know the answer, or haven't developed the language to describe what they are quantifying.

In that sense 'infinity' is like 'time' or 'dimensional space'. It is a construct use to define the indefinable.

One could argue, and indeed some very eminent men have done so, that infinity is both huge and at the same time zero.

Whenever you attempt to include 'infinity' into mathematical reasoning, you will come unstuck. You might just as well include 'happiness' in your equations.

Read:
Hoffner and Heikle 'infinity and reason'
J.Arthur Beinsdeck 'pure and simple'
and Jasson&Jasson 'beyond reconing, a direct summation of mathematical theory in a non mathematical universe'

DO NOT read Hawking's 'a brief history of time' because he confuses reality with relativity in a way most pre-grad students do.

Answer 5

I think maths geniuses would tell you about different types of infinity but I'm just going to tell you that infinity is just a concept So the normal rules such as +,-,x,/ don't always give reasonable answers.

If you wanted to try then replace infinity by a very large number
and zero by 1/a very large number. This is where the different infinities would come in to say which large number is larger.

I try not to think about it, since no one is ever going to count to infinity. It can be useful sometimes though when you want to find the behaviour of a function as one of its variables tends to infinity or zero.

Answer 6

Infinity is not a number.
All your reasoning is just based on your misconception that it is. You're trying to add, subtract, multiply, divide or exponentiate infinities as if they were numbers. You cannot. It makes no sense.

Whenever it makes sense, the sense is inherently different from regular addition, subtraction, multiplication, division or exponentiation of numbers. That's like when we're speaking about limits and derivatives, or projective geometry, or transfinite induction, or anything like that. In all these cases, mathematicians are applying some new sense to operations with infinities.

Answer 7

Infinity minus itself, when dealing with limits, is treated as zero. Both values cancel themselves out. 0/0 and infinity over infinity, is according to L'Hopital, undefined, if you're doing limits you would use L'Hopital's theorem to find it. 0^Inifinity = zero, just as 0^1 would equal zero. Infinity^Infinity would either remain written as just that, or be explicitly undefined.

Answer 8

These are all undefined as written.

However, if you're dealing with limits which yield "an expression that increases without bound subtracted from or divided by another expression that increases without bound" (e.g.), then L'Hopital's Rule allows for solution to specific instances.

Answer 9

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Answer 10

theyre all undefined. infinity can be any hugely big number so the answer could be anything so its just undefined