are infinite and undefined the same thing?

Question

1/0 = undefined


.999999999 repeating = 1

therefor,

.0000000 . . . infinite zeros . . .0001 = 0

therefor,

1/infinity = 0

and

1/(1/infinity) = infinity

so

1/0 = infinity

So are "infinity" and "undefined" the same thing?

SV, I am glad we met because you provide an educated opposition to my opinions. I wish we could correspond through another medium besides the message board, but you do not allow e-mail or IM.
Infinity is a concept, but ALL numbers are concepts. It is easier for us to grasp the concept zero because we can see it on a number line, and we can imagine zero apples. However, that doesn't mean it is any more abstract than the concept infinity.
Counting existed long before the concept zero existed, because it was much more difficult for people to grasp nothing rather than something. In the same way it is very difficult for us to grasp "without end" in the same way as it is difficult for us to grasp "without beggining".
If you want to read an interesting book, read "Zero: The Biography of a Dangerous Idea". It was a book that I have recently read, and describes the nuances of the concept.

Answer 1

I love this question. I once read a book titled 'the journey to infinity'. Half-way through it I got so confused I went to prove 0/0 = 0 and got loads of scary formulae. I remained confused till I became a math undergrad. I'm going to talk about languages first, in the end you will see this chitchat is relevant to our problem.

When someone says 'Cool', we know he is pleasantly surprised, satisfied or excited about something. If someone shouted the same word in a Victorian English street, the gentleman nearby would probably say 'Hm, certainly, it's raining'. By this example I want to show that words don't have a permanent meaning; it is people who gave it to them. When people are not clear about some concepts, the words expressing them are equally weird. Such is the case with '0', 'infinity' and '3.14159265358979323846...'.

The decimal expression as illustrated by the third example has a long-winding, far-fetched formal definition, which you would need to read a couple of books to understand. On the other hand, you can already work with finite or recurring decimals. That's what counts. The rest is bullshit. I hate every high-school teacher who solemnly announces "0.999999999... = 1". The fools don't know what they are dealing with. For now you can forget this formula as long as it didn't appear in the final paper. It's useless. The formal reason step 1 -> step 2 doesn't work is that the number ".0000000 . . . infinite zeros . . .0001" is not defined in the currently popular system of mathematics. '1/0' and '1/infinity' are in the same similar position. Just ignore them.

Let's move to the question of 'infinity'. In my opinion nobody in this world understand what this word means. The mathematicians don't understand either; they simply invented a way to work with it, a way no more mysterious than your ways working with decimals. You memorise certain computation rules. The mathematicians do this also, but they call what they take for granted 'axioms'.

Now that I have hopefully convinced you that your proof did not say anything the popular opinion agrees upon. Your original question however has a simple answer from the linguistic prospective. In mathematics, 'infinity' is undefined, but 'hamburger' is also undefined -- at least till the last time I checked. And we all agree hamburger is not infinity. Therefore 'infinity' is not 'undefined'.

Answer 2

Infinity and undefined are not the same, although they are related.

Infinity is not a number, it is just a concept, and it is meaningless to do straightforward arithmetic with it. If an arithmetic step includes infinity, it is said top be undefined. For example 1/infinity is undefined.

[Edit - see below]

Strictly speaking, 1/0 is not equal to infinity either, because it is not really valid to use infinity in equalities. Rather, 1/0 is undefined. But it is often accepted shorthand to write 1/0 = infinity.


You are correct that .9999 repeating with infinite nines is equal to 1. [For anyone who is unconvinced, one way is to consider that 1/3 = .333 etc and 2/3 = .666 etc, then add them up]

But .0000000...infinite zeros...0001 is not a valid concept.

By putting 1 at the end, you have terminated the string of numbers, so it is no longer of infinite length. To be of infinite length (that is to have an infinite number of zeros) there can be no last number - that's the whole point of infinity.

Hence your assumption that 1/infinity = 0 doesn't work.

[Edit following your later question: the limit of 1/x as x->infinity is indeed 0, but, as discussed by various people in that question, this doesn't make 1/infinity=0 a valid statement, it is just a useful, if dangerous, shorthand for the same thing.]


Consider the function 1/x.
As x tends to zero from above (x is positive), 1/x becomes infinitely large and stays positive.
But as x tends to zero from below (x is negative), 1/x becomes infinitely large and stays negative.
This would suggest that +infinity and -infinity are equal.

But what about the function 1/x^2 ?
Whether x is positive or negative, x^2 is positive. So as x tends to zero from above or below, 1/x^2 becomes infinitely large and stays positive.

This is just one example of how infinity doesn't behave like finite numbers in basic arithmetic.


Then we get to the fact that there is more than one size of infinity, but that's for a different question.

PS I have starred the question because I think this is a fascinating area of maths, where it shows its roots in philosophy.

Answer 3

The answerer who claims that .1111.... is undefined is incorrect. 1/9 = .11111..... and 1/9 is definitely not "undefined". Likewise, 2/9 =.2222...., 3/9 = 1/3 = .33333....., etc. Each example, when represented as a decimal, repeats without end (infinite), but none of these is "undefined". Therefore, to answer the original question, "infinity" and "undefined" are not the same.
Also, there is another example of undefined:
0^0 (zero raised to the zero power) is undefined, but not infinite.
This example is undefined because, 0^n, is normally 0 (unless n itself is 0), and m^0 =1, (unless m = 0), i.e., 0 to any power is 0; and conversely, anything to the 0 power is 1, except for the case of 0^0, which tries to be 0 and 1 at the same time, thus making it undefined.

Answer 4

for example u take the following sequence:

1,1/2,1/3,1/4....
in this the value gradually decreases & approches zero not accurately zero.if u have the value 0.00000000001.u consider this as zero. Infinity is a uncountable vaue ,so u may get like above ans or .0.00000000001,we are writting 1/infinity =0

Answer 5

1/0 = Infinity which simply means not finite and thus not defined in value.

0.999999..............

0.11111111111111.....

are also undefined since the last number keeps on repeating but they are not called infinite.

We need some words to define some specific concepts.

Answer 6

Not always.
For example,
y = (lim x->∞ x^2) approaches ∞. But y = x^2 is defined.

Answer 7

not everytime