What is infinity divided by infinity?

Question

i am getting 5 values..... ((undefined)^-1, infinity, and undefined, and zero...and even one....i am not confining this q to mathematics....bring in theoretical proofs if possible...try substituion......and yeah one more thing....undefined and infity are not one and the same....i could prove this theorem...

to the first answerer.....ur answer says that the final answer cannot be defined and is hence "undefined" and not infinity......infinity= a set of numbers whose end and starting point are undefined.....

umm...i need to see the proofs....

Answer 1

I dunno. I don't think an actual answer can be reached, since infinity is something we can't fully grasp. I mean, if you have an unlimited supply of something, and must distribute it evenly among an infinite amount of people...1, maybe? But even that solution won't hold up very well.

In response to your additional details, and the first answerer, infinity is not "any number". It is not even a number, so much as a concept. It means unlimited, neverending. It would be the highest number possible. Not an unknown number. That is what we use a variable for. No offense to the first answerer, but I really don't think he understands.

Answer 2

Infinity Divided By Infinity

Answer 3

Infinity Over Infinity

Answer 4

Ok. 7/3 means 7 x 1/3 where 1/3 is the rational number that we can multiply times 3 to get a 1. We say that 1/3 is the multiplicative inverse of 3 just as -3 is the additive inverse of 3. Division by 0 is undefined because 1/0 does not exist. There is no number, that is consistant with the axioms of the rational numbers, which can be multiplied time 0 to get a 1. Now "infinity" is not a rational number. We do not add it to the rationals because it is not needed. It is a useful symbol, but it is not a rational number. Writing plus and minus infinity as numbers simplifies notation but we are free to define objects like infinity divided by infinity in any way that we please so long as it is useful and its usage does not conflict with the rules of the number system we are working in. Logic is once again a growing field of mathematics and much used in the computational sciences. There are many usefuls structures yet to be created/discovered.
Have a nice day

Answer 5

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Numbers can't divide; we divide. We divide 12 by 3 to get 4; the number 3 can't divide anything; it isn't a sentient being, and neither can infinity. It depends on what you mean by infinity. If you mean potential infinity, maybe you mean something like (1/x) / (-1/x) when x = or approaches 0. If you take a number close to 0, you can perform the divisions. Simplifying the expression gives you -1 regardless of x, so if x approaches 0, this expression approaches -1. I.e., infinity divided by negative infinity is -1, not negative infinity. Actually the quotient infinity / (-infinity) is indeterminate and could be any value, including infinity. If you mean actual infinities, such as the transfinite ordinal number w (omega), your equation still does not hold, for in the surreal numbers, w / (-w) = -1, not -w. You can cancel the w's as if they were 2s or something. You need to take a higher ordinal infinity such as w^2 to get -w: w^2/(-w) = -w.

Answer 6

Everyone else seems to be wrong, so I'll clear things up here. Infinity divided by infinity is not one, or infinity, or even nothing: infinity divided by infinity is not a mathematical statement. Infinity is not a number, so the normal operations of addition, subtraction, multiplication, and, indeed, division, and composition cannot be applied. To begin with, I answer the question of, "What is infinity?" In the most basic terms I can think of, while still being mathematically correct, has very little to do with numbers. Here it is: for any number you can think of, no matter how large, I can add 1 to that number. Therefore, infinity is not even close to a number; it is a description of the real number system, which has an analogue in the complex number system. If infinity can't be defined in terms of numbers, then you can't apply division to it. Done and done.

Answer 7

As I understand the concept infinity, it is infinite and there is nothing beyond infinity because if there was, then infinity wouldn't be infinite. My apologies for belabouring the issue, but the point is this. When little kids are arguing and the one makes some claim saying he does it "infinity" and the other responds "oh yeah, well infinity +1" and the first retorts "oh yeah, well infinity squared" etc etc etc., the grammar of their statements notwithstanding, mathematically "+1" and "squared" have not increased the value. Infinity is infinite, the first kid won the argument. It seems logical that if infinity cannot be increased by addition or multiplication, then neither can it be diminished by subtraction or division. The problem, it seems to me, lies in thinking, albeit mistakenly, of infinity as actually just a really big number to which the standard operations with their mathematically defined answers (division by zero, multiplication by one, etc.) apply. But as already noted, this is not the case, infinity is infinite. Infinity may well be the ultimate identity "value" - ANY mathematical operation or series of operations performed on infinity will always yield infinity.

Answer 8

Here is the problems involved in such a question:

- People consider infinity to be a number which is wrong. Infinity is a concept.
- You can't go into the concept of infinity without discussing the levels of infinity involved. I believe it was Cantor who determined that there are different types of infinity.
- Somehow I want to believe this idea is motivated by a ratio of functions going to infinity. Chances are that in this case the real problem can be solved via methods like L'Hopital's Rule.

Simple idea that proves a contradiction based on the original statement of this question. Consider a ratio of the measure of the real numbers to the measure of the rationals. This was the basic conception. Even though both are infinite, in any interval, despite the fact that Q is dense in R, the measure of the reals is larger than the measure of rationals. Hence the ratio would go to infinity for all non-degenerative sets. Now flip the ratio and it will go to 0. If you compare the measure of the rationals to itself, using Cantor's logic, you should possibly get a ratio of 1.

All in all, without more detailed values of what infinities you are considering, it's like debating 0/0 because right now I can set the ratio to many values.

The traditional answer though is undefined.

Answer 9

Mathematically, any number divided by itself is 1 so it would equal 1. If you mean by technicality, it would be unknown because infinity is an on going number so it is always changing, so you would not be able to divide that. I also agree with Abhimanyu Khurana because 0/0=infinity/infinity.

Answer 10

Wheeeeeee!!!!!!!! This should be fun

To begin with, 'infinity' really means (in a mathematical sense) 'increases without bound' or 'without limit'. And that leads to some pretty interesting things.

If you had a large box of bolts and a large box of washers, how could you tell if you had the same number of washers as bolts? (Without bothering to count both boxes)

You'd take a bolt out of one box and put a washer from the other box on it and continue until one of three things happened:

a: you run out of washers
b: you run out of bolts
c: you run out of both

In case (a) you have more bolts than washers.
In case (b) you have more washers than bolts
In case (c) you have exactly the same number of bolts as washers.

Now consider the number line. Are there the same number of points on the closed interval [0,1] (we'll call it A) as there are on [0,2] (which we'll call B)?

If I can prove that every point on A 'matches up' with one and only one point on B *and* that every point on B 'matches up' with one and only point on A then it must be that they have the same number of points. OK let x be a point on A and y be a point on B. If I then write

y=2x and x=y/2

I'm done. All of the points of A and B are in one to one correspondence even though B is twice as long as A. In fact, B could be a million times as long as A and we'd get the same result. They'd both have the same number of points.

Try this. Draw a horizontal line. Choose a point on the line and call it '0'. That makes +∞ be waaaaayyyyyy off to the right, and -∞ is the same distance to the left. Now draw a circle above the line that just touches the point you called zero. Make a point at the very top of the circle and draw a line from it to any point on the line and it will cross the circle at one point. This gives us a way to 'map' every point in an infinitely long line into one and only one point on this circle. Best part is, +∞ and -∞ map into the *same* point (at the top of the circle). This is an example of what's called a 'conformal mapping' and they're *very* important in higher math.

You can do the same thing with a sphere setting at the origin of the x,y plane and map the entire x,y plane onto it's surface (with infinity in all directions being the point at the very top) That one is so important that it's called a 'Riemann sphere' after the German Mathematician who first developed it and used it to show some very important and useful things (again, in higher math)

So.... What is ∞/∞? It's every bit as meaningless and undefined as 0/0.


Doug

Answer 11

This question is not answerable with high school mathematics. If you want to know how to answer it, you will like calculus and you should definitely take it!

Calculus is the mathematics of limits, which is a way of looking at an expression as one or more variables tend toward the final limiting value (when it is problematic).

For example, take the expression f(x) = 1/x

If x = 0, the answer is indeterminate. However, if we use limits we can look at the behavior as x approaches the problematic value of zero.

l"imit x=>0+" means "The limit of the function as x approaches the value zero from the right". You can try this with a calculator by substituting numbers larger than zero that get progressively smaller and smaller.

if x = 0.1, f(x) = 1/0.1 = 10
if x = 0.01, f(x) = 1/0.01 = 100
if x = 0.001, f(x) = 1/0.001 = 1000
and continuing in this way we see that f(x) keeps getting bigger as we make x arbitrarily close to zero (but positive)...this means that the limit f(x) x=>0+ is +infinity

Along the same lines, we can look at the limit for negative values of x that get closer and closer to zero:
if x = -0.1, f(x) = 1/-0.1 = -10
if x = -0.01, f(x) = 1/-0.01 = -100
if x = -0.001, f(x) = 1/-0.001 = -1000

And we see that f(x) gets to be a negative number whose magnitude increases without bound. This means that the limit from the left (lim x=>0-) is negative infinity.

Generally, we say that a limit exists when the right and left hand limits agree.

You asked about dividing infinity by infinity. In calculus, with limits, there are different "degrees" of infinity. You can see this by trying to take limits:

Let f(x) = x/x (a number divided by itself)

We can't directly calculate this expression at zero or +infinity because division by zero and infinity/infinity are not defined. However, using limits, we can try simplifying it first:

limit x=> 0 (x/x)
Cancel out the x in the numerator and denominator:
limit x=> 0 (1)
And we see that the limit as x=0 of this expression is 1.
In the same way, we can see that the limit of (x/x) as x=> infinity is also 1.

It is possible to construct an expression so that the limit of "infinity/infinity" is anything you want!

What is limit x=>infinity (2*x/x)

If you simplify this one, you find that "infinity/infinity" is two!

How about limit x=> infinity (x/x^2)

This simplifies to limit x=> infinity (1/x)

Which approaches 0+ as x=> infinity...so "infinity/infinity" is zero in this case. (try bigger and bigger numbers and see what the limiting value of the expression is in your calculator)

I really hope you take calculus. If you like numbers this class will help you understand a lot of cool ways to use them.