To Infinity And Beyond!

To see that $\infty - \infty$ is undefined, let's take a look at some common sets that are infinite. First, consider the natural numbers $\mathbb{N} = \{1, 2, 3, 4,...\}$ and the subset of even numbers $\{2,4,6,8,...\}$ . Note that these sets are the same size because for every natural number $n$ there is a corresponding even number, namely $2n$ .

Now, if you remove the even numbers from the natural numbers, then you are left with only the odds: $\{1,3,5,7...\}$ , which will also be same size. So, in this sense we removed an infinite number of elements and were left with an infinite number of elements, so you might say $\infty - \infty = \infty$ .

However, suppose we now remove the set $\{2,3,4,5,...\}$ from the natural numbers. This time we are removing and infinite set from an infinite set to leave only one element, so you might say $\infty - \infty = 1$ . This shows that the difference isn't well defined in general and doesn't even take into account uncountable infinities.

To those who might be confused google Hilbert's hotel paradox it will clear the Idea

As infinity is itself ""not defined"' hence any simple operation with infinite is not defined

It can be a large infinite minus a shorter infinite, or a short infinite minus a larger infinite, or equal infinites, or a large infinite minus a larger infinite, you can't define how long will the infinite be or if it can be a normal number, then, it's undefined

(1/0) - (1/0) = 1-1/0 = 0/0 = un-defined

We can say that two given quantities are equal if we can measure them. But infinity is an abstract concept and it can neither be measured nor manipulated using ordinary mathematical laws.

One simple example to prove my point is that $\infty+1=\infty$

Infinity minus infinity could be infinity or 0. The answer varies with what function is being run or what set is being analyzed. Without a defining function(s) or set(s), there is not enough information to solve the problem - or even define it. There are also different types of infinities. The set of integers and the set of real numbers are both infinite, but differ in size

There are two circles, one with a radius of 1 and the other with the radius of 2. You can fit an infinite number of points into both, but the first circle is still smaller than the second.

Not defined

In fact we use the term "unspecified" to (infinity - infinity) not undefined, there are 7 quantities are "unspecified" from which 0/0, infinity/infinity, infinity-infinity and 0 times infinity

Infinity is not a number. Just an idea.

The easiest way I found to think about it is as such. Infinity is not a "number" so we can not perform real calculations on it. So to help ourselves understand it better we use "real" terminology to help ourselves understand it. This is just like how the "house" in the picture below is only a representation. It's a pretty crappy house and nobody would ever live in that house because it has no windows but it is only for representation purposes.

(https://d18l82el6cdm1i.cloudfront.net/uploads/9YwKJGl9fQ-image.jpg)

Now to continue with our "algebra" you must remember that these calculations are representative and however not proven fact they simply help us understand infinity better.

Let's say we have a library with an infinite number of books. Every book is named a number and there is a book for every natural number and each is named their respective number. If a customer were to enter the library and say "I wish to borrow all the even numbered books." Then there are an infinite number of even numbered books being borrowed and taken from the library. The calculation for the number of books left being :

(total library books [infinity]) - (even numbered library books [infinity]) = (odd numbered library books [infinity])

Which breaks down to [ infinity - infinity = infinity ]

Now if overnight after all of the book had been returned two burglars were to come in and one of them stole all of the even numbered book so and the other stole all the odd numbered books our calculation for the number of books left over would be:

(Total library books [infinity]) - (even numbered library book + odd numbered library books or also "total library books" [infinity]) = (no books left in the library [0])

Which really breaks down to [ infinity - infinity = 0 ]

Giving the answer to the same " algebraic" equation different outcomes. I am hesitant in using the term "answer" as it is not an answer for it is not one true fact but many. But it all sums up to the fact that infinity minus infinity can only be undefined as there is no "one true outcome"

If you took the time to read through this I thank you and would very much appreciate hearing what other people think of my method!

Everything minus everything isn't nothing. It's not defined. It's not possible to take everything from everything conceptually speaking. If you did, you would be left with nothing, which is to say something. Which obviously isn't nothing at all and therefore impossible to determine.

I'm seeing some really clever and intelligent solutions on here, but the main reason I knew it was undefined is because infinity isn't recognised as a proper number, it's a more a concept. I think it's important people know that, although there are some great solutions on here that would make it so easy to understand, so people should definitely check those out.

Suppose $n$ is some number, where $\infty-\infty = n$ .
We know that $\infty+\infty=\infty$ .
So $\infty-\infty=n$
$(\infty+\infty)-\infty=n$
$\infty+\infty-\infty=n$
$\infty+(\infty-\infty)=n$
$\infty + n = n$
$\infty = 0$
However, $\infty \neq 0$ .
Therefore, $\boxed{\infty - \infty = undefined}$

$\infty$ is nothing but number which is undetermined, even $10^{1234567}$ is $\infty$ and also $10^{100000000000}$ , so you cannot determine which number actually denotes the infinity. So that's undetermined.

Infinity is just a concept, never treat it as a real number.

we know something divided by 0 is called infinity . so if make infinity - infinity =0/0 its undefined

I think of it this way: infinity - infinity could equal 0, if the two different infinities diverged at the same rate. However without any functions the value is undefined, because we have no knowledge of the rate that the infinities were reached.

Although it wasn't an option, I think it should be indeterminate, which is different from undefined. See https://en.wikipedia.org/wiki/Indeterminate_form. Because for different infinities if you will there are different solutions rather than always being undefined it is considered indeterminate.

Any rules of mathematics is now applicable with 0 & infinity

One common mistake is that people make is the idea that infinity is a number. It's not a number but a concept. It means to go on without end. You can't treat it like a numeric value because that would be giving it an end. Thus an operation such as this isn't possible

It actually depends on preceding limiting variable. Suppose there is a variable 'a' limiting to infinity and variable 'b' also limiting to infinity. 'a' is defined as a = क(a) and b is defined as f(b). The Two different function can approach infinity differently. So a - b depends on the value of k = क(a)/f(b) i.e. if k = 1 then a - b equals 0 i.e. constant value. If k > 1 then a - b equals a. and k < 1 then it is -b.

Lol .... I remembered the solution from fault in our stars that one infinity can be greater or less than other infinity so its should be undefined

Am not sure of this. Since infinity =1/0. Therefore infinity-infinity =1/0-1/0=0/0. But 0/0=undefined. Then the answer is undefined

To be technically correct, the answer should read "Indeterminate", not "Undefined". The word "Undefined" is for expressions that have NO answer, like 5/0, since there is NO real number that you can multiply by zero to get five. The expression 0/0 is indeterminate, since ANY number times zero equals zero. Many commenters have shown how you can remove a set of infinite cardinality from a set of infinite cardinality and leave zero, some finite number, or an infinite number of elements, so there is not ONE answer, but an infinite number of possible answers and thus it is indeterminate.

i just thought of infinity as 'unknown' so if you take unknown-unknown The answer is unknown or in other words 'undefined'

$\infty$ is unreal, so any calculation with it is meaningless. Also, there's no definition of Infinity's calculation, so undefined

The infinity number doesn't represent any particular number, so you don't know the answer. That's why it is undefined. Simple.

Although it may seem that infinite minus infinite equals zero (after all, they both are the same numbers) infinite is not a defined number. It keeps growing. As taught in elementary school, x-x=0- for defined numbers. But as stated before, infinite is not a defined number. Infinity grows to a constant number, then grows again to another constant, then grows again. The other infinity is the same; it may lag behind, then catch up, etc. It's like watching a never ending horse race; one horse may take the lead, another may catch up, then the first will take the lead again, etc.

Infinity=1/0 So infinity-infinity =1/0-1/0=0/0, which is not defined

First of all "infinity" is something which is not defined.therefore any function involving "infinity "is not defined!!..that's wat I thought!;p

Infinity taken away from infinity= undefined. How could you measure infinity minus infinity? How can that be given a value? The first thought would be 0. Although, 0 is measurable. What if there wasn't a 0 to measure?

I also saw this as Reality - Reality=? (Reality, everything in literal existence, all matter). If there was nothing then you couldn't measure it because there wouldn't be anything or anyone to measure it. Thus it makes no sense, thus it's undefined.

infinity is not a quantity but rather it is a direction.

Infinite number means a very very extremely large number . So if a very large number is subtracted from another large number then we cannot actually predict the result. It can be 0 , infinite, or a finite number. Therfore the answer is undefined.

Technically, infinity isn't a real number, so anything involved with it isn't really a real number either.

We. Don't known that undefined what is number so undified minus undefined is undefined thank frnnds

Well if we see at this way that infinity is equal to 1/0 then the solution is quite clear. And answer will be unrefined

It is one of the indeterminate form

that is undefined cause when we talk about infinity , we talk about something which is very large but we can't say definitely what the number is ............ INFINITY is something which is larger than the number the number we r thinking instantaneously................ so its absurd to guess what infinity minus infinity will bw

Simply put, infinite numbers cannot be either zero or one, therefore, so subtracting infinity by itself is indeterminate.

Undefined is the only technically correct answer, as infinity is a placeholder for undefined equations and doesn't have a value. Although, in more advanced physics you can assign a value to an infinite sequence defined as {1+2+3+4+...}. This can be done in a very specific way (no, not as "done" by numberphile), which comes to -1/12. Considering that, you could subtract the values that you assigned to come to 0, but that isn't infinity minus infinity, simply the value of one sequence subtracted by itself.

Imagine it like this. A rabbit and a tortoise are in an infinite race. They will continue to run for infinity but will obviously travel at different speeds. At any point of time the rabbits infinity will be larger that the tortoises due to the rabbit being faster. This means that you can have an infinity larger than another infinity. The answer is undefined because it is unknown the 'largeness' of the infinities.

I thought infinity - infinity was indeterminate? Is that the same as undefined or am I wrong?

answer is undefined but according to Vedic mathematics its infinity

Suppose 5/0 -3/0 Then this can be written as 5-3/0 which is equal to infinity. There we can say infinity - infinity = infinity

Infinity can be any set of numbers making the answer improbable to find. Therefore making it undefined. Infinity is not a variable that can be supplemented with only one number. EG the variable "x" could be substituted with the number 5 and it would have to stay at 5. With infinity, it can be any possible number, making infinite amount of possible solutions.

Man, this is basic Calculus. Correct answer is UNDEFINED. No need to chat here if you do not have basic understanding of the limit concept. Infinite is a tool only with clear purpose to deal with functions (or other math entities) that have a tendency to grow indefinitely when argument approaches a given value (E,G, tan(Pi/2)) So please keep away from posting nonsense if you do not have the proper induction in this kind of matters.

Since infinity is undefined value for any number ,therefore the given expression can neigher be ZERO,nor any other value and it is defined....Ans

A growing number cannot be subtracted by a growing number so if we hold this to be true the answer would also be changing so it cannot be defined. Undefined is correct.

Infinities can be created from any range of numbers. For example, between the two numbers 1 and 2 there are an infinite amount of numbers but the difference between them is only 1. Also, between the numbers 1 and 100 there are also an infinite amount of numbers, however their difference is 99. So it is undefined because one infinity can actually be smaller than another

It is undefined because the size of the infinitys. Can change, we can talk about different infinitys as we don't know yet how or of what an infinity is composed. We just know it's really big and we cannot understand it

Infinity isn't really a number, so you can't do the same things with it. Infinity is "how many" of something that cannot be counted because there is no end to it. Since infinity isn't a number, what "minus" means isn't really clear.

Infinity doesn't have a value. For example, if you keep counting upwards in a number line in whole real numbers, there are infinite potential values. There are also infinite potential values between 0 and 1. Since it's impossible to define the value of infinity it must also be impossible to define infinity minus infinity.

muito fácil

Infinity=1/0 (as 1/infinity=0) We know 1/0= not defined then infinity is said to be not defined not defined - not defined = not defined Therefore infinity-infinity=not defined.

There are differents infinities ? Or is it the largest number ? We do not have any idea about infinity ? I don't understand why is undefined

There is somewhat of a flaw in this question. The issue is that infinity is not a number, but rather a concept, and thus it is inappropriate to pose the question as "what is infinity minus infinity?" A more appropriate way to pose this question would be, "What is the limit of f(x,y)=x-y as x approaches infinity and why approaches infinity?" The answer depends on the aleph of infinity of x and y. With additional information, such as x=y, or 2x=y, or x=2y, we can obtain different answers, such as 0, infinity, or negative infinity. Without additional information, the question is undetermined and therefore the solution undefined.

Infinity isn't really a number, so you can't do the same things with it. Infinity is "how many" of something that cannot be counted because there is no end to it.

Moreover since ininifty is a number , you are indefinite of what minus is then.

The answer wouldve been zero if both the numbers subtracting were same.

The answer could be infinite. If you have all numbers, and you remove all even numbers, you still have infinitely many left.

There would be more possiblities with negative numbers. Thus, subtracting infinity minus infinity isnt really definite

Couldn't you use l'hopital's rule to argue that the answer is zero?

Infinity is not a number so we dont know what we are going to subtract . Its just a idea where we cannot reach