Adding infinite and finite numbers: why doesn't 0=1?

Okay, so,

$$\infty + 1 = \infty$$ subtract infinity from both sides. $$1=0$$

At first I thought, duh, $\infty \neq \infty+1$, but, now, I'm just more confused because my brother rephrased it in terms of geometry, and it seems to hold there i.e., if you have a ray of infinite (unbounded) length, and then you start a parallel ray one unit behind it, how long is the new ray? I want to say infinite, but then, if you subtract the length of the ray beside it, then the result is the same as in the first problem.

Is there any way someone could explain why this doesn't work?

• Infinity is not a number, hence you cannot apply normal arithmetic to it. You could argue that $\lim_{x \to \infty} x=\lim_{x\to \infty} x+1$ but that holds a different meaning. Jul 16, 2018 at 19:24
• You can't subtract infinity from both sides because it is not a number Jul 16, 2018 at 19:25
• Some people work with $\infty$ as a "number", but even those won't even dare to subtract infinity from infinity.. Jul 16, 2018 at 19:45
• Many answers have mentioned that "infinity is not a number," but not many have tried to explain why. If you are the curious type, I recommend watching the VSauce video How to Count Past Infinity which is the best layman's introduction to how infinities work that I know of. It does a great job of capturing some of the wierdnesses that appear around infinities and how we try to conquer them. Jul 17, 2018 at 0:48

When we extend arithmetic to include $\pm \infty$, the arithmetic operations (and various other functions) are defined by continuous extension.

This means that we leave $\infty - \infty$ undefined (much like how we leave $1/0$ defined).

In more detail, if $x$ and $y$ are extended real numbers, then we define subtraction in the maximal way so that, whenever subtraction is defined, we have

$$\left( \lim_n x_n \right) - \left( \lim_n y_n \right) = \lim_n \left( x_n - y_n \right)$$

This means that $x-y$ is only defined if the right hand side has the same value for every pair of sequences $x_n$ and $y_n$ that converge to $x$ and $y$ respectively.

It's easy to find two sequences that converge to different limits:

• If $x_n = y_n = n$, the right hand side is zero
• If $x_n = n+1$ and $y_n = n$, the right hand side is one

Thus, we don't define $\infty - \infty$.

The whole idea of $\infty = \infty + 1$ is not properly defined, as is often the case for arithmetic and even geometry involving infinity. Length is defined to be a real number and, as has already been pointed out, infinity is not a real number. So the idea that there are rays of infinite length is not properly defined.

There are instances outside usual geometry and arithmetic where infinity (or rather infinities) are defined and equations with them can be created and make sense. I would direct you to learning about ordinal numbers (https://en.wikipedia.org/wiki/Ordinal_number#Arithmetic_of_ordinals) and cardinal numbers (https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic) if you're looking for arithmetic involving the infinite.

For example, $\omega$ is defined as the first infinite ordinal and arithmetic with it can be kind of interesting. For example, $$\omega + 1 \neq \omega$$ but $$1 + \omega = \omega$$ So some of the more comfortable and familiar properties like commutativity of numbers breaks down with infinity. And as such, thought experiments like mental pictures involving lines and rays, or sets and orderings need to have their terms carefully defined in order to have any real meaning.

Hint:

Here is a quiz: as $\infty+1=\infty$, the two sides are interchangeable.

So what is meant by $\infty-\infty$ ?

• a: $\infty-\infty$,
• b: $\infty+1-\infty$,
• c: $\infty-(\infty+1)$,
• d: $\infty+1-(\infty+1)$,
• e: none of these,
• f: all of these.
• I can't see how this is a helpful answer to OP. It seems to be a comment, since it does not answer nothing, although provides interesting insights. Jul 16, 2018 at 19:41
• I understand your frustration with this often-asked question. That said, it's hard to see how this could lead to the OP realizing "I see - the usual rules of arithmetic can't be consistent if infinity is a number. Now I understand." Jul 16, 2018 at 20:02
• @EthanBolker: no, this is not my message. The message, though not made explicit is "how could you deal with something that has several values that are at the same time different and equal" ? Infinity doesn't have a "memory" of what was added to it.
– user65203
Jul 16, 2018 at 20:07
• @rafa11111: an answer need not be written in the affirmative mode.
– user65203
Jul 16, 2018 at 20:09
• I like this answer. Makes you realize why infinity cannot be a number, instead of just telling you that it is not. Proof by contradiction of sorts. Jul 16, 2018 at 20:57

$\infty$ is not a number. On the real number line, $\mathbb{R}$ we only have numbers. The extended real numbers, $\overline{\mathbb{R}}:=\mathbb{R} \cup\{-\infty;\infty\}$ that has a property that $\forall x \in \mathbb{R}$, $\infty + x =\infty$ so you couldn't subtract it. This also holds for other fields rather than $\mathbb{R}$.