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

Question

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?

Answer 1

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 $\infty - \infty$.

Answer 2

$\infty$ 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 $\mathbb{R}$.

Answer 3

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.

Answer 4

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.