# ∞ and NaN

... What does "4+𝜖" mean? Also, it WAS coming to a close, but then it went in a completely different direction.

𝜖 (epsilon) is a constant for a super super small number (like the opposite of infinity) but not 0

In this context, it means that I've written four books (three volumes of Computer Science Logo Style plus Simply Scheme) and part of Logo Works, which is an anthology of Atari Logo projects by several authors.

So I assume there are other contexts?

Yes. 𝜖 just means 1/∞ .

Ok, but what is 𝜖^2? 𝜖^3? 𝜖^∞?

𝜖 for all of them, I think. (Though, I'm not sure about 𝜖.)

Ok.

If 𝜖 is a super small number, then powers of 𝜖 such as 𝜖^2 are even smaller -- a lot smaller. In calculus, where this notation is mostly used, you're supposed to think of 𝜖 as smaller than any finite positive number, but to understand its powers, pretend for a moment that it's 1/1000. Then 𝜖^2 is 1/1,000,000 which is 1000 times as small. Similarly, the infinitely-small 𝜖^2 is infinitely smaller than 𝜖. This actually matters because sometimes you end up doing computations such as (𝜖^2)/𝜖 which is 𝜖, same as (100^2)/100 = 100.

But I just meant "four plus a little bit" rather than anything technically precise! If it were a calculus problem, 4+𝜖 is just 4, provided that that's the final answer to the problem. If it's an intermediate result, you have to keep it as 4+𝜖 just in case later in the problem you subtract 4, and then the 𝜖 is still a dominant part of the value.

Basically you want to end up with 𝜖 gone from your final result. So if the final result is 𝜖 then you just let it be 0.

The really official way to talk about this is that 𝜖 is a variable, just as x is a variable, but you are taking the limit of some expression using 𝜖 as 𝜖 "tends to 0." So if your result is 4+𝜖 you say that the limit as 𝜖->0 is 4.

sometime i think you math guys are fantasizers. it always blows my mind. if there is infinity, it means there is infinity amount of numbers, but it means that there is also infinity amount of numbers after 0.00000..., which means because it is infinity, it never makes to 1, but then it means it is not infinity because it never makes to 1, but if it does it means it is not infinity because there is no infinity numbers after 0,00000..., and if there is-

Right that's why instead of saying "𝜖 is this infinitely small number" you're really supposed to say "do the calculation thinking of 𝜖 as an ordinary small number, but then take the limit as 𝜖 tends to 0" so you never actually have a tiny but nonzero value.

On the other hand, there is also a formal mathematical theory about sets of numbers that can handle infinite-size sets, e.g., all the real numbers, or all the numbers between 0 and 1, etc.

Hardy har har. Infinity is not a number, it is a concept. ∞ is a number, but it is a transfinite number. 𝜖 is just 1/∞.

yeah anything with infinity does not make sense and never will

0^0? 1/0? 0th root of a number? Those don't have Infinity, but they don't make sense.

all those involve infinity actually

How?

I would like to know how 0^0, 1/0, and the 0th root of a number involve infinity?

The symbol "∞" is just an abbreviation for the word "infinity." And you're right, there's no such number, which is why there's no such number as 𝜖 either. They both make sense only as limits. ∞ means "the limit of [some expression involving x] as x gets bigger and bigger." 𝜖 means "the limit of [...] as |x| gets smaller and smaller." (The point of using the absolute value is that the lower limit is zero.)

Transfinite numbers are different kinds of infinite values. The smallest one, which is the number of integers, is called ℵ₀ (pronounced "aleph null"). The number of real numbers, called C, is bigger than that, which gets us back to

The number of (real) numbers between 0 and 1 is infinite. How infinite? There are C of them. In other words, the fact that there are infinitely many numbers between 0 and 1 doesn't mean that any one of those numbers is infinitely big. They're just crammed in really tightly, so to speak. C is called C because it's the size of a Continuum, which is to say the number line, or a piece of the number line, with no "holes" in it -- every possible point represents a number.

This small message can't, of course, say everything there is to say on this topic. Find a set theory text if you want more.

Ok, but @luna.judah said that 0^0, 1/0, and the 0th root of a number involve infinity, and I don't know how that could be.

The reason why infinity doesn't exist is because it doesn't have a defined value. Just like a Quantum Bit - it can have 2 states and even an in-between state.

Infinity could be 1, could be 2, could even be 4 Quintillion.