Infinity and Negative Infinity Logical Query

Question

If infiniti is the highest theoretical number then what is the opposite of it? At first glance one might think 0 but that is not true. Research (aka a quick google search) has shown me that the opposite of infinity is called infinitesimal(sorry if I spelled that wrong) which is another unquantifiable number but it is basically zero point infinite zeros followed by a single one, as I understand it. Now if the definition of negative is “the opposite of a number” and we know that 0.000… infinitely, followed by a one, is the opposite of infinity, then we know that because of the transitive property then -infinity=0.0…1. This means that the highest theoretical negative number is actually a positive number. This leads me to think that there are no negative numbers and that all negative numbers are just decimal. Is my logic completely flawed or does it make sense?

Answer 1

Is my logic completely flawed or does it make sense?

Your question makes a lot of separate claims that each have different issues, so I think the best way to clear things up is to comment on some individual lines.

Clarifying the Logic

If infiniti is the highest theoretical number

"Infinity" (note the "y" at the end in English) is not just one idea, and the ideas related to infinity are not always called "number"s, and never have all the nice properties that "real numbers" do. For a survey of some related ideas, see the English Wikipedia page for Infinity or the Math StackExchange question Understanding infinity.

what is the opposite of it?

Dictionary.com has a definition for "opposite": "contrary or radically different in some respect common to both...". This can me used in a variety of different ways in mathematics. Here are two main ones:

1. In arithmetic, "opposite" often means the additive inverse or negative of a regular finite number. So you might hope that a given concept of infinity has something analogous, like a "negative infinity".
2. If you think of infinity as in some sense "very big (and positive)", then a different application of the definition would make the opposite "very small (and positive)". It is in this sense that the word "infinitesimal" may be used. For regular finite numbers, this kind of opposite might be the multiplicative inverse or reciprocal.

then we know that because of the transitive property...

This entire line of reasoning is invalid because you are treating the two ideas of "opposite" above as if they are the same thing. As HallaSurvivor alluded to in a comment, this is very similar to saying: "$-2$ is an opposite of $2$ because it's the same size but negative. And $1/2$ is an opposite of $2$ because multiplying by $1/2$ shrinks something by the same factor that multiplying by $2$ grows it. Since they're both 'opposites', $-2=1/2$."

Clarifying Infinity

As discussed above, there are at least two senses of "opposite" you might be interested in: something like "negative infinity" and something like "reciprocal infinity". What sense these do or do not make depends a lot on what meaning of infinity you're working with. Without getting into all of the possible interpretations in detail, I'll just outline a few key examples.

Beyond regular decimals?

zero point infinite zeros followed by a single one

This would not represent a real number, and does not have a standard definition. People have tried making up new number systems where this sort of thing would convey some meaning, but it's not easy and to my knowledge hasn't ever been shown to work out nicely. This matter is discussed a bit in answers to the Math StackExchange question Is it possible to create the smallest real positive number by axiome?.

Calculus $\infty$

In Calculus, the lemniscate $\infty$ is used to represent an idea like "a function or sequence gets (and stays) greater than any finite positive number". For instance, we might say "the limit of $(1,4,9,16,\ldots)$ is $\infty$" because the sequence stays above $100$ after the tenth term, stays above $1000$ after the $31^{\text{st}}$ term, etc.

Analogously, it's common to then use $-\infty$ to represent things that get and stay less than any finite negative number". For example, $(17-1,17-4,17-9,17-16,17-25,\ldots)=(16,13,8,1,-8,\ldots)$ might be said to have a limit of $-\infty$. Note that we cannot usefully think of this $-\infty$ as an additive inverse to $\infty$. For example, if we add the two sequences above term by term, we get $(17,17,\ldots)$ which stays at $17$ and does not approach $0$.

As for $1/\infty$, since the reciprocals of a sequence that gets larger and larger get smaller and smaller, some may write $1/\infty=0$. In this context, people would not generally call that an infinitesimal, just $0$.

Complex Calculus $\infty$

When doing Calculus with the complex numbers in "complex analysis", we are not limited to two directions on the number line since there is a whole complex plane to work with. In that context, we often use the symbol $\infty$ to represent things that get and stay further away from $0$ than any finite positive distance, no matter the direction. A way of visualizing that is the Riemann sphere. In that usage, both sequences above have numbers moving away from zero, so their behavior might both be represented by $\infty$, and there is no $-\infty$ (or perhaps we would declare $-\infty=\infty$).

Similarly to the real numbers, reciprocals of complex numbers far from $0$ are close to $0$, so it would be common to write $1/\infty=0$.

Infinite Sizes

Another common use of the ideas of infinity is in giving names to the sizes of infinite sets in the study of "cardinality". But every set has at least $0$ elements, so "negative infinity" would make no sense in that sort of context. And it doesn't make sense to have between $0$ and $1$ elements, so "reciprocal infinity" wouldn't make sense either.

Abstract Settings

In more obscure mathematics, we might have a sort of "number system" where arithmetic and order work out in a fairly normal way, but there are now new numbers larger than any positive integer. Many of these are called "non-Archimedean ordered fields". In such a system, there is not just one "infinity" so we would not use the symbol $\infty$, but we could have "an infinite element $H$" and then things like $1+H>H$, $-H+H=0$, and $0<1/H<1/100$ might all make perfect sense.

In this context, "positive infinite" would mean "greater than any positive integer". And "positive infinitesimal" would mean "positive but less than the reciprocal of any positive integer". $0$ might be considered an "infinitesimal", but $1/0$ would still not make sense.