# What does inversion mean?

I am in highschool taking some advanced math courses and I have some questions about terminology.

There appears to be more definitions to the meaning of inversion in math than I can count. I'm hearing the word invert used in seemingly different ways without warning of switching the meaning.

I thought there was a single definition which was to take the reciprocal of a number. This definition seems to get almost no acknowledgement anymore.

Instead inversions are talked about in terms of inverting a function i.e. to solve for an input variable. Or I'm told to take the inverse of a number and add it to another when the meaning is to take the opposite of the number and add it e.g. the inverse of 3 --> -3.

It seems odd that the word inverse is being used where there are already words for said operation, like the latter example in the preceding paragraph. It also confuses me because I don't know whether to take the reciprocal or the opposite of the number and do something with it.

Lastly, the symbol inverse is used in notation for expressing the arctan, arcsin and arccos functions. Is there any intended correlation to the inverse definition in this use case?

A helpful answer here would include a list of all the different uses of the word inverse and when each of them are used. If possible, a definition of the word so that I can extrapolate its meaning if it's used in a context where I forget the precise use case.

• Impressive that as a high school student you're aware that there's something here and that you can phrase your question so well. – Git Gud Apr 13 '15 at 22:05
• There are occasions when saying "additive inverse" is suggestive and useful, but dropping the "additive" part of the term should not be allowed if it could be confused with multiplicative inverse. When working at the elementary level (e.g. with rationals or reals) it's better still to just say "opposite," which is what I do and what I believe I see far more frequently. In any case, additive inverses, multiplicative inverses, and functional inverses are all special cases of inverses with respect to a binary operation - listing too many different usages of "inverse" can obfuscate the unity here. – anon Apr 13 '15 at 22:08
• OP, you are quite right that "inversion" is one of those words in math that has multiple precise meanings, that depend on context. Others include "canonical", "fundamental theorem", "natural", and even "prime". – vadim123 Apr 13 '15 at 22:10
• Other examples might include normal, regular, perfect, simple, separable, elliptic, primitive, elementary, nice, complete (all of which I stole from the relevant MathOverflow thread). – anon Apr 13 '15 at 22:16
• It means things like "standard," "prescribed by an explicit rule," "natural and unique in a certain sense." – anon Apr 13 '15 at 22:18

This is another grey area with mathematical terminology. Inversion can mean many different things in many different instances, but in general, it has to do with swapping two things, or creating something's "opposite." Each definition you give of inversion or finding an inverse can be made more precise with more words: $1/a$ is the multiplicative inverse of $a$, $-a$ is the additive inverse of $a$, $f^{-1}(x)$ is the inverse function of $f(x)$, etc. It can become confusing if one is not careful in context. For instance, suppose we are just talking about functions, and I bring up the topic of inverses. It is clear from context that every inverse we talk about is going to be the inverse function. If we want to make reference to $1/a$ or $-a$ at that time, we might instead use the words reciprocal and opposite, respectively. It just happens that saying "inverse" instead of the more descriptive "multiplicative inverse" say, is more brief and can at times be unnecessary.

All in all, there can be many different terms for a single operation, and there can be one term that applies to many different operations. This is quite common in introductory mathematics, as the different words are used to convey different meanings within context. You can see this with terms like "inversion" and "prime," but in each setting, you should strive to make their meanings precise and leave no ambiguity (but certainly within reason).

(P.S. Yes, arcsine is the same as inverse sine. Here is the functional inverse)

Wikipedia's Inverse(Mathematics) has a lengthy list of interpretations as there is also negation that can be inverted on a Boolean value(turning true to false or false to true), inverting a permutation, inverting a linear transformation and other operations beyond addition and subtraction as logarithms can be inverses of exponential functions in some cases. The reciprocal of a number would be a "Multiplicative" inverse where 0 would be the classic number without one.

I think the primary point of interest here is that many English (or really any natural language) words can have more than one meaning, and it can be frustrating. Like the comments say, the context is the only thing that can really help you identify what the words mean.

In any given mathematical context, there is a canonical meaning for each term. When speaking about rational numbers, the canonical meaning for the word inverse is that number's reciprocal. When speaking about matrices (perhaps containing rational numbers), the canonical inverse is something entirely different.

Your question seems motivated by dissatisfaction with this state of affairs, and I completely relate. However, I think the notion of canonical terms is extremely beneficial to communicating mathematics. Imagine if there were a different word for 'inverse' in every possible context. How many words would you have to remember?

Imagine sitting in a lecture/talk about a subject in which you are not completely familiar. If the speaker uses the word inverse without definition, you may not know precisely what they are talking about, but your past experience with inverses could help you fill in the gaps to sufficiently follow along with the lecture. If they used some other word that you never heard before, you could end up completely lost.