# Use of “inverse” to mean reciprocal

Oftentimes I hear people referring to reciprocals as in this example:

The slopes are negative inverses, so the lines are perpendicular to each other.

This always confuses me because the word "inverses" seems overly general to refer to reciprocals in particular.

Is this common usage ever correct, or is "reciprocal" always a better word to use?

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I would just say the product of the slopes is -1. – Emre Jun 23 '11 at 20:29
The reciprocal of a real number is its multiplicative inverse, so it is certainly the case that calling them "inverses" is correct (though perhaps not entirely precise absent context). – Arturo Magidin Jun 23 '11 at 20:29
– Bill Dubuque Jun 23 '11 at 20:35
@Ben: In the abstract, yes; but in context it may be perfectly clear. The sentence you mention, for example, is clear to me. – Arturo Magidin Jun 23 '11 at 20:59
@Isaac The usual convention is that the operation is omitted if it is clear from context. If you search Google Books, subject: mathematics, you'll find that both forms of recip...able occur less than 20 times total, vs. 300000 for invertible. Further Google ngrams shows that inverse is overtaking reciprocal in the last few decades. – Bill Dubuque Jun 23 '11 at 22:58

## 4 Answers

I cannot think of an instance where "inverse" by itself is a better choice than "reciprocal" if you are discussing a reciprocal. Of course, "multiplicative inverse" is just as good as "reciprocal."

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The word inverse needs a context. People often use it alone when they think the context is clear. There are additive inverses, multiplicative inverses and compositional inverses to name a few.

Yes, there is some ambiguity there.

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In the context of functional iteration we have "the inverse" which means then the functional inverse, $f^{-1}(x)$ for instance $\sin(x)$ and $\arcsin(x)$ while "the reciprocal" means $1/f(x)$.

In the context of matrices of infinite size the inverse should always be called "reciprocal" - if I recall right; I think the reason given was some indeterminacy, for instance we can have two infinite matrices M1 and M2 which give the (infinite) unit-matrix when right-multiplied with some other matrix X: $M1*X = M2*X=I$ (But I don't have the source of that recommendation at hand)

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A reciprocal is something that, when multiplied by the given number, leads to a product of 1. Ditto for a type of inverse known as a MULTIPLICATIVE inverse. (And a "negative reciprocal" gives you a -1).

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