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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?

Aryabhata
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Sophie Alpert
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    I would just say the product of the slopes is -1. – Emre Jun 23 '11 at 20:29
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    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
  • Isn't this simply referring to the vanishing of the dot product $\begin{bmatrix} 1 \\ m \end{bmatrix} \cdot \begin{bmatrix} 1 \\ -1/m \end{bmatrix} = 0$? – t.b. Jun 23 '11 at 20:29
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    See http://en.wikipedia.org/wiki/Multiplicative_inverse – Bill Dubuque Jun 23 '11 at 20:35
  • Yes, I understood that the reciprocal is a type of inverse, but "inverse" seems overly general to specify a reciprocal relationship. – Sophie Alpert Jun 23 '11 at 20:56
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    @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
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    Don't overlook the fact that "inverse" is easier to say. – Mike Jones Jun 23 '11 at 21:16
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    In the sentence at hand, the only other conceivable meaning for "inverse" is "additive inverse", but if two numbers are "negative additive inverses" of each other then they are more simply called equal! So after a little thought one should become convinced that multiplicative inverse is intended. But I agree that using "reciprocal" here would be better: it is instantly clear. – Pete L. Clark Jun 23 '11 at 21:41
  • Also don't overlook the fact that using "inverse" allows us to concisely speak of "invertible" elements. Contrast the tongue twister "reciprocatable" - which appears to be very little used mathematically. – Bill Dubuque Jun 23 '11 at 22:12
  • @Bill: Invertible with respect to what operation? It still needs context and/or clarification. – Isaac Jun 23 '11 at 22:32
  • ... and I suspect it'd be "reciprocable," but I don't have a good source for that. – Isaac Jun 23 '11 at 22:44
  • When you define $x/y = x y^{-1}$, what do you call that on the end? $y$ inverse? or $y$ reciprocal? – GEdgar Jun 23 '11 at 22:52
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    @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
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    @Bill: Inverse ought to be higher in the ngrams chart as it's a broader term. And, sure, most people at the graduate or post-graduate level can easily determine which inverse is meant from context, but most non-mathematicians cannot. – Isaac Jun 23 '11 at 23:05
  • @Isaac Of course, but I searched for specific instances, such as "inverse matrix", "inverse of each other", etc. All showed "inverse" becoming more frequent from about 1950-1970, as would be expected as terminology from abstract algebra and set theory percolated into the mainstream. – Bill Dubuque Jun 23 '11 at 23:09

4 Answers4

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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."

Isaac
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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.

ncmathsadist
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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)

Gottfried Helms
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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).

Tom Au
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