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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?

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    I would just say the product of the slopes is -1.2011-06-23
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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).2011-06-23
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    Isn't this simply referring to the vanishing of the dot product $\begin{bmatrix} 1 \\\ m \end{bmatrix} \cdot \begin{bmatrix} 1 \\\ -1/m \end{bmatrix} = 0$?2011-06-23
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    See http://en.wikipedia.org/wiki/Multiplicative_inverse2011-06-23
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    Yes, I understood that the reciprocal is a type of inverse, but "inverse" seems overly general to specify a reciprocal relationship.2011-06-23
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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.2011-06-23
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    Don't overlook the fact that "inverse" is easier to say.2011-06-23
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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.2011-06-23
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    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.2011-06-23
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    @Bill: Invertible with respect to what operation? It still needs context and/or clarification.2011-06-23
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    ... and I suspect it'd be "reciprocable," but I don't have a good source for that.2011-06-23
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    When you define $x/y = x y^{-1}$, what do you call that on the end? $y$ inverse? or $y$ reciprocal?2011-06-23
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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.2011-06-23
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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.2011-06-23
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    @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.2011-06-23

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