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How would one call this operation in English where we add a term $a$ and subtract the additive inverse $-a$ right away again, so $$ f(x)-g(x)=f(x)-a+a-g(x) $$ for example here where we also use the triangle inequality $$ |x|= |(x + y)−y|\leq|x+y|+|y|\iff |x| − |y| \leq |x + y| $$ where we added and subtracted $y$.

I just couldn't find a proper translation, in German one would call it for example nahrhafte Null or Nullergänzung which literally means nutritious zero and zero completion/addition respectively, though I have never read those translations anywhere, ever.

Other languages might also be interesting but I am mostly interested in English. Are there any dictionaries out there which mainly focus on mathematical terminology?

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    i would write Zero Addition this is the nearest word to German Nulladdition – 2017-02-13
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    I have often called this (and heard it be called) "adding a zero", so one uses a verb to describe doing the procedure. – 2017-02-13
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    @s.harp by any chance, have you any reference at hand? – 2017-02-13
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    I have perhaps heard it called a few different things, but there isn't a great term in English for it afaik. I would stick with "add and subtract $a$" or something of the like. If I needed to explain my process in a paper. In my mind, ***clarity >> brevity*** – 2017-02-13

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I had a great teacher in high school, named John Titterton. He called this a "propitious zero." When multiplying and dividing by the same thing, he called it a "propitious one." My guess is that he invented these phrases himself. They work well.

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    I go with that! It's funny how much the terminology deviates from language to language - it's a quite common wording in German, every school child knows this (or at least should). – 2017-02-14