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?
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.