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Not sure if I should ask this in German, English or Mathematics Stackexchange, but decided to ask here.

So I combed the Internet for the words "Mathematische Umformung" and "Äquivalenzumformung" in English but couldn't find anything that means exactly that. I am specifically asking for these words in a mathematical context.

Does anyone know what the words for "Mathematische Umformung" and "Äquivalenzumformung" in English are?

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For "Äquivalenzumformung" there is to my knowledge no english word. The process (of solving equations by using logically equivalent statements) is usually described as "balance method" or "simple algebraic manipulation", but these even leave out one of the key ideas of "Äquivalenzumformung", the reversibility. Some books call it "rearranging an equation/formula". As far as i know the term "equivalence transformation" is NOT used.

"Umformung" in a mathematical context is a manipulation of terms of which an "Äquivalenzumformung" is a special case. It can be translated by "manipulation (of terms)" or "operation (with/of terms).

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  • There is an article on German Wikipedia for [Äquivalenzumformung but as far as I know there is no equivalent article in English Wikipedia. The technique for solving equations is part of any grade school algebra class, and the article for Elementare Algebra mentions Äquivalenzumformung, but Elementary algebra only mentions properties of equality individually without giving a collective name to them.
    – RDBury
    Jun 18, 2022 at 12:26
  • Is it same as solving for unknown from an equation?
    – C.F.G
    Jun 18, 2022 at 17:08
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    @C.F.G not exactly a Äquivalenzumformung is rearanging an equation in a way that the set of solutions can not change. For example if I have the equation 4x=3 a Äquivalenzumformung would be multiplying both sides by 4 so we get x=3/4. A prototypical (and usually only rearangement tought in schools) that isn't a Äquivalenzumformung would be squaring both sides. So in our top example we would get 16x^2=9. If we now divide by 16 and take the square root we would get both x=3/4 and x=-3/4. The set of solutions has changed, therefore squaring isn't a Äquivalenzumformung.
    – tempdev nova
    Jun 19, 2022 at 12:39
  • Note that the emphasis is on can not change. There are instances where squaring on both sides results in the same set of solutions as when solved with Äquivalenzumformungen but these cases are mere coincidence and don't entail that squaring is a Äquivalenzumformung.
    – tempdev nova
    Jun 19, 2022 at 12:41
  • @tempdevnova: It is a new concept for me. and I also new in German.
    – C.F.G
    Jun 19, 2022 at 12:45
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"Mathematische Umformung" of an equation most often means to apply the same operation (function, map) on both sides. "Mathematische Umformung" of a term means to rewrite the term in an equivalent way, e.g. 1 = 2/2 = 4/4,...

So it depends on what you want to rewrite.

As German is more nominal and English is more verbal, I would not try to translate "Äquivalenzumformung" literally into English, but I would describe it in verbal terms, e.g.:

"If we apply f to both sides of equation (1), then we obtain equation (2) which turns out to be even equivalent to equation (1)."

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