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I have a somewhat general question regarding the notation of mathematical expressions. I am interested in further information, that is if you know a book, a pdf, or just a wikipedia page about my question that would help a lot. Of course, if you know a specific answer feel free to share.

My question is: Is there a normal form / convention for the notation of mathematical expressions? As an example:

  1. $\frac{\sqrt{x^2+2xy+y^2}}{z}$
  2. $\frac{x+y}{z}$
  3. $\frac{x}{z}+\frac{y}{\sqrt{z^2}}$

Those three expressions are mathematically equal but semantically different. Is there a normal form i.e. a rule set that - when given a those three expressions - defines one "normalized" way to write them down? And if there is, is there an associated normalization algorithm that would, with absolute certainty, convert these three examples (that are mathematically equal) into the same (syntactically equal) expression? The resulting expression does not necessarily have to be the "simplest form" that contains the least amount of symbols or whatever. I am also aware that there are probably many different "normal forms". I am just searching for one of them.

Normal forms exist in expression logic (disjunctive, conjunctive,...). I am searching for something analogous for math expressions.

Thanks for the help in advance :)

Edit: I realized that my question is a bit to general. Maybe it is easier to first limit the expressions to algebraic expressions, that is expressions involving integer constants, variables, and the operations

  • addition
  • subtraction
  • multiplication
  • division
  • exponentiation by a rational
vreithinger
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    This question might be related to yours. – soupless Nov 05 '22 at 12:45
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    You may want to read https://en.wikipedia.org/wiki/Beta_normal_form, https://en.wikipedia.org/wiki/Normalisation_by_evaluation, and various other pages. This has been well studied for lambda-calculus and programming languages. It is not easy to give a short answer (and I am not an expert). Note that equality of expressions is usually undecidable, except in limited theories, so converting all semantically equal expressions to the same expression is not possible. – Jean-Armand Moroni Nov 06 '22 at 21:42
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    If you read the book "The Little Typer", you will learn about one existing approach to normalization. – CrabMan Nov 09 '22 at 10:07
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    I think your example is no good: (1) and (2) are not equal, you'd have to write $|x+y|$ in (2); and similarly (2) and (3) are not equal. – ancient mathematician Nov 11 '22 at 12:01
  • How about using (2) since it is in simplest form? – Тyma Gaidash Nov 11 '22 at 19:17

1 Answers1

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Edit: The question was changed substantially after I posted this answer. It does not address the current form of the question.

Mathematical expression is far to vague and wide to ask for normal forms, it is not even clear how to define what exactly makes for a mathematical expression. You also have the syntax vs semantics issue: You don't just care about the sequence of symbols, but rather their meaning.

Your examples all denote rational functions, they are built from variables, integers, addition, substraction, multiplication and division. These do have normal forms:

Going from outside to inside, we first have a single quotient. Both numerator and denominator are sums, and each summand is a product of variables and integers.

To really standardize things, we put an order on the variables, and can then order both the factors inside the products and the summands inside the products in a canonic way.

We can compute the normal form of a given term, essentially following school arithmetic. However, the normal form will not necessarily be the most concise way to write a given rational function.

Arno
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  • you are right, the question was not specific enough and i accidentally chose my examples in a bad way. sorry for that. i edited the question – vreithinger Nov 09 '22 at 11:58