Talk:Expression (mathematics)

Algebraic expression

Should this be moved to algebraic expression? Septentrionalis 15:42, 20 September 2005 (UTC)

I think not. There are many expressions such as $\int _{0}^{t}f(x)\,dx$ which are usually not called "algebraic". --Aleph4 16:03, 20 September 2005 (UTC)

Yes, but they're not really discussed in this article. Septentrionalis 16:58, 24 September 2005 (UTC)

i think we are all missing the point. by our clumsy definitions of the basic algebraical terms and making them too technical the essence of mathematics is being taken away from the masses. it is high time we make proper changes. i request all to refer to hall and knight's elementary algebra for schools and then build up from there.

Should the link to axiomatic theory of expressions be removed? It's a poorly organized page that describes some guy's new "theory" about the foundations of mathematics. He's still developing his theory. I don't have the knowledge to adequately evaluate his claims, but it looked pretty sketchy to the untrained eye. I expect the link was added by the fellow himself.

I added the article algebraic expression. Any comments are appreciated. Isheden (talk) 11:16, 28 October 2011 (UTC)

I'm not sure about your article algebraic expression. According to your usage, expressions involving pi and e are not algebraic expressions because pi and e are not algebraic numbers. That's not a usage I've ever heard. Rick Norwood (talk) 17:41, 28 October 2011 (UTC)

It's a very good point you make. The sources I found indicate that expressions involving transcendental numbers are not considered as algebraic expressions. It's at least consistent since the algebraic numbers form a field, the algebraic closure of the rational numbers. It also fits to the informal definition in the lead of the article algebraic function. Actually, the reason I've been considering this is that I wanted to define what an algebraic fraction is—another article I've created. The background was that I wanted to find out whether all examples in the article Fraction (mathematics) should really be called fractions. The results would indicate that e.g. √2/2 and ${\frac {2+i}{2-2\cdot i}}$ are rational algebraic fractions since they involve only algebraic numbers, whereas π/4 is not, since π is transcendental. Do you know of sources claiming otherwise? Isheden (talk) 21:59, 28 October 2011 (UTC)

Numbers or numerals?

Oleg just reverted this, but I think the proper term is "numerals". An expression is a combination of symbols. A number is not a symbol, but a numeral is. Therefore, expressions contain numerals; they do not contain numbers. Other thoughts? I'll re-revert if there are no objections in the next day or so. capitalist 02:53, 14 June 2006 (UTC)

Why do you say that a mathematical expression is a combination of symbols? Do you have a reference for this? I think it is a combination of numbers, not numerals, together with functions and variables (see my comment on Talk:Complex number). -- Jitse Niesen (talk) 05:17, 14 June 2006 (UTC)

Yes, the reference is the Merriam Webster's Online Dictionary which gives this definition:

1 a : an act, process, or instance of representing in a medium (as words) : UTTERANCE <freedom of expression> b (1) : something that manifests, embodies, or symbolizes something else <this gift is an expression of my admiration for you> (2) : a significant word or phrase (3) : a mathematical or logical symbol or a meaningful combination of symbols (4) : the detectable effect of a gene; also : EXPRESSIVITY 1

So given that definition, we can conclude that a mathematical expression is a group of symbols. Given that conclusion and the fact that numerals are symbols while numbers are not, we can conclude that mathematical expressions contain numerals, not numbers. QED! :0) capitalist 03:28, 15 June 2006 (UTC)

The Oxford English Dictionary has a similar definition—unfortunately there is no Mathworld article on it. However, with the example given at Talk:Complex number of Polish notation of "+ 1 1" being the same expression, with just a different notation, as the common infix "1 + 1", I do not know that this is sufficient. This has the same numbers operated on by the same operator; and both mean what is possibly another notation "one plus one". Is this different because the symbols are different, where the Polish notation has the same symbols in a different order? Still, certainly, a expression is still a representation, and "3 - 1" is not the same expression as "1 + 1". —Centrx→talk 07:14, 15 June 2006 (UTC)

It is possible to be right and wrong at the same time. Yes, there is a distinction to be made between numbers and numerals and, yes, an expression contains the latter, not the former. However, it is equally true that 2 plus 2 is not 4. Rather, we should say that the number represented by the numeral 2 added to the number represented by the numeral 2 yields the number represented by the numeral four. That sort of excessive precission is called pedantry, and is to be avoided. Make the technical distinctions only in cases where they matter. Rick Norwood 15:34, 15 June 2006 (UTC)

No, when a word is used in language, the meaning to which it refers is always implied, that is the prime and default purpose of language. When a term is used as term, it is the exception and has special formatting to indicate it, in such examples as — Depend derives from the same word as pendulum. — and — The word "cat" designates a feline. — but that does not mean that it is false to say that "A cat is a feline." It is the reason why on Wikipedia each article begins with something like "The United States is a country" rather than "The United States is the name of a country". The fact remains that we are defining what an expression is. If an expression is, in fact, a combination of numerals,etc. rather than numbers, then defining it as number would be equivalent to saying "The United States is a continent" or the "United States is a population", not the pedantry of defining it as as term. —Centrx→talk 19:20, 15 June 2006 (UTC)

You are splitting hairs. If I did the same, I could object that "The United States" is not the name of a country. The name of the country is "The United States of America". Rick Norwood 21:09, 15 June 2006 (UTC)

You are incorrect. If an expression does not consist of numbers, then defining it as such would be false. The analogue would be stating that the name of the United States is "Northern Hemisphere". If indeed, this were simply splitting hairs, then it should be perfectly acceptable to you either way. —Centrx→talk 21:35, 15 June 2006 (UTC)

So is the number 1 or a lone variable, say x, a mathematical expression? The MW dictionary quote seems to imply yes. It says symbol, singular, not symbols, plural. The article is unclear. — Preceding unsigned comment added by 71.220.62.231 (talk) 04:37, 2 August 2011 (UTC)

Yes. The number (numeral) 1 and the monomial x are mathematical expressions. Rick Norwood (talk) 13:21, 2 August 2011 (UTC)

Well that is a lot there Redneck121 (talk) 21:16, 10 February 2016 (UTC)

explicit vs implicit expression

In Golden_ratio, there is a redlink to "explicit expression". Is this a well defined math concept? If so, it might be useful to define it in this article? &#151; Xiutwel ♫☺♥♪ (talk) 21:08, 22 December 2007 (UTC)

• See also: Closed-form expression 21:10, 22 December 2007 (UTC) —Preceding unsigned comment added by Xiutwel (talk • contribs)

Division by zero not an expression?

"... is not [an expression], because the parentheses are not balanced and division by zero is undefined." Does that make sense? That x / 0 is not mathematical expression just because it's undefined? Saeed Jahed (talk) 10:19, 15 March 2009 (UTC)

A nonsense string of symbols is not considered a (well formed) mathematical expression. As another example "3 @ 4" would not be considered a mathematical expression because the operation "@" is undefined. On the other hand, if a definition of "@" were given in the preceding text, then "3 @ 4" would become a mathematical expression. There is a large body of literature on what constitutes a "well-formed formula". Rick Norwood (talk) 14:45, 15 March 2009 (UTC)

Not sure if what Rick said is relevant. Certainly the unbalanced parens discount it from being an expression. However, division by zero (x / 0) is certainly a perfectly valid expression -- its value is undefined, but it is an expression. Therefore, I have changed the counter-example to ")x)/y", and removed the statement about division by zero. (There is even a later example of an expression being undefined due to division by zero). EatMyShortz (talk) 07:22, 31 August 2009 (UTC)

Support: Division by 0 is, correctly, undefined. However, "(x ÷ 0)" is correct when defined as a mathematical expression (which has been defined in Expression_(math).) ACredibleLie (talk) 15:15, 18 February 2010 (UTC)

Relations?

Can expressions include relations? Is x < y an expression? It seems the distinction that expressions cannot include the equals sign (often considered a relation) should at least be included in the second sentence. I usually think of expressions as more or less equivalent to "terms" in first order logic, and terms cannot include relations... Is there a source that includes relations as part of expressions? Dmcginn (talk) 16:12, 2 July 2010 (UTC)

Here is an example of an inequality being called an expression:

"The expression $a\geq b$ is synonymous with $b\leq a$." (p. 66)

• Real Analysis: A Historical Approach By Saul Stahl

--50.53.50.57 (talk) 06:47, 2 October 2014 (UTC)

In this example, a function definition is called an expression:

"... the set $B$ in the expression $f:A\to B$ is sometimes called the codomain of $f$." (p. 128)

• Introduction to Real Analysis By Michael J. Schramm

--50.53.50.57 (talk) 07:25, 2 October 2014 (UTC)

Isheden's complaint

I agree that this article (and all other articles on elementry mathematics) must be as easy for a non-mathematician to read as possible. I've shorted and simplified the first paragraph.

As best can see, the second paragraph, beginning "In algebra..." does not say anything that needs to be said here. Do I hear any objection to removing it?

On the other hand, we might want to add a little bit about the rules for well-formed formulas.

Rick Norwood (talk) 17:59, 28 October 2011 (UTC)

I think the paragraph you mention can be removed. Possibly algebraic expression could be merged into this article instead. Isheden (talk) 22:02, 28 October 2011 (UTC)

"Meaningless jumble"

This pearl of prose is hardly encyclopedic language. It should be rephrased. FilipeS (talk) 18:55, 5 April 2012 (UTC)

Agree. Possibly algebraic expression could be merged into this article if it becomes more encyclopedic. Isheden (talk) 08:12, 6 April 2012 (UTC)

Contradictory as to whether an equation is an expression

The section Semantics: meaningful expressions says

...for instance, an expression might designate a condition, or an equation that is to be solved, or it can be viewed as an object in its own right that can be manipulated according to certain rules.

But this is contradicted in paragraph 2 of the section Variables where it says

Thus an expression represents a function whose inputs are the value assigned the free variables and whose output is the resulting value of the expression.

By saying it's a function it precludes it from being an equation. Also, the last paragraph of the lead in the article Formula says

Expressions are distinct from formulas in that they cannot contain an equals sign (=).[6] Whereas formulas are comparable to sentences, expressions are more like phrases.

again contradicting the first quote above. I'll leave it to others to decide whether/how to fix this. Loraof (talk) 20:15, 29 September 2014 (UTC)

The two first quotes are not contradictory, as "=" may be considered as an operator which takes its values in {true, false}. This the case in general purpose computer algebra systems. However the second quote is wrong for at least two reasons. Firstly, an expression may represent a function only when the set of possible values of the variables (range of the function) is defined, or in other words, if the semantic of the expression is defined. Secondly, an expression may not contain any variable and its evaluation may not result in a number. For example, a matrix of integers is an expression which does not represent any function and cannot represent any other mathematical object than itself.

I agree that the quote of Formula deserve to be edited: this is the opinion of one author, not a common convention. D.Lazard (talk) 21:48, 29 September 2014 (UTC)

Inconsistency in table

In the section Different forms of mathematical expressions, the table says that arithmetic expressions can have factorials but not integer exponents. This seems contradictory: each is simply a sequence of multiplications, in one case like 4×3×2×1, and in the other case like 4×4×4×4. So they both ought to be allowed or not allowed.

Also, since the table says that polynomials can contain an "integer exponent", I think that row heading should be renamed "Positive integer exponent" or "Non-negative integer exponent".Loraof (talk) 20:25, 29 September 2014 (UTC)

Good points. Another problem with the table is that it says polynomials can have elementary arithmetic operations, but polynomials do not have division. BTW, the table is a template: Template:Mathematical expressions. --50.53.46.203 (talk) 14:29, 1 October 2014 (UTC)

Dictionary definitions of "expression"

• expression: "A very general term used to designate any symbolic mathematical form, such, for instance, as a polynomial."

The Mathematics Dictionary edited by R.C. James

• expression 1b(3): "a sign or character or a finite sequence of signs or characters (as logical or mathematical symbols) representing a quantity or operation"

Webster's Third New International Dictionary of the English Language, Unabridged edited by Philip Babcock Gove

• expression 1b(3): "a mathematical or logical symbol or a meaningful combination of symbols"

Merriam-Webster's Collegiate® Dictionary, Eleventh Edition

--50.53.50.57 (talk) 08:14, 2 October 2014 (UTC)

• "Definition 1: Let S be a set of symbols. An expression in S (or word in S) is a finite sequence of symbols of S. For example, if the set of symbols is {a, b, c}, then aabc and cba are both expressions in S." (p. 7)

An Introduction to Mathematical Logic By Richard E. Hodel

--50.53.46.137 (talk) 06:03, 3 October 2014 (UTC)

• "We call a finite sequence of (occurrences of) formal symbols a formal expression." (§ 38. Formal number theory.)

Mathematical Logic By Stephen Cole Kleene

--50.53.60.76 (talk) 14:42, 3 October 2014 (UTC)

Margaris gives a definition of "string" that appears to mean the same thing as what Hodel and Kleene mean by "expression":

• "The construction of a formal axiomatic theory begins with the specification of a finite set of formal symbols, and a string is defined to be a finite sequence of formal symbols." (p. 14)

Later, in the context of Gödel's incompleteness theorem, Margaris says of N, a formal number theory:

• "An expression of N is a string or a finite sequence of strings of N." (p. 185)

First Order Mathematical Logic By Angelo Margaris

--50.53.60.76 (talk) 19:27, 3 October 2014 (UTC)

The definitions from logic, where an expression is any string, differ sharply from the more general definitions, where an expression is a meaningful string. In general mathematics x + 2 is an expression, while +=@@#%$ is a string but not a mathematical expression. Rick Norwood (talk) 11:12, 4 October 2014 (UTC)

Thanks for pointing that out. Logicians appear to define the term "expression" differently from mathematicians, and the article should say so. Do you have any sources other than the dictionary definitions above that define "expression" in the sense that mathematicians use it? --50.53.61.13 (talk) 14:09, 4 October 2014 (UTC)

Complete rewriting needed

I agree with preceding posts that the article has many issues. Here are several ones that have not been quoted in the preceding quotes, or have only been partially quoted. Here are some of these issue

• The article does not contains anything beyond the informal dictionary definition
• The article is unreferenced
• The classification of the types of expression seems original research
• This classification is incomplete, as it does not mention logical expressions and expressions that cannot been evaluated to numbers, as matrices.
• It uses without definition "arithmetic expression" (the wikilink provided is a self redirect)
• It does not mention computer algebra at all, while this field is the basis of the modern understanding of the concept (computer algebra systems use to not manipulate mathematical objects, but only their representation as expressions, forgetting most of the associated semantic.

This list is incomplete, and shows that the article deserves to be completely rewritten. D.Lazard (talk) 11:00, 2 October 2014 (UTC)

"The classification of the types of expression seems original research"

Are you referring to the table in the section Different forms of mathematical expressions? --50.53.47.11 (talk) 14:12, 2 October 2014 (UTC)

I refer to the template and to the remainder of the article. The term "arithmetic expression" seems to not have a standard meaning. The distinction between analytic and closed form expression is also dubious. As far as I know, "analytic expression" is an old term of what is now called "closed form expression", which is no more in common use because the possibility of confusion with the "analytic" of the analytic function. D.Lazard (talk) 17:17, 2 October 2014 (UTC)

No History

There is no section describing how and when symbolic expressions were developed and replaced text. FreeFlow99 (talk) 09:49, 6 November 2015 (UTC)

The problem for that is the lack of WP:Reliable sources. In fact, as far as I know, there are two periods. The first one is the introduction of mathematical notation, which is described in History of mathematical notation. However, the concept of "expression" appeared much more recently. I suspect that it has been rarely used before the second half of the 20th century, when it has been popularized by its use in computer algebra and other aspects of the computerization of mathematics. In fact, that is the computerization that requires a clear distinction between a mathematical object and its various representations as expressions. Moreover, in relation with computer proofs, a formal definition of "expression" is often needed, and there is not yet a general agreement of what should be such a formal definition. This explains the lack of reliable sources for the history of the notion. D.Lazard (talk) 15:18, 6 November 2015 (UTC)

Missing from table

grouping (parentheses)

Equations (=), inequalities, and inequations (not =) as expressions (e.g. x = x, treated as an expression, reduces to true)

power towers, finite and infinite

nested radicals, finite and infinite

hyperoperations

multivariable and vector calculus

integral transforms and other transforms

operators (differential and integral, vector)

probability and statistics operators

sets and set operations (union, intersection, ...)

logic (and, or, not, implication, truth values, quantifiers, higher order logic...)

vectors and matrices and their operations

roots of polynomials that cannot be expressed as radicals (e.g. roots of x^5 + x + 1)

complex numbers

combinatorics (combinations, permutations, ...)

functions (e.g. f(x + 1) as an expression)

and many more...

A more general notion of mathematical expression

Objects from literally all areas of mathematics, physics, and computer science can be treated as mathematical expressions, because they can be manipulated as such. Indeed, mathematical software such as Mathematica and Maple do exactly this. Groups, graphs, geometric shapes, and computer programs are just as much mathematical expressions as 1 + 1.

75.46.182.198 (talk) 02:07, 6 June 2018 (UTC)

I totally agree with the first sentence of the last paragraph. However, the last sentence is wrong, as a mathematical object is not an expression by itself. If you omit the section "Forms", this is exactly what the article says.

I agree also that the section "Forms" is problematic. This is a classification that is far to be complete, as you have pointed. It is also wrong. For example $x+y$ is not an arithmetic expression if x and y do not denote numbers. Also, it is misleading, as suggesting that a specific named function (such as "gamma function") is an expression. Also, the distinction between closed form expressions and analytic expressions seems WP:OR. Also, the table seems WP:SYNTHESIS. My opinion is that section must be removed. However, this needs a consensus. So I will first edit the section and moving it toward the end of the article. D.Lazard (talk) 08:58, 6 June 2018 (UTC)

Expression, formula, equation, function

The second paragraph of the current article lead says that a formula can be evaluated to true or false. This does not aid understanding of the use of formulas in elementary maths or physics. Consider the well-known quadratic equation ($ax^{2}+bx+c=0$) and quadratic formula ($x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$). It is possible to substitute a, b, c and x into the formula and get a true result if x is one of the values of the expression. But to test a value of x, it is simpler to use the equation. The true use of a formula is to provide instructions for calculating a value. Formulas could use words or an assignment operator, but they are usually written starting with a variable name and equals sign, so they are often confused with equations when the term formula is being defined. For this reason, it is hard to find useful reliable sources. I also think the link to formula (see following note) is unhelpful for most readers of the article.

An expression is an anonymous formula (by analogy with an anonymous function) in the sense that it also provides instructions for calculating a value, but without giving a name for the instructions or result. It would also be useful to contrast expressions and functions (like the distinction between polynomial expression and polynomial). JonH (talk) 23:55, 15 April 2022 (UTC)

• Note that when I wrote this comment, formula redirected to well-formed formula which was unhelpful, but now it redirects to formula#In mathematics which does help. JonH (talk) 19:37, 24 September 2023 (UTC)

The paragraph says that a formula can be evaluated to true or false, depending on the values that are given to the variables. So the text does require an assignment of a value to each variable before evaluation. You have a point in that the phrasing could be more clear.

Moreover, you are right that one of the meanings for the word "formula" is "receipe to calculate a value (given values for its 'input' variables)". This meaning is quite different from what is meant in the article, and we should clarify this, too. The latter meaning is the one used widely in mathematical logic, and the inequation example illustrates it perfectly: no receipe whatsoever can be obtained from it, but it evaluates to true or false, depending on the value of x.

A reliable source for this meaning of "formula" (and "expression") is likely to be found in an arbitrary logic textbook. Unfortunately, I only have German textbooks at hand. I found the source Bergmann.Noll.1977, p.28, before Def.6.4 and 6.5, but this book has no English translation. On the other hand, in Hermes.1972, translated as Hermes.1973, I didn't find a concise remark that could be used as source. - Jochen Burghardt (talk) 17:07, 16 April 2022 (UTC)

Also, good sources for "expression" could be found in textbooks on computer algebra. In fact, this is computer algebra that has popularized the term of "expression", because of the need of distinguishing mathematical objects from their representations. Indeed, a large part of computer algebra consists of transforming expressions without changing the represented object (for example, for simplification). D.Lazard (talk) 09:01, 17 April 2022 (UTC)

References