1143:. 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.
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527:. — 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 Knowledge 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
2051:. 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).
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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
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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
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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
1939:). 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
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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.
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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
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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
2263:". Therefore, a formal definition of "formula" always includes a formal definition of "expression". Hence every logic textbook contains a definition of "expression", as part of a definition of "formula". See e.g. Hans Hermes, Introduction to Mathematical Logic, 1973, ISBN 3540058192, sect. II.1. -
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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
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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
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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
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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
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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
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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
1176:, 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.
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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)
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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.
531:. 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. —
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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.
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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
1958:) 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
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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
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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
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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
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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).
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397:. 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.
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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
1089:...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.
1641:. 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
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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". —
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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.
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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.
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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".
690:"... is not , 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?
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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
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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
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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".
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280:. 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.
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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.
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Expressions are distinct from formulas in that they cannot contain an equals sign (=). Whereas formulas are comparable to sentences, expressions are more like phrases.
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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
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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
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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
1787:. 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.
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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"
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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.
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algebraic fractions since they involve only algebraic numbers, whereas π/4 is not, since π is transcendental. Do you know of sources claiming otherwise?
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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
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The article still needs general work, but there seems to be enough general citations and inline citations to justify removing the current alerts.
2021:. 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.
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operator, but they are usually written starting with a variable name and equals sign, so they are often confused with equations when the term
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The Oxford
English Dictionary has a similar definition—unfortunately there is no Mathworld article on it. However, with the example given at
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Division by 0 is, correctly, undefined. However, "(x ÷ 0)" is correct when defined as a mathematical expression (which has been defined in
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b (1) : something that manifests, embodies, or symbolizes something else <this gift is an expression of my admiration for you: -->
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form a field, the algebraic closure of the rational numbers. It also fits to the informal definition in the lead of the article
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expression", which is no more in common use because the possibility of confusion with the "analytic" of the analytic function.
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Equations (=), inequalities, and inequations (not =) as expressions (e.g. x = x, treated as an expression, reduces to true)
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Margaris gives a definition of "string" that appears to mean the same thing as what Hodel and Kleene mean by "expression":
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expression: "A very general term used to designate any symbolic mathematical form, such, for instance, as a polynomial."
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By saying it's a function it precludes it from being an equation. Also, the last paragraph of the lead in the article
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become a mathematical expression. There is a large body of literature on what constitutes a "well-formed formula".
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1 a : an act, process, or instance of representing in a medium (as words) : UTTERANCE <freedom of expression: -->
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is—another article I've created. The background was that I wanted to find out whether all examples in the article
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hand, in Hermes.1972, translated as Hermes.1973, I didn't find a concise remark that could be used as source. -
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logic, and terms cannot include relations... Is there a source that includes relations as part of expressions?
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is being defined. For this reason, it is hard to find useful reliable sources. I also think the link to
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Oleg just reverted this, but I think the proper term is "numerals". An expression is a combination of
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again contradicting the first quote above. I'll leave it to others to decide whether/how to fix this.
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There is no section describing how and when symbolic expressions were developed and replaced text.
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I'm repeating my previous edit summary here: A formula contains expressions as constituents, e.g."
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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On the other hand, we might want to add a little bit about the rules for well-formed formulas.
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It uses without definition "arithmetic expression" (the wikilink provided is a self redirect)
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expression 1b(3): "a mathematical or logical symbol or a meaningful combination of symbols"
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term, it is the exception and has special formatting to indicate it, in such examples as —
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Yes, the reference is the
Merriam Webster's Online Dictionary which gives this definition:
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This list is incomplete, and shows that the article deserves to be completely rewritten.
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It's a very good point you make. The sources I found indicate that expressions involving
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That's a good point. If you'd like, please start a
History section talking about this.
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roots of polynomials that cannot be expressed as radicals (e.g. roots of x^5 + x + 1)
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2101:. Heidelberger Taschenbücher, Sammlung Informatik. Vol. 187. Heidelberg: Springer.
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deserve to be edited: this is the opinion of one author, not a common convention.
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should really be called fractions. The results would indicate that e.g. √2/2 and
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The article does not contains anything beyond the informal dictionary definition
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Webster's Third New
International Dictionary of the English Language, Unabridged
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are not considered as algebraic expressions. It's at least consistent since the
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Yes. The number (numeral) 1 and the monomial x are mathematical expressions.
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This pearl of prose is hardly encyclopedic language. It should be rephrased.
450:(3) : a mathematical or logical symbol or a meaningful combination of symbols
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other than the dictionary definitions above that define "expression" in the
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the untrained eye. I expect the link was added by the fellow himself.
1202:, but polynomials do not have division. BTW, the table is a template:
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The paragraph says that a formula can be evaluated to true or false,
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Also, good sources for "expression" could be found in textbooks on
1951:(see following note) is unhelpful for most readers of the article.
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and expressions that cannot been evaluated to numbers, as matrices.
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could be merged into this article if it becomes more encyclopedic.
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of N is a string or a finite sequence of strings of N." (p. 185)
1367:"We call a finite sequence of (occurrences of) formal symbols a
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In this example, a function definition is called an expression:
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Here is an example of an inequality being called an expression:
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a perfectly valid expression -- its value is undefined, but it
2116:. Mathematische Leitfäden (4th ed.). Stuttgart: B.G. Teubner.
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is defined to be a finite sequence of formal symbols." (p. 14)
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Good points. Another problem with the table is that it says
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I think the paragraph you mention can be removed. Possibly
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An expression is an anonymous formula (by analogy with an
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This classification is incomplete, as it does not mention
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Merriam-Webster's
Collegiate® Dictionary, Eleventh Edition
1932:{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}
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appear to define the term "expression" differently from
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Contradictory as to whether an equation is an expression
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depending on the values that are given to the variables
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But this is contradicted in paragraph 2 of the section
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Yes, but they're not really discussed in this article.
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2140:. Hochschultext (Springer-Verlag). London: Springer.
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As best can see, the second paragraph, beginning "In
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The classification of the types of expression seems
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2222:" is an expression, it is part of the formula "
2099:Mathematische Logik mit Informatik-Anwendungen
1687:operators (differential and integral, vector)
1424:, Margaris says of N, a formal number theory:
1993:which was unhelpful, but now it redirects to
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228:which are usually not called "algebraic". --
1711:functions (e.g. f(x + 1) as an expression)
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1985:Note that when I wrote this comment,
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1343:An Introduction to Mathematical Logic
1308:) is a finite sequence of symbols of
221:{\displaystyle \int _{0}^{t}f(x)\,dx}
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1633:The problem for that is the lack of
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447:(2) : a significant word or phrase
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109:and see a list of open tasks.
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2258:
2217:
1960:polynomial expression
1934:
1858:
1765:
1569:Are you referring to
963:By Michael J. Schramm
950:
926:
894:
840:
814:
362:
223:
2226:
2200:
2136:Hans Hermes (1973).
2112:Hans Hermes (1972).
1871:
1816:
1748:
1538:It does not mention
1240:edited by R.C. James
1058:algebraic expression
1037:"Meaningless jumble"
1019:algebraic expression
939:
903:
883:
823:
797:
389:Numbers or numerals?
322:
278:algebraic expression
259:algebraic expression
257:I added the article
180:
164:algebraic expression
158:Algebraic expression
121:mathematics articles
2215:{\displaystyle 2+2}
1991:well-formed formula
1763:{\displaystyle x+y}
1635:WP:Reliable sources
1530:logical expressions
1383:Stephen Cole Kleene
1346:By Richard E. Hodel
986:Isheden's complaint
819:is synonymous with
640:explicit expression
477:Talk:Complex number
413:Talk:Complex number
197:
2253:
2212:
1956:anonymous function
1929:
1853:
1810:quadratic equation
1760:
1662:Missing from table
1440:By Angelo Margaris
1378:Mathematical Logic
945:
921:
899:in the expression
889:
835:
809:
357:
312:algebraic fraction
308:algebraic function
218:
211:
183:
90:Mathematics portal
34:content assessment
1997:which does help.
1927:
1916:
1865:quadratic formula
1714:and many more...
1564:original research
1524:original research
1369:formal expression
948:{\displaystyle f}
892:{\displaystyle B}
750:Expression_(math)
681:
668:comment added by
651:
580:comment added by
355:
304:algebraic numbers
155:
154:
151:
150:
147:
146:
2299:
2265:Jochen Burghardt
2262:
2260:
2259:
2254:
2221:
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2213:
2195:
2158:
2157:
2133:
2127:
2126:
2109:
2103:
2102:
2094:
2049:computer algebra
2035:Jochen Burghardt
1938:
1936:
1935:
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1928:
1926:
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1893:
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1859:
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1777:
1773:
1769:
1767:
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1705:complex numbers
1678:hyperoperations
1643:computer algebra
1540:computer algebra
1056:Agree. Possibly
954:
952:
951:
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928:
927:
922:
898:
896:
895:
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663:
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1170:
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988:
937:
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901:
900:
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481:Polish notation
391:
338:
327:
320:
319:
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178:
177:
168:Septentrionalis
160:
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2077:
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2030:
2022:
2014:
2013:
2012:
2011:
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852:
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932:
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754:ACredibleLie
745:
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689:
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629:
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576:— Preceding
573:
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1581:50.53.47.11
1494:50.53.61.13
1447:50.53.60.76
1391:50.53.60.76
1276:50.53.50.57
1196:polynomials
970:50.53.50.57
860:50.53.50.57
845:." (p. 66)
733:EatMyShortz
692:Saeed Jahed
664:—Preceding
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
2281:Categories
2146:3540058192
2085:References
1964:polynomial
1941:assignment
1620:FreeFlow99
1614:No History
1430:expression
1223:Dictionary
770:Relations?
658:See also:
613:Redneck121
468:capitalist
399:capitalist
2154:1431-4657
1571:the table
1543:semantic.
1478:Logicians
1336:." (p. 7)
1306:word in S
1198:can have
1096:Variables
724:certainly
2053:D.Lazard
1789:D.Lazard
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1596:D.Lazard
1549:D.Lazard
1412:, and a
1324:}, then
1152:D.Lazard
933:codomain
746:Support:
678:contribs
666:unsigned
578:unsigned
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995:algebra
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670:Xiutwel
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636:redlink
395:symbols
373:Isheden
263:Isheden
139:on the
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1124:Loraof
649:(talk)
560:Centrx
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486:Centrx
415:). --
230:Aleph4
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1781:WP:OR
1741:says.
1490:sense
1112:says
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646:♫☺♥♪
2269:talk
2176:talk
2151:ISSN
2142:ISBN
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1968:JonH
1962:and
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1467:talk
1451:talk
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1395:talk
1357:talk
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1280:talk
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638:to "
617:talk
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1729:(
1650:(
1622:(
1598:(
1583:(
1551:(
1496:(
1465:(
1449:(
1393:(
1355:(
1334:S
1322:c
1318:b
1314:a
1310:S
1298:S
1290:"
1278:(
1210:(
1183:(
1154:(
1126:(
1064:(
1045:(
1025:(
1006:(
972:(
943:f
919:B
913:A
910::
907:f
887:B
862:(
833:a
827:b
807:b
801:a
779:(
756:(
735:(
710:(
694:(
672:(
615:(
600:(
584:(
562:→
558:—
535:→
488:→
419:(
375:(
352:i
346:2
340:2
335:i
332:+
329:2
284:(
265:(
216:x
213:d
208:)
205:x
202:(
199:f
194:t
189:0
143:.
42::
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