# Must an equation contain at least one variable? Can we call 1+1=2 an equation?

According to the Wikipedia and Encyclopædia Universalis, an equation must contain at least one variable but there is no such condition mentioned in other definitions.

Thus can we call the following equalities equations? Columbia Encyclopedia says yes but this contradicts with Wikipedia and Encyclopædia Universalis definitions.

$$1+1=2$$

$$9+4=13$$

In mathematics, an equation is a statement of an equality containing one or more variables. -Wikipedia

The original 2 citations mentioned in the Wikipedia article are mentioned later in this question

Equation, Statement of equality between two expressions consisting of variables and/or numbers. -Encyclopedia Britannica

An equation is a mathematical expression stating that two or more quantities are the same as one another -Wolfram Mathworld

a mathematical statement in which you show that two amounts are equal using mathematical symbols -Cambridge Dictionary

A statement that the values of two mathematical expressions are equal (indicated by the sign =) -Oxford Dictionary

Equation, in mathematics, a statement, usually written in symbols, that states the equality of two quantities or algebraic expressions, e.g., x+3=5. (...) A numerical equation is one containing only numbers, e.g., 2+3=5 -Columbia Encyclopedia, 6th ed

The Wikipedia definition cites 2 different sources. I will quote them here:

An equation is an equality between two mathematical expressions, therefore a formula of the form A=B, where the two members A and B of the equation are expressions in which one or more variables, represented by letters, appear -Encyclopædia Universalis, French-language general encyclopedia published by Encyclopædia Britannica, Inc (Translated by Google Translate, emphasis mine.)

"A statement of equality between two expressions. Equations are of two types, identities and conditional equations (or usually simply "equations")". « Equation », in Mathematics Dictionary, Glenn James [de] et Robert C. James [de] (éd.), Van Nostrand, 1968, 3 ed. 1st ed. 1948, p. 131.

So, I am still confused. The first definition of the above two definitions, says that there must be a variable and the 2nd one has no such condition.

The reason, I am asking the question because, in India, some popular textbooks have mentioned that an equation must contain a variable. Here is the definition used in the NCERT class 7 mathematics book (Page 79).

Here is another Government published book, WBBSE class 7 mathematics textbook (language: Bengali) where they instructed the students to find out which of the followings are equations and which are not. In the solutions, they didn't consider (f) and (g) as equations.

People having the idea that an equation must contain an unknown, can be found often though. For example, let's consider this similar unanswered question on this forum. There are only 2 comments and they contradict each other. Also, this question have some answers where the users believe that an equation should have an unknown.

An equation is meant to be solved, that is, there are some unknowns

You solve an equation, while you evaluate a formula.

• I have never heard that equations are required to have variables. Of the definitions you cite, only Wikipedia requires it. The Britannica quote use "and/or" which means that variables are not required. I would say $1+1 = 2$ is an equation. – Jair Taylor Feb 16 '19 at 7:10
• To make things more complicated: An equation is an equality between two mathematical expressions, therefore a formula of the form A=B, where the two members A and B of the equation are expressions in which **one or more variables**, represented by letters, appear--Encyclopædia Universalis, French-language general encyclopedia published by Encyclopædia Britannica, Inc. (universalis.fr/encyclopedie/equation-mathematique) – Sourav Ghosh Feb 16 '19 at 7:25
• This is not a question about math but about English language. As a matter of fact, other languages just have one word for both equation and equality. – Dirk Feb 16 '19 at 8:12
• @mrtaurhoMr The point of my comment was the fact that distinction of words "equality" and "equation" in other languages does not matter. Observe however that in English there exist at least three words (+ identity). It would be surprising if they all had absolutely the same meaning. – user Feb 16 '19 at 23:20
• Possibly helpful, possible duplicate: math.stackexchange.com/questions/2738360/… – Ethan Bolker Feb 17 '19 at 13:00

The definition on Wikipedia is unusual and does not match the usual usage of the term "equation" by mathematicians. Mathematicians regularly use the term "equation" to refer to any statement that two things are equal written with the symbol $$=$$, regardless of whether any variables are involved. It is usually used as an informal term, but can be given precise formal definitions in various settings. I do not know of any precise formal definition that requires there to be at least one variable.

• I have also added the 2 original sources. Wikipedia article actually cited Encyclopædia Universalis where they mentioned about the presence of at least one variable. – Sourav Ghosh Feb 16 '19 at 8:04
• @EricWofsey The symbol $=$ does not automatically indicate an equation. There is an important distinction between equations and identities. I rarely meet a mathematician who would call $1+1=2$ an equation. If you establish "zero variables", then it's an identity. Otherwise, the most common term I hear is "equality". An equation is either solvable or insolvable by definition. You can't "solve" $1+1=2$, even in the most abstract of terms. – C. Melton Feb 16 '19 at 8:54
• @EricWofsey The definition on Wikipedia is NOT unusual. It's the formal definition, but everyone is so used to using it incorrectly that they say "anything with the symbol $=$ is an equation". – C. Melton Feb 16 '19 at 9:30

I have collected different definitions from different references and tried to find out whether 1+1=2 satisfies all the conditions to be called an equation according to that definition. (The definitions and the URLs are mentioned in the question)

All of the above references except Encyclopædia Universalis can be used to call 1+1=2 an equation. Encyclopædia Universalis article mentioned about the presence of 'one or more variables' in the equality to call it an equation. But as I continued reading the article on Encyclopædia Universalis, I found another statement which can be used to contradict their definition of the equation.

An equation that is true regardless of the values ​​of the variables is an identity.

Equations are of two types, identities and conditional equations (or usually simply "equations")".

I think from the above 2 references, it is safe to say that identities are a subset of equations. We can call identities equation but not conditional equation (which most people simply refer as 'equation' and it should have at least one variable). We all know the famous Euler Identity ($$e^{i\pi}=-1$$) and it has no variable. Wolfram Mathworld and Wikipedia article mentioned Euler Identity as Euler Equation.

In mathematics, Euler's identity (also known as Euler's equation)... --Wikipedia

$$e^{i\pi}+1=0$$ an equation connecting the fundamental numbers i, pi, e, 1, and 0 (zero), the fundamental operations +, ×, and exponentiation, the most important relation =, and nothing else. --Wolfram Mathworld

I have also found another article, Equations And The Equal Sign In Elementary Mathematics Textbooks, written by Sarah R. Powell and was published in The Elementary school journal 2012 Jun; 112(4): 627–648. The manuscript is available on PubMed Central. They have used the following equation terminology.

A mathematical equation is an equation with zero or one variables (e.g., 9 = 6 + 3; 9 = x + 3), whereas an algebraic equation is an equation with two or more variables (e.g., x − 3 = y).

Some may argue that we can just call 1+1=2, an equality but not an equation, I shall quote the definition of Identity from the Mathematics Dictionary.

Equality: The relation of being equal; the statement, usually in the form of an equation, that two things are equal.

Thus, equality is the relation and we often express/write it as an equation. In my opinion, whether to call it an Equality or an Equation, is about English language but not about Mathematics.

I have also liked the Vacuous truth idea by Eric Wofsey, posted in one of the comments. 1+1=2 can also be considered as an equation in the variable x, in which x happens to not actually appear.

• Your research on this is exemplary. You have my upvote. I stand by my answer, but I must commend you on starting this discussion and adding so much insight to it. – C. Melton Feb 18 '19 at 4:38
• @C.Melton Thank you for standing by your answer because yours is the only answer here explaining the different viewpoint. Mike R.'s comment on calling it just an equality also has about 3 upvotes. I have also edited my question yesterday to show that many users on this Q&A forum think that an equation should contain a variable. Most importantly, NCERT(National Council of Educational Research and Training), India textbook says that an equation should contain at least one variable. I guess that their perspective is similar to yours. – Sourav Ghosh Feb 18 '19 at 6:44

I found this on the Equality Wikipedia page:

There is no standard notation that distinguishes an equation from an identity or other use of the equality relation: a reader has to guess an appropriate interpretation from the semantics of expressions and the context. An identity is asserted to be true for all values of variables in a given domain. An "equation" may sometimes mean an identity, but more often it specifies a subset of the variable space to be the subset where the equation is true.

$$1+1=2\;$$ is an equality. It is a relation asserting that $$\;1+1\;$$ is the same object as $$\;2$$. This is always true. $$\;x=x\;$$ is an identity because the statement is true for all $$x\in\mathbb{R}$$. There is no confusion or question about the value of $$x$$, because it could be any value in $$\mathbb{R}.$$

There is an important difference between "equation" and "equality".

$$x^3=x+1\;$$ is an equation, since it contains varibles (or unknowns). It is not true for all $$x\in\mathbb{R}$$, so there is a question about the value of $$x$$. Here we have the concept of "solution" to an equation. It doesn't make sense to say that the equality $$\;1+1=2\;$$ has a "solution", because it is an unchanging, forever true statement to begin with. However, there is a solution to $$\;x^3=x+1.\;$$

You could consider a statement an equation if the concept of "solution" applies to it.

EDIT: Ask yourself why they call $$e^{i\pi}=-1$$ Euler's identity and not Euler's equation.

• Okay. Now it makes sense. So, your proposal is similar to Mike R.'s comment below another answer of this question. You are saying that all equations are equality but not all equality are equations. 1+1=2 is an equality but not an equation because there is nothing unknown to solve. am I correct? – Sourav Ghosh Feb 16 '19 at 7:58
• @SouravGhosh Essentially, yes. From what I understand, the key difference is "unknowns". As you said, there is nothing to solve for in an equality; in an equation, however, there are variables whose values are unknown but come from a specific domain. – C. Melton Feb 16 '19 at 8:04
• This usage may sometimes be found in elementary algebra texts but is not standard at all in higher mathematics. To be clear, it is an important distinction, but mathematicians regularly refer to both of these types of equalities as "equations". Note that the second type of equation does not need to actually have any variables: $1+1=2$ can be considered as an equation of the second type in zero variables, which is solved by the unique (vacuous) assignment of the zero variables. (It can also be considered as an equation in the variable $x$, in which $x$ happens to not actually appear.) – Eric Wofsey Feb 16 '19 at 8:11
• @EricWofsey If you assign zero variables to $1+1=2$, then the equality would be true for all values in its domain, making it an identity, not an equation. Equations are meant to be solvable or insolvable. $e^{i\pi}+1=0$ is not an equation; there is no dimension of solvability. That's why they call it Euler's identity and not Euler's equation. – C. Melton Feb 16 '19 at 8:25
• FYI, neither of the two books I have that are primarily devoted to $e^{i\pi}=-1,$ Nahin's book and Stipp's book, appear to use the phrase "Euler's identity". However, I would not rely on the usage in a couple of semi-popular books anymore than I would rely on State/National education materials or Wikipedia. I agree with the comment (just above) by @Eric Wofsey which, incidentally, your follow-up comment doesn't refute because the premises your refutation relies on are the premises being contested. – Dave L. Renfro Feb 17 '19 at 13:14

Nice question. An equation is basically any mathematical expression involving the equality sign.

So $$X + 2 = 5$$ is an equation. That's true.

But isn't $$1 + 5 = 6$$ also an equation?

• I think, $1+5=6$ can be just called an equality. – Michael Rybkin Feb 16 '19 at 7:13