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

Question

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.

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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.

Answer 1

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?

Answer 2

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.

Answer 3

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.

Answer 4

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.

Also from Mathematics Dictionary, James and James, 5th edition, Page-147:

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.}