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What is the definition of an equation (as a mathematical terminology)?

I have been using this term, equation, for a long time. I don't even remember when and where I have learned this term (Possibly, I have learned about it in my grade 4 in a different language.) Now I am preparing for teaching Introductory Algebra to undergrads and found that the text book has a different definition for this term, equation.

Could you share how you have learned about it and what is the definition of it?

Here are two definitions of an equation that I found:

(1) From Wikipedia: a statement of an equality containing one or more variables (this is the one I have learned.)

(2) From the text book that I am using: a statement that two quantities are equal. (this seems to be the one taught in elementary schools in Canada)

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  • $\begingroup$ The difference being that the latter need not involve a variable? $\endgroup$ – Adam Jul 7 '16 at 18:26
  • $\begingroup$ Breaking the concept of "=" into "equations" and "identities" doesn't really seem to be a useful distinction. $\endgroup$ – Adam Jul 7 '16 at 18:40
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    $\begingroup$ @Adam Your latter comment -- that it seems not useful to distinguish between "equations" and "identities" -- is directly at odds with the article to which I linked in my answer below. Perhaps you could also write up a response to the OP so that multiple viewpoints are described! $\endgroup$ – Benjamin Dickman Jul 7 '16 at 20:23
  • $\begingroup$ Perhaps Adam's last comment above should be moved below Benjamin's answer. $\endgroup$ – Daniel R. Collins Jul 8 '16 at 6:27
  • $\begingroup$ MESE 10673 on What is a variable? may be of interest, too... $\endgroup$ – Benjamin Dickman Jul 8 '16 at 7:14
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Your question is, in some sense, purely mathematical: What is "the" definition of an equation? You identify two different definitions, so I do not know if any single one of them can be claimed as canonical. Instead, here are some remarks about defining 'equation' from the perspective of the teaching and learning of mathematics.

The first name that comes to my mind around a question such as this (i.e., related to equations or equality) is Carolyn Kieran. For example, see:

Herscovics, N., & Kieran, C. (1980). Constructing meaning for the concept of equation. The Mathematics Teacher, 73(8), 572-580. JSTOR.

Here is the first page:

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And here is the definition suggested, though you would need to read the full article to see what is meant by, e.g., arithmetic identity (although I think you could well guess it):

enter image description here

This is pretty much in keeping with, e.g., the Common Core State Standard 6.EE.B.5:

Grade 6 » Expressions & Equations » Reason about and solve one-variable equations and inequalities. » 5

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

Probably it does not jibe so well with dictionary definitions; e.g., Merriam-Webster on equation:

mathematics: a statement that two expressions are equal (such as $8 + 3 = 11$ or $2x – 3 = 7$)

I think that this is a pretty common definition (asserting that two expressions are equal) although the reference above distinguishes the two Merriam-Webster examples by referring to the former as an arithmetic identity and the latter as an equation.

It is also worth pointing out, as Herscovics and Kieran do later in the article, that the definition provided around hiding a number in an arithmetic identity leads only to equations that have solutions. Here are the related remarks, which concludes with their proposed "general definition of equation":

enter image description here

As a closing note, again from the mathematics education literature, here is one more related Kieran reference from around that time, although this one is specifically about $=$ as a symbol:

Kieran, C. (1981). Concepts associated with the equality symbol. Educational studies in Mathematics, 12(3), 317-326. Springer Link.

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    $\begingroup$ Your information answers to my question very well. Now I need to think about how to present this notion to undergrads. At least now I can tell my students that there are variations for this terminology and they need to understand this within the given context. I think it will be too much to bring up the distinction between arithmetic identity and algebraic equation. Thank you so much. $\endgroup$ – Jessie Jul 7 '16 at 19:20
  • $\begingroup$ Yick, I must say that argument is both confusing and totally unpersuasive. $\endgroup$ – Daniel R. Collins Jul 8 '16 at 6:29
  • $\begingroup$ @DanielR.Collins The identity versus equation (with solutions) one above? Or the general one: any algebraic expression of equality containing a letter or letters? Or something else? Maybe you could post a counterpoint, or link to a different reference? I'd be glad to learn about more successful approaches to broach equations -- success measured in quality research or practice! $\endgroup$ – Benjamin Dickman Jul 8 '16 at 7:07
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    $\begingroup$ Having trouble getting the point here. Is it that the most common usage of the term is in the instruction "solve this equation", and thus "equation" must mean that there is an unknown? Therefore we need another name for a statement that things are equal but which has no unknown? $\endgroup$ – DavidButlerUofA Jul 8 '16 at 14:56
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I'd say that an equation is a statement that two things are equal. This is slightly more general than Amy B's answer, since I don't insist that the statement be mathematical, and it's more general then the second definition in the question, since I don't insist that the two things be quantities. So for example, I consider $$\text{Andreas Blass}=\text{the author of this answer}$$ to be an equation. (It's not important that this particular equation is true; replace "Andreas Blass" with "Napoleon" and you still have an equation --- a false equation, but an equation nevertheless.)

In the special case where the equation contains one or more variables (as in the first definition in the question), the context, and in particular the meaning of the variables, can affect how the equation is to be understood.

One common situation is that the variables are understood as unknowns, and the intended meaning of the equation is the problem: "Find all values for the variables that make the equation true."

Another common situation is that the variables are understood as universally quantified, so the intended meaning of the equation is that it's true for all values of the variables. For example, some people state the commutative law of multiplication as $xy=yx$, meaning $(\forall x)(\forall y)\,xy=yx$. Equations with this interpretation are what I'd call identities.

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Having taught math in elementary school, I can assure you the second definition is correct.

According to Math Open Reference (and many other math sites)

"An equation is a mathematical statement that two things are equal"

Of course this definition includes equations with variables. It also includes:

  • 1+3 = 4
  • 1 + 3 = 11 - 7
  • 4 = 19 - 15

This question is a great example of why you shouldn't rely on Wikipedia.

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    $\begingroup$ I agree with you that we shouldn't rely on Wikipedia. However, it provides an excellent resource. Also, as the reference above (provided by Benjamin Dickman) points out, the second definition is the result of the mixing of two notions: arithmetic identity and algebraic equation. These two notions are rigorously distinguished in "Logic". Thank you for letting me know how the notion, equation, is taught in elementary schools. This somehow gives me a clue for why some undergrads have hard time to understand the notion of function with respect to equation. $\endgroup$ – Jessie Jul 7 '16 at 19:31
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    $\begingroup$ @Jessie: Skeptical that that these notions are distinguished in "Logic". If you could provide a reference that would be interesting. $\endgroup$ – Daniel R. Collins Jul 8 '16 at 6:30
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I believe there are different things we use the term 'equation' to mean over the course of education, and students can get confused because it's usually never explained where the distinctions come in.

At a simple level, an equation is any statement that two things are equal. It is a logical statement with a true/false value that uses 'equals' as the relation, rather than, say, less-than. It's worth remembering that 'equals' means different things in different contexts.

So then different things can happen. One option is that the logical statement is false, e.g. $2+1=5$. We don't use these that much in education, so students may well get confused unless it is made clear that they are being taught to evaluate rather than being taught facts.

Another option is that the statement is true, e.g. $2+1=3$ or $2x+x=3x$. We start teaching simpler forms of these at primary school, and since this is pretty much the only form around, 'equation' does the job, if needed at all.

The third option is that we have decided to assume that the equation is true by hypothesis, and wish to determine the consequences of that hypothesis, e.g. $2x+1=3$. We introduce these later as 'algebra'. Here students will see both cases: $2x+x=3x=5$. These are generally simple enough that they can distinguish between 'true' and 'true by hypothesis' without realising that there's a distinction. In this context we tend to use 'solve' to mean the action of determining the value(s) of the variable. But we also use 'solve' for questions that aren't obviously of this form, so students may not pick up the technical meaning.

After a bit, we start to introduce other statements like $\sin^2(x)+\cos^2(x)=1$, which are of the always-true type, but are not obviously so. We call these 'identities'. These are, in a sense, equations that 'cannot be solved' (more accurately, solving them would not yield much useful information about the value of the variable). Some students will not pick up this distinction, so 'identity' becomes a word they don't understand but think they should (and therefore possibly scary).

Along the way, we also introduce 'equations' with other relations, e.g. $2x\leq 3$. Since we typically use only a limited range of relations, and may not supply an alternative term, these might also get categorised as 'equations'. After all, basically the same methods are used to 'solve' them.

In addition, we bring in questions where the solution requires a sequence of logical statements rather than simply a sequence of equal expressions. Again, students may not notice the change, and come to think of '$=$' as meaning 'the next bit of the solution'. So 'equation' is no longer 'a statement with $=$'.

So then they reach university, and start getting picked up on these distinctions. Maybe they then start to catch on. Meanwhile, they meet definitions. And after a bit one might ask 'what is the definition of 'equation'?'.

Perhaps, then, the answer is partly that there isn't a single definition, because 'equation' is both a mathematical term but also a meta-mathematical term. That is, we use it to talk about doing maths, as well as about the content of maths. And education terms aren't that well suited to rigorous definitions.

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The Wikipedia definition seems very nonstandard in my experience (and also unnecessarily complicated). The definition with which I'm familiar, and which seems much simpler, is as follows:

  • Equation: A statement that two expressions are equal.

We may then observe three subcategories of equations:

  • Identity: An equation true for all possible values of any variables.
  • Conditional: An equation true for some possible values of any variables.
  • Inconsistent: An equation true for no possible values of any variables.

See, for example: Ratti & McWatters, Precalculus, Section 1.1.

Edit: See also: OpenStax College Algebra, Section 2.2.

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  • $\begingroup$ Please note that all these subcategories you listed involve the notion of variable. Thank you for the reference. I will look it up. $\endgroup$ – Jessie Jul 8 '16 at 20:41
  • $\begingroup$ @Jessie: Note that those definitions are all perfectly serviceable even if the number of variables is zero. For example, is $3 + 5 = 8$ true for all possible values of the (zero) variables? Yes; so it is an identity. And likewise $\sin^2(t) + cos^2(t) = 1$ (one variable), $a+b = b+a$ (two variables), etc. $\endgroup$ – Daniel R. Collins Jul 8 '16 at 22:54
  • $\begingroup$ @Jessie: In fact, recognizing purely numerical identities is how one can tell whether any other equation is an identity (after algebraic manipulations). Ratti & McWatters, Section 1.1, Example 4 (Solution): "We have shown that $0 = 0$ is equivalent to the original equation... Thus, the original equation is an identity." $\endgroup$ – Daniel R. Collins Jul 8 '16 at 23:01
  • $\begingroup$ Another example from Martin-Gay, Introductory Algebra 3E (Sec 2.3, Example 7): Having solved an equation and produced the statement $-12 = -12$, she writes: "Again, the final equation contains no variables... thus [the equation] is an identity...". $\endgroup$ – Daniel R. Collins Jul 15 '16 at 16:02
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"Equation" and "solve" are among the words most frequently thrown about promiscuously by people without understanding. They write things like "How can I solve this equation: $\sum_{n=0}^\infty r^n$?" (which is not an equation at all, and which admits evaluation rather than solution.)

An equation is an equality that has some solutions but of which not every number or function or other appropriate object is a solution. Thus $x^2 - 10x + 24 = 0$ is an equation of which $4$ and $6$ are solutions. And identity is an equality that is true of everything (every number or every function or every other appropriate object). For example, $\log(ab) = \log a + \log b$ or $\cos(a+b)=\cos a\sin b + \sin a \cos b$ or $(x+1)^2 = x^2+2x+1$ are identities. Both equations and identities are examples of equalities, which are statements that one thing equals another.

(In the field of typography, the word equation often means what illiterate laypersons think it means: anything written in mathematical notation.)

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