Problem: Is $3+2=5$ a equation ?

Solution As we know that that $3+2$ is a arithmetic expression.

So $3+2 = 5$ is a arithmetic equation.

But my friend said that $3+2=5$ is not a equation as it should contain variable

I told that $3+x$ a algebraic expression and $3+x=5 $ is algebraic equation

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    $\begingroup$ An equation is, by definition, something with an "=" (or, more generally, a statement that two things are equal). Your friend is wrong. $\endgroup$ Jun 2, 2016 at 10:38
  • $\begingroup$ But in Wikipedia it is written that "an equation is an equality containing one or more variables" $\endgroup$
    – rst
    Jun 2, 2016 at 10:42
  • $\begingroup$ It is an expression (which is always true) IMO. $\endgroup$ Jun 2, 2016 at 10:45
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    $\begingroup$ Judging by the answers and comments, this is a surprisingly nontrivial question. $\endgroup$
    – Dejan Govc
    Jun 2, 2016 at 10:48
  • $\begingroup$ See Peter Cameron, Introduction to Algebra (2nd ed - 2008), page 23: "A formula, or expression, is some collection of symbols [like: $x^2 +3$]. This formula contains a variable $x$, and the assumption is that if we assign a numerical value to $x$, then we can in principle evaluate the formula and obtain a number. An equation is a mathematical statement of the form $F_1 = F_2$, where $F_1$ and $F_2$ are formulae." $\endgroup$ Jun 2, 2016 at 10:59

4 Answers 4


Your friend is:

I do not think it is worth debating the meaning of the word "equation" - just agree upon a meaning. If you want you can use the word "equality" in general and "equation" for an equality with variables. But as you see these words are not exactly set in stone.

EDIT: As pointed out in the comments and what I neglected to address is, that Wikipedia (and other Wikimedia) pages are of course not exactly authoritative sources. If I take those sources out of the equation it seems your friend is more likely wrong than right. Although there is a chance, that my selection of sources is heavily biased, so the above paragraph still applies in my opinion.

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    $\begingroup$ Interesting answer (and perhaps confusing the OP more than before)...:-) $\endgroup$
    – Soham
    Jun 2, 2016 at 10:52
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    $\begingroup$ @tatan One can provide an easy answer or an approximately honest answer. I decided in favor of the latter. $\endgroup$ Jun 2, 2016 at 11:16
  • $\begingroup$ Wikipedia and Wiktionary should be summarily struck out of this answer. Then it's only Encyclopedia of Mathematics saying your friend is right. $\endgroup$
    – David R.
    Jun 4, 2016 at 21:37
  • $\begingroup$ @DavidR. Perhaps you do have a point. I'll think about it. $\endgroup$ Jun 4, 2016 at 21:38
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    $\begingroup$ Theoretically, any one of us can go on Wikipedia right now and change what it says to match what we think an equation should be (don't know how long the changes would stick, but that's another topic). Besides, how do we know the expertise of any of the Wikipedia editors? Paul August claims to be the holder of advanced degrees and a writer of arcane dissertations, but he could just as easily be a bored 7-Eleven assistant manager. $\endgroup$ Jun 5, 2016 at 0:01

Is $e^{i \pi} = -1$ an equation? We agree that $e$ is the natural logarithm base, $i = \sqrt{-1}$ and $\pi \approx 3.14159$ has something to do with a circle. Thus it does not contain any variables either.

And yet we call it "Euler's identity" (some people prefer to express it as $e^{i \pi} + 1 = 0$, which still contains no variables).

Identity is just another synonym for equality or equation. So yeah, both $3 + 2 = 5$ and $e^{i \pi} = -1$ are equations. It just has to tell us that what's on the left of the equal sign is the same as what's on the right of it.


I would ask your friend whether he considers this an equation: $$x = x$$ It does have one variable...

Now, I'm sure our learned colleagues can point out some esoteric domain in which the equality of an object to itself is not a foregone conclusion. But for most practical purposes, it is an useless equation, as it does not really tell us anything we didn't already know (before you say that neither does 3 + 2 = 5, it does tell us that the base of numeration is not 2, 3, 4 or 5).

However, by the Merriam-Webster definition, $x = x$ is not an equation, because it involves only one expression ($x$).

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

The big difference between $3 + x = 5$ and $3 + 2 = 5$ is that, although they are both equations by the dictionary definition, only one of them needs to be solved, the other one already is.

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    $\begingroup$ So what about $x = x-1+1$? That equality consists of two expressions. $\endgroup$ Jun 5, 2016 at 8:05
  • $\begingroup$ @Stefan That's a brilliant point, as the dictionary didn't say anything about efficiency of expression. $\endgroup$ Jun 6, 2016 at 2:48

As a former math teacher, I would say "You're both right." An equation simply needs an equal sign to be an equation (thus making it different from an expression, which lacks the equal sign). But in high school, equations will (almost) always have variables in them. So, at this point, the important parts will not deal with the definition of equations, but in making sense of them.

P.S. Single variable equations are easy. Wait until Chemistry or Physics, when some equations have a half dozen variables -- or more!


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