Is $3+2=5$ a equation?

Question

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

Answer 1

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.

Answer 2

Your friend is:

• right, according to the Encyclopedia of Mathematics (and Wikipedia and Wiktionary)
• wrong, according to the Oxford Dictionary, the Cambridge Dictionary, dictionary.com, merriam-webster.com, planetmath.org, mathworld.wolfram.com, Math dictionary

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.

Answer 3

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!

Answer 4

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.