# Can we have an equation for any graph?

I was wondering, can we have an equation for any graph even the most zigzagged one?

• define equation – Dan Rust Sep 4 '13 at 22:14
• It depends on exactly what you mean by graph and equation. – mrf Sep 4 '13 at 22:14
• There is someone that pick up a number randomly (1,20,8,-5,7..), we graph it, can we find its equation? Merci. – Bonjour Sep 4 '13 at 22:16
• If we pick even a single number randomly (say by throwing a dart at a one-dimensional dartboard and measuring where it hits) can we describe with complete accuracy with some (finite) expression? Probably not. So I think one may as well ask the question in one dimension, and the answer will still be "no". – Trevor Wilson Sep 4 '13 at 22:31
• Graph as in map into an xy plane, xyz plane, a 9 dimension space, or what? There is also the question of how well defined over what domain and range are you plotting here. If you are plotting over the Real or some other complete number system there are a lot of points to define here. – JB King Sep 4 '13 at 22:35

## 3 Answers

There are many, many more graphs in the plane than there are formulas we can write down. In fact, since equations must be written in finitely many characters from a finite alphabet, the number of possible equations - or definite descriptions of any kind - we can write is a countable infinity. However, there are uncountably many functions with different graphs on the plane.

I would have a hard time, of course, describing to you in a definite way one of the uncountable multitude of graphs for which there exists no definite description. They're everywhere, though.

• In some sense, if you can describe their graph entirely, you probably have a formula. – Dan Rust Sep 4 '13 at 22:23
• Yeah, and zig-zags really aren't the issue. We can't even describe all the horizontal lines, i.e., graphs of constant functions. – G Tony Jacobs Sep 4 '13 at 22:26
• It seems quite unfair to say "I have a graph here, but I can't finitely describe it" but then say "I don't accept equations that can't be finitely described as candidates for matching this graph". – Ben Millwood Sep 4 '13 at 22:31
• @BenMillwood To me, a graph means the graph of a function (set theoretically, this is the same as a function itself.) That is, any subset of the plane that passes the vertical line test. There is no definability requirement. On the other hand, "equation" usually means something that can be expressed in a finite language. – Trevor Wilson Sep 4 '13 at 22:40
• There's an interesting question in the comments, too - does $\forall x\exists! y R(x,y)\implies\exists F(x)(\forall x R(x,F(x)))$? This is a core foundational question, and not one that has a unique answer; different formal systems provide different answers, and some versions of this question are unknown (for instance, if $R$ is a polynomial-time predicate and $F$ is intended to be as well, then the question is a slight variation on the P=NP question...) – Steven Stadnicki Sep 5 '13 at 22:22

If you specify any finite collection of points with distinct $x$ co-ordinates and say you want a graph that will go through them, there is an equation, a polynomial even, that will go through all your points. This is called an interpolating polynomial and is not too hard to construct. I wrote an article on interpolating polynomials that explains the process a bit.

If you draw a line on a page with a pencil (so the line has nonzero thickness), there is even a polynomial that stays within that pencil line, as long as it's reasonable (no gaps in the line, no going back on yourself, etc.). That is to say, any continuous graph can be approximated as close as you like (e.g. within a pencil stroke's thickness) with polynomials; one method of doing so is using Bernstein polynomials.

Hence, I suspect the answer to the question you were trying to ask is "yes".

But that's sort of besides the point. The important thing to understand is that "equations" can really be whatever you like. I can write an equation, if it pleases me, of the following form:

$y = \text{the vertical distance of Graph A from the x-axis, at the point distance x from the y-axis}$

As long as I also show you what Graph A is, this is a statement of equality between two well-defined numeric expressions: an equation. So by definition, there is an equation associated with every graph.

You may be more interested in questions like "suppose I have a graph with property $P$, can I match it with an equation with property $Q$". The answers to these questions of course depend on what $P$ and $Q$ are, but as shown in the beginning of my answer, there are a lot of good results you can find.

• Okay, I see you point now, but I think that usually "equation" means an expression in some finite language that is fixed in advance, whereas for your answer you need to expand the language to contain new symbols for any graphs you might want to talk about. – Trevor Wilson Sep 4 '13 at 22:46
• Then how to find an equation for any graph? – Bonjour Sep 5 '13 at 11:25
• I mean is there a general formula? – Bonjour Sep 5 '13 at 11:44
• @Bonjour: I expanded on my answer a little. – Ben Millwood Sep 5 '13 at 21:47

I would say the apt way of representing graphs are not equations, but functions (a function can reduce to an equation in x for 2d, if the function is smooth and continuous, by using some sort of spline interpolation or using nth order polynomial fit). On the other hand any curves in 2d plane atleast, you can write for the worst case, as discontinuous functions representing the graph, even if the function has a limit going to infinity at some point. But one may not want to do this if the graph is too complicated.

• So are you saying the answer is yes or no? – Trevor Wilson Sep 4 '13 at 22:35
• In principle as I said, you should be able to fit the curve to a nth degree polynomial. I hope I am not overlooking anything here. – Vaidyanathan Sep 4 '13 at 22:38
• The equation $x^2 + y^2 = 1$ has a graph, but it's not a function, in the sense of $y$ being a function of $x$. – G Tony Jacobs Sep 4 '13 at 22:39
• Also, there are graphs of functions - most functions in fact - that are discontinuous at every point in the domain. No amount of polynomial interpolation to any degree will get you that. – G Tony Jacobs Sep 4 '13 at 22:40
• yes, i agree discontinuous functions cannot be represented by interpolation. There you can write it as piecewise continuous functions. My point was just to say that, equation may not be a right word to represent a graph. Also can you not represent a circle by 4 piecewise continuous functions, one in each quadrant? – Vaidyanathan Sep 4 '13 at 22:45