# How to know what the letters mean in math formulas?

For example, see this Wikipedia section on Newton's laws of motion:

Newton's Second Law states that an applied force, {F},
on an object equals the rate of change of its momentum,
{p}, with time. Mathematically, this is expressed as:


Based on this chart of symbols I understand they are saying A / B = C / D and that d is a function and mv is the input to that function, but how do I figure out what "dp", "dt", "d", and "mv" mean? Please don't give me the answer, but instead, pretty please tell me what method a math person would use to always know what there mean (without asking anyone).

I can sort of guess that mv might mean motion/velocity or something, but surely you're not just supposed to guess at the symbols. A see formulas like this all the time and I can never find a key that explains what the letters mean. What am I missing?

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Well-written mathematical exposition explains all the symbols it uses. Not all mathematical exposition is well-written. But the more math you learn, the more symbols you'll recognize. For example, $dp/dt$ refers to the derivative of $p$ with respect to $t$. That won't make any sense to you, until you've done some calculus, by which point you'll know it better than the back of your hand. – Gerry Myerson Nov 27 '12 at 1:00
In addition to the nice answers you received, people have tried to catalog all symbols, but as Gerry M. stated, authors can define their own as long as it is explained in the exposition. For example, see: rapidtables.com/math/symbols/Basic_Math_Symbols.htm or en.wikipedia.org/wiki/List_of_mathematical_symbols – Amzoti Nov 27 '12 at 1:29
Thanks, @Amzoti - I had found the Wiki link, but that first one is much easier to read and better organized. – user353885 Nov 27 '12 at 1:33

A mathematical text would define the notation it uses, either within the body of the text or in an index of symbols at the start or end of the text, unless the notation is really common. For example, a high school text would not define the symbol for ordinary addition or subtraction.

If you are ready to see what the notation in your example means, read on. Otherwise, stop here.

In your example $F=\frac{dp}{dt}=\frac{d(mv)}{dt}$, $t$ is time, $m$ is mass, $v$ is velocity, $p=mv$ is momentum, and $\frac{d}{dt}$ is not a fraction but the derivative function. You will learn about taking derivatives in a calculus course.

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Thanks. I guess I need to learn calculus. – user353885 Nov 27 '12 at 1:12
I don't understand why you accepted this answer when you specifically requested that people not tell you what the symbols mean. – Qiaochu Yuan Nov 27 '12 at 1:12
Additional information that the OP specifically requested that you not give. I'm not saying this because I don't think this is a sensible answer but because I want clarification about what the question is. – Qiaochu Yuan Nov 27 '12 at 1:14
@QiaochuYuan Yes, the method he gave that I accepted was that the terms should be defined in the text unless they are really common (so, look for the key). That tells me that even in the case of "dp" for example, I should be able to Google "what is dp in formula" and get an answer, where before I thought they were arbitrary letters made up for that specific formula. – user353885 Nov 27 '12 at 1:23
@user: Googing "what is dp" won't get you a useful answer because you're parsing the formula into the wrong conceptual chunks. The smallest conceptual chunks are $F, p, t$ (force, momentum, and time), and the next smallest conceptual chunk is the entire expression $\frac{dp}{dt}$, which denotes the derivative of momentum with respect to time (and which is in particular not the result of dividing something called $dp$ by something called $dt$). This is not the kind of thing you can Google for until you already know more or less what you're looking for. – Qiaochu Yuan Nov 27 '12 at 1:41

You take classes, and hopefully the people teaching those classes explain what the symbols they're using mean. Alternately, you read textbooks, and hopefully the people writing those textbooks explain what the symbols they're using mean. In this particular case, taking a physics class and a calculus class (alternatively, reading a physics textbook and a calculus textbook) would tell you what all of the relevant symbols mean.

(In particular, in this case $d$ is not (quite) a function and the bar does not (quite) denote division. This is not something you can just figure out.)

I don't understand how you're supposed to know what symbols mean without asking anyone. Do you expect that you can learn what Chinese characters mean without asking anyone (not even a dictionary)? Did you learn what English characters meant without asking anyone?

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I can't afford classes. There are no online resources you know of? Seems like reading the letters formulas would be a pretty basic part of math. There are no books you can recommend? Why not just tell me what the method for looking up the letters is? Or if is there is no method, if it is all knowledge handed down from father to son, please at least tell me that. – user353885 Nov 27 '12 at 1:07
It is knowledge handed down from teacher to student. You can probably learn all of the relevant background on Khan Academy (khanacademy.org). You did not ask for recommendations in your question. You asked for a method, and this is the method: you learn what symbols mean by people explaining to you what they mean. – Qiaochu Yuan Nov 27 '12 at 1:08
Cool, thanks for the link. – user353885 Nov 27 '12 at 1:16
Also, to answer your question, Chinese is arbitrary, so someone would have to explain what all the symbols mean and how to combine them, etc. I thought math was more universal and logical, not just another hard-to-understand arbitrary language like Chinese or English. – user353885 Nov 27 '12 at 1:30
@user: whether or not mathematics is universal and logical (I am not sure what this means), mathematical notation is definitely not. Like all human languages, it was invented by humans to communicate with other humans, and like all natural languages, it has evolved considerably over time. – Qiaochu Yuan Nov 27 '12 at 1:36