Order of Variables and Constants in Mathematical Expressions

[Cspeed]

Does anyone know if there is a list of the order that goes in writing a mathematical expression (i.e. 7 $$\pi$$ $$\sqrt{5}$$ vs 7 $$\sqrt{5}$$ $$\pi$$ )?

Is there a generally accepted method?

 

[Mark44]

Does anyone know if there is a list of the order that goes in writing a mathematical expression (i.e. 7 $$\pi$$ $$\sqrt{5}$$ vs 7 $$\sqrt{5}$$ $$\pi$$ )?

Is there a generally accepted method?

I don't know if there is an explicit rule other than by what looks better. For expressions with radicals, factors are usually collected in front of the radical, like this (using your example):
$$7 \pi \sqrt{5}$$

Tip: When you create expressions or equations in LaTeX, it's usually best to put everything inside one pair of tags rather than use a bunch of pairs of tags.

 

[HallsofIvy]

The reason for that is that if you write something like $\sqrt{2}x$, rather than $x\sqrt{2}$ is that it can be hard to distinguish whether or not the "x" is supposed to be inside the square root or not. Of course, as long as you are careful about that, $x\sqrt{2}$ and $\sqrt{2} x$ mean exactly the same thing- it's not like the ordering of operations where a+ bc is different from (a+b)c.

 

[Tac-Tics]

Speaking in (incredibly) general terms, the problem you have just described is that of finding the "normal" form of a term.

In the study of formal systems, equivalent terms usually have several different representations. For example, if we're working with polynomials, x^2 + 1 + 2x is equal to 2x + x^2 + 1, even though how they are written on paper ("exs squared plus one... etc" versus "two exs plus ... etc") are different.

So if you have two terms, how can you tell if they are equal if they can be written out different? One practical example of this would be: If you were a math teacher and the answer your student gave you was different from your answer key, how could you tell if their answer was incorrect? Your answer key may say the answer is "x^2 + 2x + 1", but the student put "1 + 2x + x^2". Obviously these terms are different, yet they are "equal" inside the system we're working with.

The answer is you convert them to normal formal. The definition on "normal form" differs from system to system. For polynomials, we usually use the following definition:

A polynomial is in normal form when it is written $$c_nx^n + c_{n-1}x^{n-1} + c_{n-2}x^{n-2} + \cdots + c_1x + c_0$$ for some (constant) integers $$c_0, c_1, c_2, \cdots, c_n$$ and (constant) integer n.

Some systems do not have normal forms. This was one of the big discoveries in the early 20th Century. For example, if instead of polynomials (a kind of mathematical object), we used computer programs in some programming language (another kind of mathematical object), we would find there is no useful way to put a program in any sort of "normal" form. The major consequence of this is that it's impossible to mechanically decide whether two programs are equal to each other. (To show two programs are equal, you need look for a custom proof -- it's effectively a very difficult guessing game to show two arbitrary computer programs are equal).

In your particular case, you're working with algebraic expressions. There are no hard-and-fast definitions for what constitutes a normal form (I wouldn't imagine there is one at all). However, there are lots of loose conventions. Generally, you keep integral constants on the far left. Variables on the right. But in the case of the radical, those rules can be bent to avoid confusion (since some people are sloppy with the "bar" of the radical). The two primary concerns are readability and ease of manipulation. If you can achieve those, it really doesn't matter how you write them.

 

[Cspeed]

Thanks for the explanations, but I am a little surprised. I always felt that there was some "right" way to write algebraic expressions. For example, in published mathematical papers, I'm sure they strive for consistency, so that from article to article you don't have radicals coming first one place and last somewhere else.

I've been searching, but couldn't find a list like this (just made up):
1. Integers
2. Variables, in alphabetical order
3. Greek letters, in alphabetical order
4. Radicals
etc.