Use of "the" for math variables

Question

It seems clear to me that this is correct:

1. Let positive real numbers a,b,c satisfy abc<1. Find ...
   (No "the" before "positive".)

Should "The" be used here?

2. [The] Positive real numbers a,b,c satisfy abc<1. Find the minimum of a+b+c.

And what if the values are determined or even unique?

3. [The] Positive real numbers a,b,c satisfy abc=1 and a+b+c=3.
   (Only a=b=c=1 can make both equations hold.)

Are they more or less the same?

Answer 1

In this case you should use the definite article in all the above cases. Algebra is a compact, manipulable and more easily understandable form of whatever natural language you are using and if you were writing the first part out in English you would say

Let the positive real numbers a, b and c satisfy the condition that the product of a, b and c is less than one.

If you were working in a language without articles, such as Han Chinese, you would, of course, not use articles but if you then translated into a European language you should insert them. The algebra is independent of the language but the linguistic part defining the variables is not and the whole thing should be grammatical.

I would not actually use the first form anyway unless a, b and c had been previously defined. I would say

Let a, b and c be real numbers such that abc<1

because that defines a, b and c formally. In that case you don't need the article.