# How to write ordinal expressions in mathematical texts?

I was wondering if there is a rule or recommendation how to write ordinal expressions if instead of a number, we have a variable or a formula. For example, it is common to come across to the following cases in mathematical and computer science textbooks and papers:

• The ith equation (i=1,2,3,...,n) is solved by...
• Let X(n) denote the best solution found in the nth iteration of the algorithm. We will form a set containing X(n) and all the solutions tested in the nth iteration.

If instead of i and n there were n-1 and i+2, how would we write that in English: n-1st or n-1th, i+2nd or i+2th? How would we deal with more complex expressions such as [n/5]? Is it possible to form such expressions at all or we need to rephrase the sentence?

• IMHO this question is better at home at Math SE than here. – Glorfindel Jan 24 '17 at 18:45
• Yeah this is an excellent question but you might have more luck over there – Joe Pinsonault Jan 24 '17 at 19:04
• Is it possible to migrate the question there? :) – Milos Jan 24 '17 at 19:09

In my opinion, when referring to lists/sets/sequences we use the standard notation.

Example

1st element, 2nd element, (n+1)th element, ...

The element at (2n+5)th position

For more complex expressions, we change the sentence structure.

Example

Evaluate the expression at k=(3n-5)/(2n+1)

And not Evaluate the expression at (3n-5)/(2n+1)th position.

• I would avoid "(2n+5)th" or "(2n+5)th position" and similar because of the "th" part. I would go with "... the expression at position k=(3n-5)/(2n+1)" and "... the expression at position (3n-5)/(2n+1)". – AdrianHHH Jan 26 '17 at 12:30
• I will write nth element if I know that n will be an integer. – Aritra Saha Jan 26 '17 at 14:25
• OK, I was not clear. I intended to distinguish a simple integer (or integer placeholder such as "n") from an expression. The spoken words "evaluate the n plus oneth widget" sounds unpleasant to my UK ears. The text "evaluate the (n+1)th widget" looks ugly to my UK eyes. – AdrianHHH Jan 26 '17 at 14:33
• We often find these type of expressions in science text books. In mathematics we often come across statements like 'nth root', 'nth term of a sequence' etc. What I observed is that the 'th' part is added only if the placeholder n is an integer. Ugliness is, in my opinion, entirely subjective. Also, most mathematicians do not care if they are writing correct English or not, they just want to get to the end of their problem. – Aritra Saha Jan 26 '17 at 16:27
• I think "correctness" in language is especially important to mathematicians. Mathematical writing tends to be terse, but strictly correct. – James K Feb 27 '17 at 23:09