# ‘A sum‘ or ‘THE sum’?

``````6 = 1 + 1 + 1 + 1 + 1 + 1 (a sum of 6 numbers)
= 6 (1 number)
= 2 + 4 (2 numbers)
= 3 + 3 (2 numbers)
= ...
``````

a natural number n, i.e. a positive whole number greater than zero such as six, is [a/the] sum of up to n natural numbers less than or equal to it

I’ve been given conflicting advice on the topic and would appreciate any insight that could clarify the issue for me. It seems to me that because there are several different ways to add up numbers to produce a number to any one of which this explanation applies in the case of this dummy example – or rather in the case of what I’m actually trying to talk about (matrices) there are infinitely many ways to add up rank-1 matrices to produce a matrix of rank higher than one – ‘a sum’ is sometimes used:

If only we had a representation of the data matrix A as a sum of several ingredients… Stanford

Recalling that the SVD expresses a matrix A as a sum of rank-1 matrices… Stanford

…we can see that the product is a sum of rank-1 matrices Prof. Richard Wilkinson

Any real m×n matrix A can be expressed as a finite sum of rank 1 matrices… Brown

Finally, we represent matrix A as a sum of rank 1 matrices… Brown

Equation (1) can also be written as a sum of rank-1 matrices… Oxford

[SVD] Gives a formula for A as a sum of rank-1 matrices… UBC

I’ve also been told however that unless I’m working with multiple definitions for the word sum, I should stick to the definite article, and that ‘a matrix of rank higher than one is the sum of rank-1 matrices’ sounds decisively more natural than ‘…a sum of rank-1 matrices’.

I’m struggling to reconcile these two seemingly conflicting accounts – when should I use each article and why? Also what part of my reasoning on/analysis of the use of the indefinite article with sum is flawed and how?

• You give a lot of examples with "a sum", but how many have you found with "the sum"? Do you want to know which is better, or whether it's possible to use "a sum"? Basically, if your tutor or someone similar is telling you to use "the sum", do it. Jan 5 at 10:26
• @StuartF i wanna know when each should be used and why. also yeah, is it possible to use ‘a sum’? if that sounds weird, what kind of weird? broken english weird or bad prose weird? Jan 5 at 11:23
• Please do not use i for I and wanna for want to and english for English and no caps at the beginning of a sentence. This site uses standard English not text messaging etc. English. Jan 5 at 15:22

The two articles in English are a and the. A is the indefinite article and refers to non-specific nouns. The is the definite article and refers to specific nouns.

6 = 1 + 1 + 1 + 1 + 1 + 1 (athe sum of 6 numbers)

You should choose the because the sum is a specific sum -- the sum of those six numbers -- and only one such sum exists. A non-specific sum might be:

Choose a sum that adds up to seven.

It's non-specific because (a) it might not even exist; and (b) there might be several choices.

I would simply apply that logic to your other examples. Many of them depend on the context, and your choice of a or the might ultimately influence the meaning of the sentence more than any other word!

Consider

8 = 5 + 3

The word sum is used both for the additive expression (5 + 3), and for the result of the addition (8). The two entities are equal, so in some sense they are the same, but the expressions are different.

a natural number n, i.e. a positive whole number greater than zero such as six, is [a/the] sum of up to n natural numbers less than or equal to it

Using the states that there is only one result for the addition.

Using a states that there is at least one additive expression for the given result. In some contexts it might state that there is at least one result for the addition, but not here.

In mathematical writing authors often try to avoid saying any more than they mean. Using a follows this approach.

For your first example I would normally prefer the, i.e.

a natural number n is the sum of up to n natural numbers less than or equal to it.

For the second meaning I would usually prefer something like

a natural number n can be written as a sum of up to n natural numbers less than or equal to it

as this makes it clearer that I am considering a range of additive expressions.

For all of your other examples I would prefer a.