# “Both of” vs. just “both” with mathematical symbols

In mathematical writing we use letters to denote the mathematical objects we are writing about. I wonder how to use "both of" in the following phrase:

Both of I and I' are irreducible ideals.

vs.

Both I and I' are irreducible ideals.

I am writing in American English. More abstractly, should I treat mathematical symbols as nouns?

• possible duplicate: “Both which” or “both of which” – Matt E. Эллен Aug 13 '12 at 9:29
• Best just to leave it off: I and I' are irreducible ideals. – Brian Hooper Aug 13 '12 at 11:37
• @MattЭллен, the questions overlap to extent of both asking about both vs both of, but this question seems to allow either form (possibly with the first preferred), and the other question primarily the second form. Ie, a combined question could ask both things, but as is they aren't duplicates – James Waldby - jwpat7 Aug 13 '12 at 15:43

Because I and I' are nouns in this sentence, your second example is the correct usage.

Both I and I' are irreducible ideals.

To see how this works, compare with:

Both of cat and dog are animals

vs

Both cat and dog are animals

• I agree. I don’t recall ever having seen both of used in mathematical writing. – Daniel Harbour Aug 13 '12 at 11:27
• @DanielHarbour, I disagree with Rory, but haven't downvoted, preferring to reserve judgement. Rory, have you got a shred of evidence for what you say, other than the irrelevant cat and dog? – James Waldby - jwpat7 Aug 13 '12 at 15:34
• The cat and dog could have been any nouns - my choice was arbitrary to highlight basic sentence structure. – Rory Alsop Aug 13 '12 at 16:22
• Rory, you have still offered no evidence to support your claim that I, I′ being nouns makes a difference. Here is my reasoning: Either of “Both I and I' are...” and “Both of I and I' are...” is acceptable, and the latter sounds better; your “rule” says the latter isn't acceptable; so your “rule” is wrong; hence your answer is wrong and should be downvoted. (By the way, I wouldn't say either phrase, but instead something like “I and I' both are irreducible ideals” or “Ideals I and I' are both irreducible.”) – James Waldby - jwpat7 Aug 13 '12 at 16:52
• Hahahaha. You may not like it but it is still correct. Your second example also works, but your first one is not common phrasing. – Rory Alsop Aug 13 '12 at 17:05