# Why is the apostrophe positioned differently in "ones' complement" than "two's complement"?

There is a concept in computer science which deals with how to demonstrate negative binary numbers. Two methods for achieving this goal are ones' complement and two's complement.

Since I got acquainted with these methods, this question never stops coming to my mind that why the apostrophe is placed differently in ones' vs two's? I also saw many articles using one's instead. Even Google suggests that I might be mistaken!

So I want to resolve this issue once and for all. Is it a stylistic preference? What causes this difference? And please provide more examples of similar usage.

Donald Knuth, that doyen of computer science, says in Art of Computer Programming, Vol 2.:

Detail-oriented readers and copy-editors should notice the position of the apostrophe in terms like "two's complement" and "ones' complement": a two's complement number is complemented with respect to a single power of 2, whereas a ones' complement number is complemented with respect to a long sequence of 1s. Indeed, there is also a twos' complement notation, which has radix 3 and complementation with respect to (2...22) (base 3).

• Well, thanks. So it was all about the word "binary" from the beginning :-) A series of twos is no longer a binary number. Commented Dec 18, 2019 at 7:49
• No, it's that two's complement is based on looking at a single binary digit at a time (a "two", so two's means "of a two" so it's conventionally written as two's) and ones' complement is based on looking at more than one digit 1 in a string (several "ones", so ones' means "of several occurrences of one" which is conventionally written as ones'). If the word ends in an 's' then ownership is specified by a trailing apostrophe with no extra s, if the word doesn't end in an 's' then ownership is specified by a trailing apostrophe and a terminal 's' is added as well. Commented Dec 18, 2019 at 8:28
• Because that isn't confusing at all. It's a good thing nobody actually uses "twos' complement notation" for anything because that would cause so much chaos when trying to describe things vocally. "Did you mean 'two's complement' or 'twos' complement'?" "You just said the same thing twice..." And then the satellite crashes into the atmosphere because two different systems had competing opinions on number formatting... Commented Dec 18, 2019 at 18:57
• I have a copy of Knuth's Vol 2, second edition, right in front of me, can't find that statement, and need to cite it. If you have a page number easily available, you can save me a lot of searching. (It's not in the obvious place on p. 187.) Commented Jul 28, 2020 at 23:05
• ... from following your link and noting the context, I fear it isn't in the second edition at all. The only reason I have a second edition is that some nefarious person borrowed my first edition a quarter-century or so ago. Commented Jul 28, 2020 at 23:21

According to yourdictionary.com's "Apostrophe Rules - English Grammar Rules & Usage" (with my edits indicated with [square brackets]):

[SINGULAR FORMS:]

Use an apostrophe + "s" ('s) to show that one person/thing owns or is a member of something.

• [Two's complement]

• Amy's ballet class

• Robert's car

• Ross's room

• Ross's sports teams

Even if the name ends in "s," it's still correct to add another "'s" to create the possessive form. It is also acceptable to add only an apostrophe to the end of singular nouns that end in "s" to make them possessive. In this case, you can show possession for Ross either way:

• Ross'

• Ross's

It makes no difference whether the item owned is singular or plural. We use "Ross's" to say that the room (singular) is his and that the sports teams (plural) are his.

[PLURAL FORMS:]

Use an apostrophe after the "s" at the end of a plural noun to show possession.

• [Ones' complement]

• The parents' bedroom

• The Smiths' lives

It is not necessary to add another "s" to the end of a possessive plural noun.

If a plural noun does not end in "s," add an apostrophe + "s" to create the possessive form.

• The children's rooms

[SUMMARY:]

Remember, a possessive noun [should have] an apostrophe and an "s" at the end. [Alternatively,] if there's already an "s" there, you can just add the apostrophe. If there's no "s," you have to add both - first the apostrophe, and then the "s."

• This should be the accepted answer based on the question "why the apostrophe is placed differently in ones' vs two's"
– Ben
Commented Jun 27, 2023 at 14:44