In math, particularly in plane geometry, there are lots of simple statements that one implies another, and that implies another, and so on. So, "hence" is frequently used.

For example, let's say A=B implies CD=CE, and CD=CE implies that F,G,H are collinear. Which of the following is correct, and what is the best expression?

Since A=B,

1 ) we have CD=CE, and hence F,G,H are collinear.

1') we have CD=CE, hence F,G,H collinear.

1'') we have CD=CE, hence F,G,H being collinear.

2 ) hence CD=CE, and hence F,G,H are collinear.

3 ) CD=CE, and hence F,G,H are collinear.

If none of them are very good, can you give the best expression for it?

  • 1
    You can usually think of "hence" as a synonym for "therefore," though it almost always comes at the beginning of the clause, unlike "therefore." 1 and 3 are fine for our purposes, though to certain readers the "hence" may look wrong without being set off by commas on either side. "Hence" is also sometimes used as a substitute for "that explains." For example: "I didn't have time to cook. Hence the carry-out on the table." Under this alternative use of "hence," 1'' above is merely awkward and not incorrect (though it probably needs a semicolon instead of a comma). 1' and 2 are wrong.
    – cruthers
    Commented Sep 15, 2021 at 2:54
  • The alternative "hence" may actually more be a substitute for "which explains," which suggests that there is actually no semicolon needed in 1''. There is some conflict between the stated definition and the examples here: dictionary.cambridge.org/us/dictionary/english/hence
    – cruthers
    Commented Sep 15, 2021 at 3:03

4 Answers 4


There are a number of synonymous expressions to 'hence' in regular writing, but with different frequency in mathematical writing.

'Hence' is common in math but here are some common alternatives:

  • Since A, it follows that B
  • A therefore B
  • A, and so B
  • A. From this B follows.

An even better method for finding alternatives is, instead of relying on a list, to look at what mathematicians actually write in practice.


Statement 1) is the best, and perfectly good mathematical English.

Statements 2) and 3) are acceptable, but stylistically not as good as 1).

Statements 1') and 1'') are incorrect.


Do you need to spell this out in words at all? As it's maths, I'd use mathematical syntax with either "therefore" or "implies" symbols. For example:

⇒ F,G,H are collinear

I find this a lot easier to parse than a wordy statement, and it doesn't rely on the reader knowing English (a bonus, as words like "hence" are not commonly encountered and might not be known, especially if English is not the reader's first language).

  • 4
    I disagree. In mathematical writing, at least if it is intended to be read by others, do not stick things like ∴ and ⇒ in there. Use words.
    – GEdgar
    Commented Sep 14, 2021 at 14:49
  • I'd use three lines: A=B / ⇒ CD=CE / ⇒ F,G,H are collinear. Commented Sep 14, 2021 at 15:45
  • GEdgar, I wouldn’t randomly insert symbols into a sentence mid-paragraph, but as a standalone line I find it works well. Edwin, I agree and that’s how I drafted it, but the formatting was lost when I saved it and I couldn’t work out how to sort it out.
    – Fiona
    Commented Sep 14, 2021 at 19:39
  • This answer omits groupings and separators between statements. For instance, does the "therefore" symbol apply to just "CD=CE" or to the whole phrase "CD=CE ⇒ F,G,H are collinear"? It's not clear. Commented Sep 15, 2021 at 0:11
  • 1
    OK, figured out the formatting (apologies for any confusion, I'm new to markdown and still learning).
    – Fiona
    Commented Sep 15, 2021 at 6:52

1 correct
1' incorrect (the auxiliary is missing (are))
1" incorrect (would be incorrect even if the auxiliary were not missing ; no continuous tense here) 2 not correct (redundancy)
3 correct

Only "1" and "3" are correct. However, the general way which consist in saying "F, G, and H" instead of "F, G, H" is usual in mathematics too.

  • This answer is mistaken about 1') and 1''). Statement 1') omits a verb in the second clause. Statement 1'') uses a verb there, but it is the wrong verb ("being"). Also, this answer is mistaken about Statement 2). Statement 2) does not contain "redundancy". It contains repetition. Repetition is not ideal stylistically, but it is not wrong. Commented Sep 15, 2021 at 0:15
  • @DanielAsimov thanks for pointing out this crude error in "1'" and "2'"; however I do not agree entirely with you. I do believe this is a case of redundancy: The definition of a redundancy is something that is repeated unnecessarily or something that is not useful because there is already another or more advanced version. google.com/search?client=firefox-b-d&q=redundancy
    – LPH
    Commented Sep 15, 2021 at 7:32

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