Use of "hence" in mathematical English

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?

Solution 1:

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.