# By definition, by the definition

I am confused how to use "by definition" in mathematics.

When I want to say that some fact follows from Definition of A, then can I say:

1. This fact follows by definition.
2. This fact follows by definition of A.
3. This fact follows by the definition of A.

If I use the sentence 1, does the meaning different from 2 or 3 (if these make sense).

It's worth noting that the use of 'by definition' requires a particular set-up when describing the steps in a mathematical problem, i.e. there has to be an actual definition involved, and tends to adhere to the following format.

A rectangle is a polygon with four sides and four right angles.

A square is a polygon with four equal sides and four right angles.

Therefore, by definition, a square is a type of rectangle.

In other words, the expression 'by definition' has a narrow scope It means that what you are about to say must be true. Not just is true, or happens to be true, but must be.

To return to the examples you have given. In my opinion, both 1 and 2 are valid statements, though they aren't overly natural in their structure. 2 is certainly more specific - it directly references A, but if A is implicit in the overall statement, then it can be omitted from the sentence.

For example:

1. A rectangle is a polygon with four sides and four right angles.

A square is a polygon with four equal sides and four right angles.

Therefore, a square is a type of rectangle. This fact follows by definition.

2. Fact A: A rectangle is a polygon with four sides and four right angles.

Fact B: A square is a polygon with four equal sides and four right angles.

Therefore, a square is a type of rectangle. This fact follows by definition of A.

The construction seems odd, but it works. The format in my first example would arguably be more recognisable though.

Mathematical problems, particularly mathematical papers, tend to follow a specific syntax and structure. the following paper provides a nice collection of these, and my prove useful.

Writing Mathematical Papers in English - A Practical Guide

• Thank you for the answer. Is "by the definition of A" incorrect?
– Eng
Jan 4, 2017 at 6:09
• 'by definition' is idiomatic, 'by the definition' is not.
– mike
Jan 4, 2017 at 6:26
• If a definition is numbered, like "Definition 1.2", then can I still say "by Definition 1.2"?
– Eng
Jan 4, 2017 at 8:56