A template for syntax of Mathematics

1. Basic Definitions and Theorems

We use AsciiDoc’s custom roles to describe definitions and theorems.

Example 1. Definition 1 (Prime Number)

A natural number greater than 1 that has no positive divisors other than 1 and itself is called a prime number.

Example 2. Theorem 1 (Fundamental Theorem of Arithmetic)

Every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers, and this representation is unique, up to the order of the factors.

Example 3. Proof

(Proof omitted) It can be shown by mathematical induction.

2. Propositions and Lemmas

Example 4. Proposition 1

Any even number is divisible by 2.

Example 5. Lemma 0

If \( n^2 \) is an even number, then \( n \) is also an even number.

Example 6. Proof

We prove the contrapositive. If \( n \) is an odd number, then \( n \) can be written as \( n = 2k + 1 \) for some integer \( k \). Then \( n^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1 \), which is an odd number. Since the contrapositive is true, the original proposition is also true.

3. Describing Corollaries

Example 7. Corollary

There are infinitely many prime numbers.

4. Application: Euler’s Formula

Example 8. Theorem 1 (Euler’s Formula)

For any \( \theta \in \mathbb{R} \), the following holds:

\[e^{i\theta} = \cos\theta + i\sin\theta\]

Example 9. Proof

Defined using Taylor series expansion. Derived by substituting \( x = i\theta \) into the Maclaurin series expansions of \( e^x, \sin x, \cos x \) and comparing.