Cheat Sheet to Write Mathematics in English for Chinese

The following is by no means ‘the’ introduction on how to write mathematical proofs in English – English is my third tongue, therefore I do not master it very well myself. Just take the following as building blocks so that I can follow your proofs. After a while, you may improve by looking at proof arguments in professional English textbooks – I am not a reference on that topic. That’s also why I recommend you to read only textbooks in English! At the beginning I accept – even if I hate it – the use of logical symbols such as \(\Rightarrow\), \(\Leftrightarrow\), \(\forall\), \(\exists\), etc. in text. Try however to replace them by their English counterparts, it is nicer to read.*


Basic math vocabulary:

Writing proofs:

Exemplary sentences of the lecture:


Let \(n\) and \(m\) be two integers.Suppose that \(n\) is even and \(m\) is even. Then, \(nm\) is also even.

令 \(n\) 和 \(m\) 是两个整数. 假设 \(n\) 是偶数 \(m\) 是偶数. 那么 \(nm\) 也是偶数.

(not nice but okay at the beginning) 证明(不完美的写法)
\(n,m \in \mathbb{Z}\) are even \(\Rightarrow\) \(\exists p,q \in \mathbb{Z}\) such that \(n=2p\) and \(m=2q\) \(\Rightarrow nm=2p2q=2(2pq)=2r\) where \(r:=2pq\in \mathbb{Z}\) \(\Rightarrow\) \(nm\) is even. CQFD.

\(n,m \in \mathbb{Z}\) 是偶数 \(\Rightarrow\) \(\exists p,q \in \mathbb{Z}\) 使得 n=2p 且 m=2q \(\Rightarrow nm=2p2q=2(2pq)=2r其中r=2pq\in \mathbb{Z}\) \(\Rightarrow\) \(mn\) 也是偶数. 证明完成.

Let $n$ and $m$ be two even integers. By definition, it follows that $n$ and $m$ are divisible by two, that is, \(n=2p\) and \(m=2q\) for some integers \(p\) and \(q\). Hence, \(nm=2p2q=4pq=2(2pq)\). It follows that \(nm=2r\) where \(r=2pq\) is an integer. Thus, \(nm\) is even, which completes the proof.

令 \(n\) 和 \(m\) 是两个偶数. 根据定义,\(m\) 和 \(n\) 可以被 \(2\) 整除,这说明 \(n=2p\) 且 \(m=2q\) 对某些整数 \(p,q\) 成立. 那么 \(mn=2r\) 其中 \(r=2pq\) 也是整数. 那么 \(nm\) 是偶数,证明完成.