It is important to write in a neat and legible manner. The following lists some frequently confused characters:
On a side note, Greek letters are extensively used by mathematicians (26 English letters are often insufficient!). One should learn all the Greek letters in order to master the mathematical language.
Always write in clear order (avoid writing in ‘two columns’ on the same page as it usually hinders reading) and cross out unwanted materials (cross out those and only those words which you don’t want; crossing out one word more or one word fewer can lead to a totally different meaning). Highlighting the final answer sometimes also helps make the overall presentation neater.
Also, there are certain expressions such as $\dfrac{1}{2x}$ and $\dfrac{1}{2}x$ which you need to distinguish carefully. Sometimes it also helps by writing with proper indentation. For example,
is more readable than
We began this writing guide by saying that complete sentences should be used in mathematical writing. We conclude it by saying that complete paragraphs should be used. In short, this means we should avoid writing a few isolated equations without explaining the logical relationship between them. For example, consider the following:
\[\begin{align*} 5x+1 &=16 \\ 5x &=15\\ x&=3 \end{align*}\]This is probably how primary and secondary school students ‘write mathematics’. Indeed these can be considered as complete sentences. (Try to read them: ‘5 times $x$ plus 1 is equal to 16. 5 times $x$ is equal to 15. $x$ is equal to 3.’ Three complete sentences!) However when put together this makes no sense — what are we trying to say? Are we assuming the first equation holds? Or is it true that the first equation does hold because of some reason? Furthermore, what is the relationship between these equations? These are important issues and must be clarified by using proper connectives or symbols. For instance, two possible ways to connect these equations together are as follows:
According to the question, we have $5x+1=16$. Since $5x+1=16 \Leftrightarrow 5x=15 \Leftrightarrow x=3$, we conclude that the value of $x$ is 3.
It follows from our previous discussion that $5x+1=16$. This is equivalent to $5x+1=15$, or $x=3$.
If you are still not convinced of the importance of avoiding isolated equations, consider the following (wrong) demonstration:
\[\begin{align*} \sqrt{x-2} &=x-4 \\ x-2 &=x^2-8x+16\\ x^2-9x+18&=0\\ (x-3)(x-6)&=0\\ x&=3\text{ or }6 \end{align*}\]Yet if we plug $x=3$ into the original equation, the two sides are not equal. Some would argue that since we have squared both sides, it is necessary that we check the ‘solutions’ obtained at the end. However this explanation is neither complete nor convincing — apart from naturally asking why (squaring both sides would matter), you can easily find many other examples in which you would end up with such ‘wrong solutions’ even though you haven’t squared both sides in the process. Ultimately, it is the relationship between different equations that matters.
The above can be rewritten as follows:
To solve the equation $\sqrt{x-2}=x-4$, we note that
\[\begin{align*} \sqrt{x-2}=x-4\ &\Longrightarrow\ x-2 =x^2-8x+16 \\ &\Longrightarrow\ x^2-9x+18=0\\ &\Longrightarrow\ (x-3)(x-6)=0 \end{align*}\]Hence the only possible values of $x$ are 3 and 6. When $x=3$, the left hand side of the equation is 1 while the right hand side is $-1$; when $x=6$ both sides are equal to 2. Thus we conclude that $x=6$ is the only solution.
As the last piece of advice, try to learn mathematical writing by reading how professional mathematicians write. Pay special attention to how words and symbols are integrated to form complete sentences, and how complete sentences are linked together by connectives to form coherent paragraphs that present rigorous mathematical arguments.