Although there is no well-defined rules in writing mathematics (unlike the subject of mathematics itself in which everything seems to be rigorously defined), there are some basic elements which should be remembered.
One of the misconceptions students have about writing mathematics (which probably arises from their writing habits in secondary school) is that the writing should be composed mostly of equations and mathematical symbols. This is not true at all. By contrast, a piece of mathematical writing should contain mostly words, supplemented by equations and mathematical symbols. Take any mathematics textbook to verify this. Even in their worked examples, the solutions contain mostly words.
One of the very first basic principles in writing mathematics is therefore to write structurally in complete sentences. Take what you are reading now as an example. You can see that the whole subsection on ‘using complete sentences’ is divided into several paragraphs, each divided into several sentences. Each sentence is a complete sentence.
There can be equations and mathematical symbols, but still they form a complete sentence when properly read. Some examples are as follows.
| Sentence containing equations and mathematical symbols | How it is read as a complete sentence |
|---|---|
| Hence $x+1>3$. | Hence $x$ plus 1 is greater than 3. |
| Since $x>2$, it follows that $x^2+3x>2^2+3(2)=10$. | Since $x$ is greater than 2, it follows that $x$ squared plus three $x$ is greater than two squared plus three times two which is equal to 10. |
| Thus $n\ne0\ \forall n\in A$. | Thus $n$ is not equal to 0 for all $n$ that belongs to $A$. |
On the other hand, the following lists some examples which are not complete sentences and hence should not appear in a piece of mathematical writing.
This sounds pretty much natural, as this is how you write an English passage. However, when it comes to writing mathematics, sometimes we may be tempted to start a sentence with a small letter because it is a symbol or a variable with which we want to use to start the sentence. Here are a couple of examples, and how they may be modified:
| Sentence starting with a small letter | How it may be modified |
|---|---|
| $p$ is not a prime number if it is divisible by 3 and greater than 3. | If the number $p$ is divisible by 3 and greater than 3, then it is not a prime number. |
| $x^2+x+1=0$ has no real root since its discriminant is negative. | The equation $x^2+x+1=0$ has no real root since its discriminant is negative. |
Some even suggest that we should never start a sentence with a symbol or a variable, even if that is in capital letter. In that case similar modifications as above may be applied.
We often separate variables by commas, for instance
However, consider the following sentence:
What’s the problem? As we read, one naturally sees ‘in addition to $p$, $q$’, so we expect that $p$ and $q$ have some property, and apart from them there is yet another number with the same property. But then this is not what the sentence intends to convey. This is somehow related to the previous subsection when we mentioned that we should not start a sentence with a variable in lower case — it turns out that starting a clause with a lower case can be confusing too when the previous clause also ends with a lower case variable.
Again, we may apply some modification to make the sentence more readable:
Here are two more examples. Can you rewrite them?
Since $x>1$, $x-1>0$ and so $\sqrt{x-1}$ is well-defined.
As a quadratic equation in $x$, $x^2y+xy^2+xy+1=0$ has discriminant $(y^2+y)^2-4y$.
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