While the use of English does not affect the actual mathematical reasoning, it is still an important aspect students should be aware of. A piece of mathematics written in good English helps the reader follow the argument more easily. The examples here are taken out of their original contexts for simplicity. A lot of these mistakes are made probably because most of the students are not native English speakers. We try to split the errors into several categories, but often one will find that some of the examples fit under multiple categories.
Grammar refers to the way words are put together to form phrases and sentences. A grammatical error is when these rules for grammatical structure are broken.
The definite article the indicates that its noun is a specific one which the reader should know. There is only one binomial theorem and therefore we say ‘the binomial theorem’. On the other hand when we define $x$ to be some non-negative number, very often it does not tell the reader which specific number we want it to be, and thus the indefinite article ‘a/an’ should be used instead. Whether ‘a’ or ‘an’ should be used depends on the pronunciation of the word or letter following it.
| Wrong | Correct | Comments |
|---|---|---|
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Construct a $m \times n$ table. | Construct an $m \times n$ table. | When the letters $m$ and $n$ are pronounced there is actually a vowel sound /e/ at the beginning of each letter. This means that the article ‘an’ should be used. |
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| When referring to a specific theorem (e.g. the Fundamental Theorem of Calculus) use ‘the’ (the definite article). However, when referring to a theorem from a named person use the zero article. |
| By (a), the $g$ is continuous. |
| When naming something with a letter (e.g. $g$) use the zero article. When using the noun function use the definite article (e.g. the function $g$). |
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| When using the phrase ‘negative’ or ‘non-negative number’, use the indefinite article because ‘a’ means any number. When using ‘non-negative’ as an adjective then use the zero article. |
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| Let $x$ be positive. | Use the adjective form and not the noun form. |
In general, most nouns have more than one form depending on the corresponding quantity. If there are more than one object, the plural form is required; whereas the singular form is used if there is only one.
| Wrong | Correct | Comments |
|---|---|---|
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Throwing two fair dices once, we have $6 \times 6 = 36$ different possible outcomes. | Throwing two fair dice once, we have $6 \times 6 = 36$ different possible outcomes. | The singular form is ‘die’, the plural form is ‘dice’ (without ‘s’). In modern English sometimes ‘dice’ is also accepted as the singular form. |
| Every square are rectangles |
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| Assume $P(k)$ is true for some positive integers $k$, i.e. $k(k+1)$ is divisible by 2. | Assume $P(k)$ is true for some positive integer $k$, i.e. $k(k+1)$ is divisible by 2. | There is only one integer here (namely $k$); therefore it should be in singular form. |
| Let $A$ be a square matrice. | Let $A$ be a square matrix. | ‘Matrices’ is the plural form of ‘matrix’. |
| Let $A$ be the vertice of the pyramid. | Let $A$ be the vertex of the pyramid. | ‘Vertices’ is the plural form of ‘vertex’. |
| $(0,0)$ is the only local maxima. | $(0,0)$ is the only local maximum. | ‘Maxima’ is the plural form of ‘maximum’. |
Similar to nouns, most verbs have more than one forms, and you need to be aware of which ones are correct and which ones are wrong.
| Wrong | Correct | Comments |
|---|---|---|
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| Make sure the subject agrees with the verb. Whether to use ‘exists’ or ‘exist’ depends on whether the noun that follows is in singular or plural form. |
| There exist no integer $k$ such that $3k=2$. | There exists no integer $k$ such that $3k=2$. | ‘Integer’ in this example is singular. |
| Replace $a$ by $-a$, we have $f(-a)=(-a)^2=a^2=f(a)$.& |
| ‘Replacing’ has the same meaning as ‘If we replace’; ‘-ing’ clauses can go at the beginning of a sentence and are a dependent clause. |
| Hence we get $x<0$, contradicts to (1). |
| The verb form ‘contradict’ does not have a dependent preposition, i.e. ‘contradict something’. The noun form ‘contradiction’ does have a dependent preposition ‘to’. The relevant phrase is ‘which is a contradiction to’. |
| Thus contradiction. |
| In the wrong version there is no verb form in the sentence. The two correct sentences contain a verb. |
Two of the most frequently used verbs in English are the verb-to-be and the verb-to-do. They are also two of the most frequently misused verbs in English as well as mathematical writing.
| Wrong | Correct | Comments |
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| The phrase ‘there is’ is usually used to introduce new information (which does not refer back to previous information); when we say ‘it is’ there should be something (in this example a rational number) introduced before. ‘There has’ is a grammatical mistake; always use ‘there is/are’ (the verb-to-be). |
| A real number $x$ whose square is negative is not exist. | A real number $x$ whose square is negative does not exist. | Use the auxiliary verb ‘do’ to form the negative. |
| This sequence is not converge. |
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Most words can be changed slightly to convert from one word form to another, such as converge (verb) and convergent (adjective). You have to be aware of the differences between the different word forms.
| Wrong | Correct | Comments |
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| Besides completing the square, we can use differentiation to find the minimum value of $x^2-4x+8$. | In this sentence ‘besides’ is used in a dependent clause, and should be followed by the -ing form of the verb. |
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| The noun form and the verb form are the same (‘increase’), albeit pronounced differently. The word ‘increasement’ does not exist. But in the verb form ‘increase’ is followed by the dependent preposition ‘by’, i.e. ‘increase by an amount’. |
| The maximal possible value of $f(x)$ is 3. | The maximum/largest possible value of $f(x)$ is 3. | The adjective form of the noun ‘maximum’ is the same (maximum). In this example the superlative ‘largest’ can also be used’. ‘Maximal’ has a slightly different meaning in mathematics. |
| The slanted height of the cone is 10 cm. | The slant height of the cone is 10 cm. | ‘Slant height’ is a compound noun (two nouns combined) like ‘traffic light’. |
| The function $y=x^2$ concaves upward. | The function $y=x^2$ is concave upward. | ‘Concave’ is an adjective rather than a verb. |
| For every rational number $x$, can be written as $x=p/q$. | Every rational number $x$ can be written as $x=p/q$. | When using this phrase only use one clause. The phrase ‘every rational number $x$’ is the subject of this clause. |
| One plus one equals to two. |
| This is a confusion between the verb form and adjective form. The verb form does not have a dependent preposition. |
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| ‘The following’ is used in phrases such as ‘we have the following’. ‘As follows’ is used in phrases such as ‘the properties/values/results are as follows’. In the incorrect example ‘follow’ is used in passive voice which means followed by something. In the correct examples ‘follow’ is used in the active voice because after this expression comes the evidence for the result(s). |
A lexical error is making the wrong choice of word for the stylistic context. The sentence is not necessarily grammatically wrong, but does not mean exactly what the author wants to convey.
The wordings in the following examples can be slightly revised so that it reads more smoothly.
| Wrong | Correct | Comments |
|---|---|---|
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| ‘Let’ is used with a bare infinitive form of a verb in the first example. In the second and third examples the convention is ‘let + $x, y,$ etc + be + the value which is assigned to $x,y$, etc’ . |
| Since the divisor $(x-1)(x-2)$ is of degree 2, 1 is the maximum degree of the remainder. | Since the divisor $(x-1)(x-2)$ is of degree 2, the maximum degree of the remainder is 1. | The description of the value (the terminology) goes first in a clause, the value/figure goes last. |
| The smallest value of $x^2$ possible is 0. | The smallest possible value of $x^2$ is 0. | The word order is ‘smallest/largest + possible + value’. |
| Let $x$ be a real positive number. | Let $x$ be a positive real number. | The word order should be: ‘positive/negative + real + number’. |
| For all even numbers $N$, we will show that $3+N$ is odd. | We will show that $3+N$ is odd for every even number $N$. | The phrase ‘we will show that’ usually begins a sentence, although the other way round is not wrong. |
| If $f$ is a strictly continuous increasing function, … | If $f$ is a strictly increasing continuous function, … | Keep the words ‘strictly’ and ‘increasing’ together when describing a function. |
A common type of problems in mathematical writing is the use of wrong words. In this part we look at some general examples of such in English. The wrong usage of mathematical terminology in a particular subject area will be dealt with in the later section on the use of terminology.
| Wrong | Correct | Comments |
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A prime number is a natural number that has no positive factors other than 1 and itself. For instance, if $k$ is a natural number and $k=pq$, where both $p,q>1$, then $k$ is not a prime number. | A prime number is a natural number that has no positive factors other than 1 and itself (for instance, 2 is a prime number). In other words, if $k$ is a natural number and $k=pq$ with $p,q> 1$, then $k$ is not a prime. | ‘In other words’ is used to introduce an explanation or clarification of an idea or concept. ‘For instance’ is used to give an example of the idea or concept. |
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| ‘e.g.’ (from the Latin phrase exempli gratia) means ‘for example’; whereas ‘i.e.’ (from the Latin phrase id est) means ‘that is’. The former uses an example to illustrate a concept, while the latter gives an alternative explanation. |
There are often more than one grammatically correct way to express something, but sometimes some choices of wordsings are more suitable than the others.
| Wrong | Correct | Comments |
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Assume the contrary that not all of them are zeros. |
| ‘On the contrary’ is used as an adverb phrase with the verb assume. On the other hand, ‘the contrary’ is a noun and the object of the verb ‘assume’. In this case add another clause which gives information about the assumption. |
| Let $m$ and $n$ be odd and even. |
| The wrong example is ambiguous because it does not precisely state which is odd and which is even. The two corrected examples show two possible variations which are much clearer. |
| This equation has finite solutions. |
| Here ‘finite’ is used to describe the \emph{number} of solutions, rather than the solutions themselves. |
| The equation $x+3=2x+4-x-1$ has infinite solutions. | The equation $x+3=2x+4-x-1$ has infinitely many solutions. | Again, ‘infinite’ refers to the number of solutions rather than the solutions themselves. |
In this section we collect some other miscellaneous issues as well as some conventional issues. In English language, conventions are a courtesy to the reader, making writing easier to read by putting it in a form that the reader expects and is comfortable with. It includes things such as sentence formations (e.g. complete sentences, punctuation) and conventions of print (e.g. spelling, capitalisation). Many of these do not exist in oral language, so you have to consciously learn them in written language.
Some words sound the same (or similar) when pronounced, but are in fact spelt differently, and may have different meanings.
| Wrong | Correct | Comments |
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The number $x$ can not be rational. | The number $x$ cannot be rational. | ‘Cannot’ and ‘can not’ have different meanings. For example ‘he cannot do it’ means he does not have the ability to do it, while ‘he can not do it’ usually implies he has the ability to do it but also has the option of not doing it. |
| Without lost of generality, we have… | Without loss of generality, we have… | The noun form ‘loss’ is used in this context. |
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| ‘Proof’ is a noun whereas ‘prove’ is the corresponding verb. |
Some words are often misused in mathematical writing.
| Wrong | Correct | Comments |
|---|---|---|
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Rotate $\triangle ABC$ clockwisely by $90^\circ$. | Rotate $\triangle ABC$ clockwise by $90^\circ$. | ‘Clockwise’ is already an adverb. The word ‘clockwisely’ does not exist. |
| The numbers $a$, $b$ and $c$ are pairwisely different. | The numbers $a$, $b$ and $c$ are pairwise different. | Again, the word ‘pairwisely’ does not exist. |
| Here is a counter example. | Here is a counterexample. | Here ‘counter’ is not a word, but rather a prefix attached to ‘example’ meaning ‘the opposite’. |
| Let $x$ be a non zero number. | Let $x$ be a nonzero number. | Again, ‘non’ is not a word, but rather a prefix attached to ‘zero’ meaning ‘not’. |
| From this we deduct that the equation has no solution. | From this we deduce that the equation has no solution. | Although ‘deduction’ is the noun for both the verbs ‘deduct’ and ‘deduce’, the two verbs have different meanings. ‘Deduct’ means ‘subtract’, while ‘deduce’ means ‘draw a logical conclusion’. |
Connectives are very important in mathematical writing as they show the logical relationship between different sentences. There are some conventions and rules on the use of such words.
Note also that there should only be one verb in a simple sentence. When there is more than one verb, we need a conjunction (e.g. ’and’) to properly link the phrases together.
| Wrong | Correct | Comments |
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If $x>0$. Then $2x>0$. | If $x>0$, then $2x>0$. | When using ‘if’ for assumption, always follow it up with something in the same sentence. ‘If $x>0$’ is not a complete sentence. |
| When $x>0$. Then $2x>0$. | When $x>0$, $2x>0$. | Same as using ‘if’, when using ‘when’ for assumption, always follow it up with something in the same sentence. ‘When $x>0$’ is not a complete sentence. |
| Since $x$ is non-negative. We have $x+1>0$. | Since $x$ is non-negative, we have $x+1>0$. | ‘Since’ is used to join dependent clauses to independent clauses and therefore must go in sentences which have two clauses. ‘Since $x$ is non-negative’ is not a complete sentence. |
| Let $x$ be non-negative, then $x+1>0$. | Let $x$ be non-negative. Then $x+1>0$. | When using ‘let’ and ‘then’ to list steps of proofs use two sentences, one for each step. |
| Suppose $x$ is non-negative, then we have $x+1>0$. | Suppose $x$ is non-negative. Then we have $x+1>0$. | When using ‘suppose’ and ‘then’ to list steps of proofs use two sentences. |
Make sure that you spell the words correctly. Apart from those mentioned in the pronunciation and frequently confused words sections, some words tend to be misspelt a lot. Surprisingly, it is not uncommon for students to misspell true as ture and false as flase, two words which occur a lot in mathematical writing.
Next: The Use of Symbols