The mathematics of Mandarin

Not too long ago I started talking about the so-called “vagueness” of Mandarin. This is the perception voiced by some — both native and non-native speakers — that the language is more vague, well, than English at any rate. It’s a perception I’d usually call an intellectual hors d’oeuvre*, a mostly untestable idea of questionable origin that has a whiff of plausibility and some tasty examples to whet your appetite — but is fundamentally unsatisfying. Still, I thought it might be possible that the difference between certain grammatical structures — between Chinese and English — might smear a bit of plausibility on the hors d’oeuvre.

Bruce Humes in a comment on another post expanded on the vagueness perception:

Many Chinese really DO believe that the Chinese language lacks grammar, and having been taught that English does have a very demanding grammar, many Chinese professionals (not just lawyers but also engineers, etc.) then proceed to either 1) Ignore finer Chinese grammar points (which they obviously do know, as native speakers), and/or 2) Consciously or sub-consciously apply English grammar to written Chinese in the belief that this would raise their status in the eyes of others.

But he also mentioned an instance of very precise Mandarin:

Before I went to Taiwan and worked briefly for Tsar & Tsai in the late 1980s, several Hong Kong lawyers told me it was virtually “impossible” to discuss the subtleties of law in Chinese.

Imagine how surprised I was to sit in on Tsai & Tsar meetings (my boss was a Harvard-educated Chinese), and never hear one word of English. The briefs and high court decisions I read and translated tended to be quite impressive.

And that got me thinking, what’s an easily accessible (e.g. not legal decisions or quantum mechanics) subject that requires linguistic precision? Where could we find a good place to demonstrate Mandarin’s ability to render concepts precisely, leaving aside the more sociolinguistic question of whether people want to speak precisely in any given situation?

So today I offer you: Second Grade Math!


Roughly translated: In a division problem with a divisor of 6, Xiaoming swapped the tens and ones digit of the number to be divided**. He got 4 for the answer. What should the real answer be? How do you know?

No, sorry, I’m not giving out the answer here (c’mon — my daughter did it; but being myself a product of the “no pressure” educational mindset that persisted in the US in the 1980s, I have to say it’s way harder than anything we ever did in second grade). If you see some way in which the problem is impossibly imprecise, let me know. Maybe this can be part of Sinoglot’s publishing arm: books teaching Mandarin to foreigners using second grade story problems.


* By that I mean: orderve. What has a script come to if its users must universally use computer assistance to look up the spelling of a common word that is problematic for spellcheckers and that not a single person on the planet can actually remember how to spell?

** The “number to be divided” (被除数) has a proper name in English: “dividend.” But as an undergraduate math major I feel obligated to note that I’ve never used the word for that purpose. According to my daughter, they use 被除数 in class all the time.

30 responses to “The mathematics of Mandarin”

  1. Zev Handel says:

    When it comes to “hors d’oeuvre”, I think the claim that “that not a single person on the planet” can spell it correctly is unfair to the French. You really can’t lay the blame for this spelling on the English orthographic system!

    By the way, I love French spelling. Maybe as much as I love English spelling. It’s the method to the madness that makes both of them not just great fun, but highly functional. I guess you could say that they put the “fun” in “functional”. Heh.

  2. Sima says:

    Now this is what I come here for! But it’s a bit of a shock to arrive and find Zev, of all people, stretched out on the sofa, blowing smoke rings and singing Jacques Brel.

    I’m always amused by the idea that one language might be more precise (or whatever) than another. I can say for sure that my Chinese is less precise than my English, though it it’s considerably more precise than my French.

    I studied engineering as an undergraduate (poor, misguided soul that I was) and so ought to have encountered plenty of maths but, like you, have never heard dividend used in that way. So what was the term we (must have) used? And, indeed, do you think that “number to be divided” is an adequate substitute for “dividend”? I suspect that my own second grade maths would have referred to “the number at the top”, but it’s an awfully long time ago.

  3. Brendan says:

    This is a great post. The “Chinese has no grammar” meme has always irritated me — do these people really think that in any given conversation, there might be some confusion as to whether the dog had eaten the homework or the homework had eaten the dog? — but the “Chinese is vague” is a lot more insidious. It’d be interesting to try to come up with a way of testing that. One thing i’ve noticed is that native speakers of English, at least, tend to use personal pronouns a lot more than is really natural when speaking Chinese. I haven’t got any impression of whether or not the same is true of speakers of other languages that do more to explicitly mark agency than Chinese does, but will keep my ears open.

    Incidentally, I do seem to remember my textbooks referring to the ‘dividend’ and the ‘divisor,’ even as the illustrations continued to put these things in terms of colored squares and pieces of candy. Go, Philadelphia public school system!

  4. Rui says:

    Hi, first time poster here, but I’ve been lurking for quite a while. This is a very interesting post!

    Where could we find a good place to demonstrate Mandarin’s ability to render concepts precisely, leaving aside the more sociolinguistic question of whether people want to speak precisely in any given situation?

    Since you seem to be asking about Mandarin’s linguistic capability, rather than its users’ decisions…

    One thing jumped out at me here: To prove that Mandarin is capable of expressing all concepts precisely is extremely hard (and how do we measure “precise”, anyway?), but to disprove it all you have to do is find a counterexample. If there is a concept that Mandarin unable to express precisely, using English as a yardstick, shouldn’t we look at English to Mandarin translations? Are there times when Chinese translators of English have great difficulty in conveying the same ideas in their mother tongue? And isn’t concision more useful to measure than precision? Since if Chinese can’t say it in a word or a sentence, all they have to do is qualify it with another sentence, but this wouldn’t be interesting. (Am I understanding your purpose correctly?) Actually, I think it’d be pretty hard to show that Chinese isn’t concise, given that Chinese-to-English translations always end up way longer on the receiving end.

    Conversely, are there any concepts that Mandarin can express more precisely than English? To avoid confirmation bias, I think you’d have to ask this as well.

    My gut feeling is that it runs both ways, and would depend very much on the subject matter. It could be true that Chinese lacks the modern, technical vocabulary to rival English in subjects such as law, Western philosophy, and sciences, but in fields that the Chinese have been familiar with for thousands of years, such as Chinese arts and literature, I’m pretty sure that they can say whatever they want, even with more concision than English (e.g. some chenyu are basically content-dense 4-syllable words).

    Another question I have is to what extent is linguistic precision/concision a function of grammar vs. common usage vs. vocabulary. Vocabulary can be enriched by borrowings or new coinages, but for a grammar to be deficient in some way, that’s probably a more serious issue.

    I remember once reading a blog written by a lawyer (can’t find it again, sorry) who bemoaned the inability of Arabic to write rigorous business contracts, and came to the conclusion that Arabic is vague. Maybe it’s because it’s seen as a less modern language, but some would argue that Arabic can be more precise than European languages because of its ability to derive very specific words (?). I’ve also read Minkaohan forums that talked about the impossibility of writing higher level science textbooks in Uyghur compared to Chinese. I’m pretty sure this is primarily a vocabulary problem though.

    (Just as I finished writing this I noticed that this post talks about the ability of Chinese to be vague, instead of the inability to be precise. I think this argument could have more going for it, since you don’t strictly need to include information about number, definite/indefinite articles, logical connectives, etc. Although you could if it were absolutely relevant.)


    One thing i’ve noticed is that native speakers of English, at least, tend to use personal pronouns a lot more than is really natural when speaking Chinese. I haven’t got any impression of whether or not the same is true of speakers of other languages that do more to explicitly mark agency than Chinese does, but will keep my ears open.

    Could this be related to pro-drop languages? English speakers of Spanish and Japanese tend to overuse personal pronouns too.

  5. jdmartinsen says:

    I’ll second Brendan about “dividend” and “divisor” (plus “quotient” of course), which were introduced around the time of long division and the DMSB system…

    But “vagueness” usually doesn’t refer to vocabulary-based ambiguity, anyway. In fact, it’s not all that uncommon to read essays about translation that emphasize how Mandarin’s rich word stock makes it difficult to precisely render the original author’s meaning into a comparatively word-poor foreign language.

  6. For what it’s worth, we used dividend as well, though I’ve managed to not so much as touch a calculator since junior year in high school so I may be the wrong person to talk to about maths (or “math” as it’s more properly called).

    I particularly love the idea of no grammar since every time I ask a professor about some particular subtlety in 文言文 the answer is always “because of the grammar”.

  7. I don’t remember what we called it in my second grade math class, or if we even studied division then; I was much too busy happily staring out the window.

    But one can’t possibly say that Chinese is imprecise. The math that 7th graders here study is beyond anything I encountered in high school. I avoided math as much as possible in school but of course was forced to learn math up to a certain level, but these kids are forced to learn it at a much higher level much earlier. (I learned to like math later.)

    This whole vagueness thing reminds me of the intro of a book I have on Japanese. The writer mocks some dialog that was used in a novel called Shogun. It’s definitely worth the read.

  8. Zev Handel says:

    People’s views on whether Chinese has “grammar” or not depend on what they mean by “grammar” — as is so often the case, it’s easy to talk past one another when we don’t specify how we are using terms.

    For many people, especially Westerners whose first experience of learning a foreign language is studying another Western language, “grammar” tends to mean inflection — that is, changes in the form of a word for grammatical purposes. Verbs changing form for person, number, and gender agreement, nouns changing form for number (singular-plural), adjectives changing into comparative and superlative forms, verbs changing forms for tense/aspect, and nouns changing forms for case all fit into the category of inflection. To put it more simply, many Westerners think of grammar as all those conjugations and declensions they spent so much time trying to memorize. After all, that stuff seems far more time-consuming and complex than word order.

    And Chinese has none of it. Aside from personal pronouns, Chinese completely lacks inflection. Words never change form — if a Chinese word is pronounced X, it’s always pronounced X. There is no plural form of X, no future form of X, no accusative form of X, etc.

    If you substitute “inflectional morphology” for “grammar”, then “Chinese has no grammar” is a reasonably accurate statement. If you substitute “syntactic rules” for “grammar”, then the statement is patently false — all natural languages have rules governing the way phrases and sentences are formed out of words.

  9. Zev Handel says:

    Here’s an example of a Chinese grammatical distinction that English lacks. I think you can make the argument that Chinese is more precise than English in this instance.

    (1) 他们高高兴兴地玩儿。Tāmen gāogāoxìngxìngde wanr.
    (2) 他们玩儿得很高兴。Tāmen wanr de hěn gāoxìng.

    (1) and (2) have distinctly different meanings — manner vs. result. But in English you’re more or less stuck with “They played happily”.

  10. Chris Waugh says:

    I read this post yesterday, but for some reason it didn’t remind me of an incident at lunch on Friday until this morning: A colleague who is still in the early stages of learning Mandarin asked me if Chinese is as expressive as English.

    And yes, I am writing this specifically for the translators out there.

    His problem, it seems, is that direct word-for-word translations of the Chinese sentences in his textbook come out sounding like “Me Tarzan, you Jane” to him.

  11. Bruce says:


    “Words never change form — if a Chinese word is pronounced X, it’s always pronounced X. There is no plural form of X, no future form of X, no accusative form of X, etc.”

    Can’t really agree with you! A fair number of Chinese words change in tone depending on their function: 教书 and 教科书 is a typical example, with “jiao” (a verb) first tone in the “jiaoshu”, and fourth tone as a noun in “jiao ke shu.”And of course there are many, many hanzi that have more than one pronunciation.

    As for plurals, since the 1950s the use of “men” has blossomed. Surely you’ve heard of 学生们 or 同志们? Ugh, I don’t care for it either, but it is omnipresent in formal written Chinese nowadays.

  12. ETH says:

    Re: the morphology discussion between Zev and Bruce.
    們 seems to me like a good candidate for a Chinese inflectional morpheme. I’d be curious to know if it’s really considered that by linguists and if there are others in Mandarin like it. Thoughts?

    As to changing tones depending on function, I would like to know more about this. I know of some instances (e.g., 數 shu3, count(v.); 數學 shu4xue2, math (n.)), but are there very many that do that? And is it a rule-governed change or are these tone changes ones that a person has to learn (and store) on a case-by-case basis?

  13. Sima says:


    Actually, I think it’d be pretty hard to show that Chinese isn’t concise, given that Chinese-to-English translations always end up way longer on the receiving end.

    How are you measuring this? How does this compare with English to Chinese?

    I like the 玩儿 example. I wonder whether, by putting the two examples into a familiar situation, we might be able to see quite a clear distinction in English.

    1 (I left the kids with their grandmother and) they played happily.
    2 (I left the kids with their grandmother and) they had a great time.

    In number 2, maybe English focusses on the ‘great time’ at the expense of the playing, whilst Chinese gives greater weight to the playing (in comparison with English). Though 玩儿 strikes me as pretty empty in this, and many other, contexts.

    More examples of distinctions made in one language, but harder to make in the other, would be interesting. If only I could get my brain in gear.

    You mention that the use of 们, as in 同学们 became widespread from the 50s. I’d love to know more about this. Do you know how this came about? Does anyone know the history of its use with pronouns – was that a relatively late addition to the language anyway?

  14. hsknotes says:

    I know I’ve either seen this, or made the same comment before, probably in regards to Mair’s asking people to write out 打噴嚏 or something like that, but I imagine in Chinese and English, the number of commonly used words that most people couldn’t spell/write correctly for the life of them (hors d’oeuvre has got to be an extremely rare, if not close to unique case in English, and even it is not incredibly popular or indispensable in English) is insignificant. If it’s 100 in English and 500 in Chinese, I hardly see how that matters to anyone in a world where we talk about tens of thousands or hundreds of thousands (perhaps even millions)of concepts.

    們 has never been a true plural marker, nor was its predecessor in 文言文. 們 declares connection, intimacy, cohesion, connection, relationship, etc, not in any way related to the neutral notion of plural found in English.

    And I’m sure I’ve said this before also, but I don’t why making a “grammatical distinction” is important if a language finds another way of achieving the same effect or explaining the thing.

    Just like the two versions of the children playing happily can be explained in English with no problems (I hope), I imagine “a thing of beauty” and “a beautiful thing” can be explained as well in Chinese.

  15. Bruce says:


    “們 has never been a true plural marker, nor was its predecessor in 文言文. 們 declares connection, intimacy, cohesion, connection, relationship, etc, not in any way related to the neutral notion of plural found in English.”


    I quote to you from 《新世纪现代汉语词典》published by 京华出版社:


    Not quite sure what you mean by a “true plural marker.” Please explain.

  16. Just gotta echo what Bruce said about words changing tone to denote different function. Sounds like inflection to me.

  17. Youngjun Kwon says:

    Chinese completely lacks inflection.


    You assert that the pronunciation of 教 depends on the function. In Chinese, except few words as 玻璃、蜘蛛、etc., one Hanzi represents one meaning, and it means that a Hanzi is a morpheme.
    Like many words in Chinese, “jiao1” in 教书 itself is a word. However, we cannot say that “jiao4” in 教科书 is a word, it is merely a morpheme. So you cannot say that “jiao4” is an inflection of “jiao1”, and vice versa.

    们 has been used as plural marker since Tang period, but it has restriction: only after personal pronouns or nouns about person.

    (…) 见于唐代,附着在人称代词或人的名词后面表示复数。(…) 这种用法,宋、元时也借“懑”“瞒”“每”“门”等来表示,明代后用“们”的多起来。

    谷衍奎,《汉字源流字典》,北京:语文出版社,2008,p. 174

    I think the reason why hsknotes say that 们 has never been a true plural marker is like below. Let’s translate your example letter by letter.

    教书 teach book [to teach]
    教科书 teach subject book [book(s) to teach subject(s); textbook(s)]

    I found that I need to add “(s)” because no one knows if 教科书 is singular or plural. Does 们 solve this problem? No(*教科书们、*教科们书们). 们 cannot substitute plural marker “s” in English. So 们 is not true plural marker.

  18. Brendan says:

    The tone-changing thing is awesome, and almost certainly represents some earlier form of inflection that’s now lost — the 買/賣 distinction’s the easiest example of this, or 王 (wáng)/王 (wàng), or 数数 (shǔshù). I don’t have the historical linguistics chops to say anything about it, but I know that this is the kind of tantalizing thing that drives those guys crazy.

  19. hsknotes says:



    is simply not enough. To understand why the two rules listed here would lead to an infinite amount of problems and non-authentic usages of the character, one needs to understand that there about about 100 qualifications/additional rules to the two rules listed in the “definition” above. If you start to explore why you can’t just willy-nilly use 們 in many cases, and explore why 者 or 那些、一些、某些 or other ways are chosen to express 複數 concepts, you start to see what 們 actually does imply, why it’s allowed/commonly used in certain situations, and why other things are chosen in other situations.

    So, yes the two notes in the “definition” above are technically true, but they are also somewhat useless.

    For example, comparing the usage/distinction in meaning expressed by 父母,父母們,父母者, (when indicating the plural), can only be done through an understanding of the relationship of the speaker to the the group under discussion, and the intention of the speaker. All I’m saying is it’s not just a simple “X indicates plural” story. Kill, murder, and assassinate all have the the same result, but the story is more complex than that. In my opinion, a definition would hopefully make that clear.

  20. Youngjun Kwon says:

    Words never change form — if a Chinese word is pronounced X, it’s always pronounced X. There is no plural form of X, no future form of X, no accusative form of X, etc.

    Everytime I engaged in the discussion on ancient Chinese grammar, I wonder if ancient Chinese people have concept about the separation of verb and noun. Conjugation and declension begin from the infinitive separation. Most of Indo-European languages have this kind of inflection. Verbs and nouns are separated from the infinitives.

    Agglutinative languages such as Korean and Japanese add suffix to noun to make the noun verb. For example, “운동(undong; 運動)” is a noun, “운동하다(undong-hada)” is a verb. The past form is “운동했다(undong-haetta)”, the suffix is slightly changed. However, Chinese language doesn’t seem to have a keen interest about this kind of variation. Noun, verb, past form of 運動 is 運動. Considering 訓読(kundoku) practice and development of Kana in Japan, Korean idu and development of to or gugyeol, we can infer that this situation in the past was same as now.

    買/賣 seems to have nothing to do with inflection. Perhaps, we can relate it to 複音字(區別詞) such as 受/授, 反/返, 坐/座, 弟/悌.

    Nevertheless, I can’t get the problem out of my head that 去聲 seems to be a verb indicator.

    王:平聲(noun), 去聲(verb)
    數:上聲(noun), 去聲(verb)
    畫:去聲(noun), 去聲(verb)

    But how about “食shí/sì” which both cases are verb(eat / make someone eat), and shí has both noun and verb. And sì also was a noun.

    論語鄉黨:“君賜食,必正席先嘗之。”特指飯(舊讀 sì)
    王力,《王力古漢語字典》,北京:中華書局,2000,p. 1660

    So we do need historical research whether ancient Chinese people have concept about separation of noun and verb. It means that we need to clarify ‘verb-去聲’ relation was just a reading tradition (later spread to spoken language) or derived from colloquial language.

  21. Chrix says:

    yes the poyinzi do represent a remnant of inflection Chinese used to apply much more productively in earlier days.

    Have a look at this table, based on Baxter and Sagart:

    Since the table is written in German, some info:

    1. voicing of initial: derivation of intransitive verbs from transitive ones

    2. -s suffix (this is held by most scholars to be one of the motivations behind fourth tone): derivation of nouns, verbs, derivation of exogenic verbs from endogenic ones

    3. -r infix: derivation of exogenic verbs from endogenic ones

  22. Chrix says:

    I forgot to add, there’s much more going on in Pre-Classical Chinese, if interested you should have a look at Schuessler’s Etymological Dictionary, it comes with a lengthy description of Pre-Classical morphology (for instance, apparently the final -t in 月 and 血 is a suffix to form nouns (no longer productive in Classical Chinese), based on comparisons to Tibeto-Burman languages!)

  23. Chrix says:

    sorry, another note: “r-infix”. Actually I don’t exactly remember its function and I don’t have Sagart handy right now.. Schuessler says that it was a causative affix…

  24. Zev Handel says:

    Since I seem to have started this big can of worms with my post on inflection, let me clarify a few things:

    1) Inflection is not derivation. All languages have derivation, which are processes that derive new words from old. This is different from inflectional processes that change the form of a word for grammatical reasons. To a linguist, “discuss”, “discusses”, and “discussed” are forms of the same word. The changes are required by grammar. In contrast, “discussion” is a different word, a noun derived from the verb.

    Chinese has many derivational processes today (prefixes fēi- kě-, suffixes -huà, -xìng, etc.); it had different derivational processes in the past, some of which are still preserved today as tonal differences or other differences.

    The examples Bruce gave are examples of derivation, not inflection.

    2) As I said earlier, Chinese does not have a general singular-plural distinction. But it does have such a distinction in pronouns. It is grammatically necessary to say nǐmen when referring to more than one person, and nǐ when referring to one person. That’s standard inflection for number. This phenomenon must be distinguished from the use of -men after other nouns, where it is not grammatically required and serves other functions — and is therefore not an inflectional process.

    3) @Youngjun; in languages with inflection it is very easy to define word classes like “noun” and “verb”, and to assign words to those classes. For example, any English word with a plural form is a noun; any English word with a past tense form is a verb; any English word with a comparative form is an adjective. In Korean and Japanese, it is easy to define and identify verbs because they have certain inflected forms.

    In languages lacking inflection, like Chinese and Vietnamese, word classes must be defined in other ways. One common way is by collocation. In Chinese, any negatable word is a verb — or in other words, if you can put bù in front of it, it’s a verb. Any word that can be put into the frame “Number + MW + _____” (e.g. yìzhǒng ____ “a kind of ____”) is a noun.

    Similar tests can be devised for Old Chinese (or, if you like, Classical Chinese) to distinguish nouns and verbs.

    Although some of the qù tone pronunciations are artificial reading pronunciations developed in the medieval period, there is no doubt that the qù-tone derivational process originated in the actual Old Chinese spoken language, where it was suffixation with -s.

    4) It is indeed the case that many derivational processes in the old Chinese period involved prefixes and suffixes (and possible infixes) that did not add additional syllables to words; because of sound changes over time, the modern manifestation of these processes can be differences in tone (‘seed’ vs. ‘to plant’) or in aspiration of initial consonant (‘grow’ vs. ‘long’), or both.

    5) Sagart proposes a number of different functions for the *-r- infix, including iterative verbal actions.

  25. Chrix says:

    While I agree in principle with you, if you look at the typological literature on inflection vs. derivation you’ll see that the distinction isn’t as clear-cut as you seem to suggest.

    And not all languages have derivation. The number might be small, but they do exist. McWhorter has written on languages without morphology, including derivation.

  26. Zev Handel says:


    Are all of McWhorter’s examples pidgins? It’s hard to see how a language can manage for several generations without derivational processes, as the only way to bring new words into the lexicon would be to borrow them.

  27. Chrix says:

    no, he is a creolist all right, but for instance he wrote an article called “Why does a language undress?”. Granted, he also includes languages that unexpectedly simplified their morphology. To be sure, this is extremely rare (according to his theory, even just a reduction of morphological complexity shouldn’t happen without a language shift situation, in another book he writes about how English, Malay and Mandarin are less complex than their linguistic relatives).

    Not just borrowing, there is also compounding as a word-formation process…

  28. Chrix says:

    Sorry, to make this clear: the article “why does a language undress?” is not about creoles (not pidgins!), but about non-creole languages. Most of them seem to be from Indonesia and East Timor, and some of the languages cited still have morphology but with reduced complexity without there being such a contact situation. Two languages from Flores have no morphology at all, inflectional or derivational. No scenario is known that would motivate this in terms of McWhorter’s theory.

  29. Zev Handel says:

    @Chrix, I see. I was considering compounding to be a type of derivation, but I do understand that it that type of word formation is often classified distinctly.

    It makes sense to me that languages might not have derivational affixes as long as they have compounding; after all, it seems likely that most if not all affixes ultimately derive historically from reduced/grammaticalized forms of roots.

  30. Youngjun Kwon says:

    Reference for “Classical Chinese had no grammar?” and “derivation by tone change”.
    Norman, Jerry, Chinese, Cambridge Univ. Press, 1988, pp. 84-85.

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