Wednesday, June 04, 2008

Base-100 Arithmetic


In The Number Sense, Stanislas Dehaene says that in Cantonese and Mandarin, the sounds for the numbers are much shorter than in Western languages, and so native speakers of those Chinese languages can speak numbers quicker. He argues that this enables them to do mental math quicker than speakers of Western languages. In many parts of Asia including China, learning mental math is considered very important for children.

Dehaene also writes elsewhere in his book that many people who can do fast mental math not only practise the many calculation shortcuts, but also often memorize the products of 2-digit numbers. I've wondered if people memorizing such products would be better off to use a base-100 instead of base-10 system, that is, to create a hundred digits and map them to the numbers from 0 to 99. After some initial memorization, it would be easy to convert back and forth between them. Even better is if Chinese sounds were used for the base-100 digits, taking advantage of the short sounds. The Chinese group digits into groups of four, unlike English-speakers' groups of three, making Chinese numbering even more suitable.

The first ten digits already exist: 0零, 1一, 2二, 3三, 4四, 5五, 6六, 7七, 8八, and 9九. There's already characters for some of the other 2-digit numbers: 10十, 20廿, 30卅, and 40卌. Perhaps also 木 for 80 (from Chinese riddles) and 半 (meaning ½) for 50. Maybe in some cases these characters for multiples of ten could be used as radicals in associated numbers, for example, digits related in some certain way to 80 could be represented by characters with the 木 radical (eg, 相枩來枳林柬朿朾朽朳朲朰東杰, etc). There's many more existing sequences that could be used in some way, like the 10 stems (甲乙丙丁戊己庚辛壬癸), the 12 branches (子丑寅卯辰巳午未申酉戌亥), or the Yi Ching characters. What is most important, though, is that the sound of each digit from 0 to 99 be different. Because there's about 400 different sounds in Mandarin Chinese, that would be possible.

The easy part for those learning such base-100 arithmetic would be memorizing every mapping between a 2-digit base-10 number and the matching base-100 digit. Children could learn that before they're 3 years old. To do any effective mental math, they would need to memorize many sums and products of pairs of base-100 digits, far more difficult. If they memorized sums by putting the higher number first, and products by putting the lower first, they wouldn't need to remember whether a sequence of four base-100 digits was a sum or product, they would only memorize the sequence itself. If the two numbers were the same, it would be the product. This gives 5050 different ways two base-100 digits can be multiplied together and 4950 ways they can be added: 10,000 combinations in total.

Many of those 10,000, though, could be worked out using shortcuts based on patterns. For example, to multiply two numbers, such as 93 x 98, by using the complement (on 100) of each number, 7 and 2, we can calculate the complement of their sum, 91, followed by their product, 14, giving the final result 9114. This particular example is really only useful in base-10 for numbers quite close to 100, but in base-100, it can be used for all numbers over 50. At the cost of memorizing 50 pairs of complements (1+99, 2+98, etc), we can reduce the 10,000 combinations down by 1275, to 8725.

There's many other shortcuts that could be utilized to reduce that number down considerably further. I suspect those shortcuts would be based on the common divisors of 100, i.e. 2, 4, 5, 10, 20, 25, and 50. For example, when adding 25 + 22, in my mind I calculate it as 25 + (25 – 3) = (2 * 25) – 3.

Of the four-character sequences that would need to be memorized, if many of them bore some pictorial or phonetic resemblance to the thousands of four-character proverbs (成语) that Chinese children already learn by rote, they'd find it much easier to memorize them. In Chinese proverbs, only the content words are recited, not the grammar words, so English proverbs in the Chinese style would be "Stitch time, save nine", "Stone roll, no moss", "Bird hand, two bush", etc. This is what would make it far easier for native Chinese speakers to do base-100 mental math than Westerners learning such arithmetic.

Here's an example of this technique, but using an English proverb instead, with associations 13=bird, 19=hand, 2=two, and 47=bush. To multiply 13 x 19, there's no shortcut, so we'd recite the associated sounds, with the lower number first for multiplication, i.e., 13 x 19 = “bird hand”. We'd automatically finish it in our heads, i.e., “two bush” = 0247. Viola!

I don't know of any existing base-100 arithmetic in China, having never seen any websites or books on the subject. What such base-100 arithmetic needs is for a native Chinese speaker with a background in computing and linguistics to design and run the intensive computations necessary to assign the best possible mapping between 2-digit numbers and base-100 digits, so the memorizations will be easiest for native-speaking Chinese children. It would be a time-consuming input-intensive programming task with a deliverable of only 90 ordered Chinese characters. An example of the future of computing, perhaps?

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