Ask HN: Why is the number of greatest magnitude on the left? | Hacker News Reader

Ask HN: Why is the number of greatest magnitude on the left?

17 pointsblintson29 comments

There's a lot of people knowledgeable 'bout math here so I thought I'd ask. Is their any reason when writing base-whatever numbers that the number of greatest magnitude's on the left and the number of least magnitude's on the right?

Since English is read and written left-to-right, it seems like it'd be more intuitive to do it the other way.

I was thinking it might be a hold-over from the Arabic language.

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From Knuth vol 2 section 4.1:

Our decimal notation...was developed first in India within the Hindu culture...The earliest known Hindu manuscripts that show decimal notation have numbers written backwards (with the most significant digit at the right), but soon it became standard to put the most significant digit on the left."

[later] "It is interesting to note that the left-to-right order of writing numbers was unchanged during [translation from Hindu to Arabic to Latin], although Arabic is written from right to left while Hindu and Latin scholars generally wrote from left to right. A detailed account of the subsequent propagation of decimal numeration and arithmetic into all parts of Europe during the period 1200-1600 has been given by David Eugene Smith in his History of Mathematics I, chapters 6 and 8."

I thought it was a smart question and I am severely impressed that Knuth addressed it. I really need to read those books.

I'm speculating, because everybody is about this point, but written the way we write it, the most important number is the first one your eyes encounter. In most human numbers, you can immediately tell the magnitude from the length without actually counting the digits (since there won't be more than about six), so the first number your eye hits tells you most of what you need to know about the number, as the difference between "five thousand" and "six thousand" is far more significant than "one when taking modulo ten of something-thousand" vs. "two when taking modulo ten of something-thousand".

Written your way, you'd have glance to the end to tell the most important digit, then work your way backwards along the digits of significance, then jump back to the text.

Human language is pretty sensitive to this sort of consideration, all things considered. It looks chaotic but there's a lot of order to it, usually, under the hood.

Left to right numbers are actually more natural in European languages than in Arabic, which is written right to left. It's very inconvenient writing numbers in Arabic, because if you write things in the order you say them (which most people do) you have to write them in the opposite direction to the way you write the words.

Left to right numbers are so inconvenient in Arabic that I expect it's something they copied from Hindu numbers, which were the original source.

You shouldn't infer from English on all European languages. In some languages you would say "three (and) twenty" instead of "twenty three".

In English you can say "three and twenty" as well, though it's an archaic usage. But is there any language in which you'd say 1,786,942 starting with "two?"

It's said that way, but still written the regular way.

English does it as well by the way: thir-teen, four-teen, fif-teen, six-teen etc. After twenty it gets more regular.

For a really interesting one look at Frech saying 89: quatre-vingt neuf

Four times twenty + 9

English equivalent: four-score and nine.

I think its just convention, in essence thats the way it started so it stayed that way.

But as a counter argument to your statement that it would be more intuitive the other way given the way english is read, consider that you want the largest magnitude digit first since it contains more information about the number than the rest of the digits.

With that reasoning, the way we write dates should be YY/MM/DD.

For some apps, that would make sense, but most use of dates is scoped within this week, this month, then this year. If you look at your calendar, you are mostly concerned with upcoming events and not next year's. So the Aug 1, 2009 date gets the important information up first, or possibly 1 Aug 2009.

We effectively do work that way, when speaking.

"Next month, I'll be out of town from the 15th to the 20th," contains a month (and implicit year), followed by the day. You'd be less likely to hear "I'll be out of town from the 15th to the 20th of September this year."

It's really common to hear "I'll be out of town from September 15 to 20 this year", though, and that's the way most (USian) people write out the date, too: Sept 15, 2009.

We use ISO dates in our company so that neither the US or European employees get confused. It's also how you express dates in Chinese ( 2009 nian 8 yue 1 ri ). It bothers me that we express dates any other way.

i.e. the way most of the world writes the date

In India, the dates are written as DD/MM/YY, whereas other people choose to do it differently, which quickly becomes confusing. That's why I always prefer something like <Name of the Month> <Day>, <Year> (i.e, August 2, 2009).

I'm also one of the crazy ones who always use a 24 hour clock. IMO, 1730 is much easier to say than 5:30 PM. Also, it's fairly unambiguous when you're dealing with someone on the other side of the globe.

The Roman numerals also had the largest symbols on the left for instance MDCLIX (1659) reads as:

M = 1000

D = 500

C = 100

L = 50

IX = 10 - 1

So I doubt it is strictly an arabic holdover.

Check out the p-adic numbers - http://en.wikipedia.org/wiki/P-adic_number :

"Informally, most people are comfortable with non-terminating decimals because it is clear that a real number can be approximated to any required degree of closeness by a terminating decimal adequately expressed for its intended application. If two decimal expansions differ only after the 10th decimal place they are quite close to one another, and if they differ only after the 20th decimal place they are even closer."

"10-adic numbers use a similar non-terminating expansion, but with a different concept of "closeness" (which mathematicians call a metric). Whereas two decimal expansions are close to one another if they differ by a large negative power of 10, two 10-adic expansions are close if they differ by a large positive power of 10. Thus 3333 and 4333 are close in the 10-adic metric, and 33333333 and 43333333 are even closer."

Since English is read and written left-to-right, it seems like it'd be more intuitive to do it the other way.

How is it more intuitive to write 3-20 when you mean twenty-three?

You would say 3 and twenty, and doesn't french say its numbers that way?

"when you mean"??? This is just how you grew up, it has no bearing what what is more or less intuitive.

French: vingt trois (twenty three) German: drei und zwanzig (three and twenty)

I thought it was french that did something like "three twenties and a ten" for seventy? Am I thinking of some other language?

That's true enough (93 = "quatre-vingt-dix trois" = "four twenties and three"), but it's still the case that the more significant digit is listed first - it's just composed by multiplying.

For the record, 93 is quatre-vingt-treize ("four twenties and thirteen")

the Swiss French dialect has a far nicer system - they say neufant trois, this is fairly colloquial though.

English: Sing a song of sixpence a pocket full of rye, Four and twenty blackbirds baked in a pie.

The quoted portion to which I responded specifically referred to English.

As others have pointed out, I believe that it probably has something to do with significant information appearing first (on a left to right interpretation). So, 4,732 tells us that the number is in the 4700 range first, followed by the less-significant 32.

I don't know if this explains how it became this way, but it would make sense in explaining why it remains this way.

But it didn't "start" that way. It came to us not from the Hindus, but from the Arabs, through Al-Andalus and the cultural exchanges at Córdoba and Toledo. The Arabs read right to left, so the most significant digit is at the end, not the beginning of the number. At the time Arabic numbers were adopted in the West, we were ignorant of the Hindu practice.

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