

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

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?<p>Since English is read and written left-to-right, it seems like it'd be more intuitive to do it the other way.<p><i></i>I was thinking it might be a hold-over from the Arabic language.
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grandpa
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."

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

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jerf
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.

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pg
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.

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

~~~
pg
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?"

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bshep
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.

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

~~~
byrneseyeview
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."

~~~
randallsquared
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.

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jacquesm
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.

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defen
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."

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byrneseyeview
_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?

~~~
ars
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.

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

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

~~~
kgrin
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.

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

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dkokelley
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.

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pseingatl
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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pseingatl
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.

