
Multiplying and dividing with Hindu-Arabic numerals and with Roman ones - Hooke
https://thonyc.wordpress.com/2017/02/10/the-widespread-and-persistent-myth-that-it-is-easier-to-multiply-and-divide-with-hindu-arabic-numerals-than-with-roman-ones/
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shripadk
Thank you for this:

"I would point out, as I have already done in my nineteenth-century style over
long title, that one should call them Hindu-Arabic numerals, as although we
appropriated them from the Islamic Empire, they in turn had appropriated them
from the Indians, who created them."

"The Hindu-Arabic number system developed sometime in the early centuries CE
and our first written account of it is from the Indian mathematician,
Brahmagupta, in his Brāhmasphuṭasiddhānta (“Correctly Established Doctrine of
Brahma“) written c. 628 CE. It came into Europe via Al-Khwārizmī’s treatise,
On the Calculation with Hindu Numerals from 825 CE, which only survives in the
12th-century Latin translation, Algoritmi de numero Indorum."

You wouldn't believe me if I said so but in Indian schools we were taught it
to be "Arabic numerals" with no context of the origin of the modern-day
numeral system. I'm not sure if this has changed in the recent years but that
is what I was taught anyways. As a child I assumed that the origin of the
modern-day numeral system was thanks to the Arabs. It was this whitewashing of
anything "Hindu" that really surprises me (especially in a country like India
with 70% Hindu populace). It's almost as if Hindus never contributed anything
to the field of Science. It was quite later in my life that I was able to read
up various articles online and realize that most Mathematical concepts
originated from India.

~~~
jacobolus
“Hindu” is even somewhat misleading, as plenty of early Indian mathematicians
were Jain or Buddhist. Likewise “Arabic” is misleading, considering Islamic
mathematicians working with such numerals were from a wide variety of ethnic
backgrounds, such as Al-Khwarizmi who was Persian.

Anyway, if you’re interested in Indian mathematics, check out this lecture
[https://www.youtube.com/watch?v=2HmMzhZ8zJg](https://www.youtube.com/watch?v=2HmMzhZ8zJg)
(Sadly I don’t know of any other good online lectures about the subject.)

~~~
nileshtrivedi
"Hindu" refers to people living across the Sindhu (Indus) river. It's a
geographic label, not religious one, given by the Persians.

~~~
devnonymous
This needs to be emphasised more. In India, the right wing Hindu brigade try
very hard to appropriate ancient Indian history as that of the Hindu religion
(while also conveniently even denying that it is indeed a religion).

Most modern day Indians do not take the trouble to learn about India (the
notion of which was created by the British). If they did they'd realise that
speaking about ancient Hindu culture as something that they can associate with
is as silly as the British claiming they invented Democracy just because they
were part on the roman empire at some point (as also was Greece who actually
could make that claim)

~~~
shripadk
> This needs to be emphasised more

This is a well known fact for many in India.

> In India, the right wing Hindu brigade try very hard to appropriate ancient
> Indian history as that of the Hindu religion (while also conveniently even
> denying that it is indeed a religion).

This is where you are wrong. There is no "ancient Indian history". "India" was
formed in 1947 and if at all there is any history for India it is the past 70
years. We were called Hindustan before 1947. The word Hindustan was not based
on the religion that we know today as "Hinduism". The religion was called
Sanathan Dharma. The word Hindu however, as was rightly said in the
grandparent comment, a name given by the Persian invaders. What the right
wingers do propagate is "Hindutvawadi" or that we are all Hindus -> Hindu-
Christians, Hindu-Muslims, Hindu-Sikhs, Hindu-Buddhists, Hindu-Jains, Hindu-
Sanatanis etc. Hindu being the geographical name of the Country we reside in
and Sanathan Dharma/Islam/Christianity/Jainism/Buddhism etc being the actual
religion. Does this make sense?

Mahatma Gandhi said the following: "I call myself a Sanatani Hindu, because I
believe in the Vedas, the Upanishads, the Puranas, and all that goes by the
name of Hindu scripture, and therefore in avataras and rebirth; I believe in
the varnashrama dharma in a sense, in my opinion strictly Vedic but not in its
presently popular and distorted crude sense; I believe in the protection of
cow. I do not disbelieve in murti puja.". Was Mahatma Gandhi a right winger to
call himself a "Sanatani Hindu"? Why call himself a "Sanatani Hindu" if
Sanatan and Hindu both mean the same? It is very evident that the word Hindu
in pre-independence India mean't geographical boundary and the word Hindu
post-independence India means religion.

> Most modern day Indians do not take the trouble to learn about India (the
> notion of which was created by the British). If they did they'd realise that
> speaking about ancient Hindu culture as something that they can associate
> with is as silly as the British claiming they invented Democracy just
> because they were part on the roman empire at some point (as also was Greece
> who actually could make that claim)

The problem is that no one claims to be "associating" with ancient Hindu
culture because the context of the word "Hindu" changed drastically over the
past century. What was once a geographical association pre-Independence has
now turned into a religious association. It's extremely sad that it has been
distorted in this manner all in the name of Secularism.

------
userbinator
_You now add the results from the right hand column leaving out those were the
number on the left is even i.e. rows 2, 4 and 5._

In other words, it's just the shift-and-add algorithm:
[https://en.wikipedia.org/wiki/Multiplication_algorithm#Shift...](https://en.wikipedia.org/wiki/Multiplication_algorithm#Shift_and_add)

...which can really be done in any number base (and is even "easier" in base
2), so I'm still not convinced.

 _All we need to carry out the multiplication is the ability to multiply and
divide by two! Somewhat simpler than the same operation in the Hindu-Arabic
number system!_

Maybe it is "simpler" but certainly not faster, because it glosses over the
fact that division by 2 of Roman numerals is itself not exactly
straightforward.

Roman numerals are a mix of positional and unary-ish "repeated symbol" \---
that's what makes them hard to manipulate.

~~~
jacobolus
Both doubling and halving on a base ten counting board (e.g. of the style
commonly used in Europe up until at least the 16th century; Roman counting
boards were presumably recognizably similar, though not much physical evidence
remains) are very fast and easy.

To halve any even group of pebbles, just remove half of them. To halve a lone
pebble resting on a line (representing some power of ten), just move it
downward into the adjacent space. To halve a number in a space (representing 5
times a power of ten), replace it by 2 pebbles on the line below, and one
pebble in the space below that. These operations can be done very quickly and
fluidly with a bit of practice. Maybe not quite as fast as operating a
soroban, but not inordinately much slower either.

Historically nobody performed arithmetic calculations by writing down a bunch
of intermediate steps on scratch paper using Roman numerals (for one thing,
paper wasn’t available in the time of the Romans and both papyrus and
parchment were expensive). Roman numerals are just a way of recording the
finished answers to calculations done on an abacus (counting board).

~~~
chopin
>Historically nobody performed arithmetic calculations by writing down a bunch
of intermediate steps on scratch paper

When I was in elementary school we used a slate and chalk. I think this was
quite common when paper wasn't abundant.

~~~
jacobolus
From what I understand ancient Roman schoolchildren did most of their writing
on wax tablets. It’s not clear when writing slates were first used; one of the
earliest known examples is apparently 11th century India, though I’m not sure
if they used chalk or some other writing material. See
[https://en.wikipedia.org/wiki/Slate_(writing)](https://en.wikipedia.org/wiki/Slate_\(writing\))
[https://en.wikipedia.org/wiki/Blackboard](https://en.wikipedia.org/wiki/Blackboard)

So to correct your statement: this was quite common in the 18th–20th century,
well after the introduction of “arithmetic” and Hindu–Arabic numerals, which
were popularized in Europe by Leonardo of Pisa’s 1202 book _Liber Abaci_
(mostly material translated from Arabic sources).

Before that (and for a few hundred years after in most parts of Europe), most
calculation was done with counting boards.

Here’s a famous 1503 woodcut showing a comparison between arithmetic
(“Boethius”) vs. abacus (“Pythagoras”):
[http://www.maa.org/sites/default/files/images/upload_library...](http://www.maa.org/sites/default/files/images/upload_library/46/Plimpton-
Smith/8056Reischarithmetic.tif)

------
my_first_acct
If you want some up-to-date (and very in-depth) information on ancient
Mesopotamian arithmetic, you might be interested in the paper "Floating
Calculation in Mesopotamia" [1]. By "floating" the author means what I would
call floating-point. If I understand the paper correctly, there were actually
two base-60 systems: one for counting (addition and subtraction) and a
different one for multiplying and dividing.

[1]
[http://cdli.ucla.edu/pubs/cdlp/cdlp0005_20160501.pdf](http://cdli.ucla.edu/pubs/cdlp/cdlp0005_20160501.pdf)

------
arjie
Nothing could have demonstrated the advantage of modern numerals any more than
this has. What an awfully cumbersome algorithm shift-and-add is to hand
execute.

With things like this (where conversion is trivial), internal use of Roman
numerals would be widespread if they were advantageous. They were not,
previously. and now the numeral system we use is irrelevant with ubiquitous
computers except for readability .

I think the evidence is quite strong that Roman numerals are a poorer numeral
system.

------
Strilanc
I think it's a fair point that the reason dividing MCMLXVI by XXXIX seems
daunting is mostly due to lack of familiarity with the notation. Roman
numerals jump up in value by factors of 5 then 2, which is 10, so translation
to/from base-10 Arabic numerals can be done inline... assuming you ignore the
god-awful reverse-order-means-subtraction rule, and the fact that you don't
have any letters for fractional numbers or huge numbers.

The post seems to imply you can't do the doubling algorithm with Arabic
numerals, but of course that's wrong. It also associates roman numerals with
abacus computations, which is probably historically sensible, but to me the
columns of the pictured abacus are literally base-10 positional notation and
so more akin to Arabic numerals. For the abacus to "really" be like roman
numerals, you'd have to remove one bead from each section of the pictured
abacus. Representing four (or one, for the top sections) would be done by
moving a bead to the next grove in sequence. You'd be able to tell it was a
bead from the previous section because every section would be required by law
to use a different kind of bead.

~~~
jacobolus
It’s not clear how much the “hand abacus” pictured, with its sliding counters
in grooves, was used in practice. Most calculations in Roman times were
probably done with a free-form counting board with some kind of tokens (for
instance “calculi”, i.e. pebbles). We don’t have a lot of surviving evidence
though, since a wood/cloth/drawing-on-the-ground counting board with pebbles
for counters isn’t exactly the most permanent artifact.

The way such counting boards worked in Europe in the Middle Ages was to put
pebbles on horizontal lines representing powers of ten, or in the spaces
between the lines representing five times some power of ten. The Roman ones
probably weren’t too different.

------
partycoder
A proficient abacus user can learn how to do math quickly by exploiting muscle
memory and imagining the abacus rather than physically using it.

[https://www.youtube.com/watch?v=Px_hvzYS3_Y](https://www.youtube.com/watch?v=Px_hvzYS3_Y)

------
labster

        perl6 -e 'say Ⅶ * 2'
        14
    

Nothing hard about this nowadays.

~~~
linschn

      $  perl6 -e 'say VII * 2'
      ===SORRY!=== Error while compiling -e
      Two terms in a row
      at -e:1
      ------> say VII *⏏ 2
         expecting any of:
             infix
             infix stopper
             postfix
             statement end
             statement modifier
             statement modifier loop
    

EDIT: One should use
[https://en.wikipedia.org/wiki/Numerals_in_Unicode#Roman_nume...](https://en.wikipedia.org/wiki/Numerals_in_Unicode#Roman_numerals_in_Unicode)

------
meitham
Decimal numbers along with zero ٠١٢٣٤٥٦٧٨٩ were invented in India. The arabs
used them until early days of Islam. Then due to fraud issues with the zero,
being just a dot that can easily be overwritten to anything else, the arabs
set to invent their own symbols. The new symbol is angles based, zero being a
circle has no angles, 1 has a single angle and so on ... The symbols
0123456789 is yet to be adopted by many Arab countries because it was invented
post writings of Quran, where it uses the original Hindi symbols.

------
Taniwha
We (the West) were screwed over by those who brought Arabic numbers into
Europe ... They should have reversed their order as they did it ... It makes
addition and subtraction easier if you work them right to left as in the
original Arabic

~~~
arjie
How? Isn't that essentially adding and subtracting from the most significant
to least significant? Could you demonstrate why it's simpler with an example?
1532 - 879 will be adequate.

~~~
TimonKnigge
Not OP, but I guess it's easier because you work from left to right. At least,
I find subtraction/addition easier to do from least to most significant digit,
since the results of the less significant digits affect the more significant
digits, but not the other way around.

EDIT: To be clear, going from left to right is not inherently 'easier' but it
would be more consistent with our direction of writing.

~~~
arjie
Isn't that what we actually do, though? Everyone is taught to subtract or add
least to most. Why would we want to flip anything.

~~~
TimonKnigge
Because then 'least' is on the left, and you work from left to right, in the
same way that you read from left to right.

~~~
arjie
Ah I see. Thought there was more to it consider the "screwed over". Bit
dramatic, I thinnk, that fellow.

~~~
Taniwha
Yes but it's cause all sorts of angst in the computer architecture world -
"little endian" architectures are "pure" in the sense that arabic is with it's
embedded numbers, while big-endian systems are this weird mixed artifact of
our language with text bytes going from lower to higher addresses and integers
going the other way.

Over time we all realised that big-endian systems (IBM, 68k, etc, and our
native western languages) were a hack and we've moved to little endian systems
(intel, arm, DEC, ...) as sane

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rahil31
Hindu is the follower of Hinduism , much as muslim is the follower of islam
.... but Hindi is a language much as Arabic is one .... It should be Hindi-
Arabic numerals :)

~~~
pavanky
Arabic and Hindu also refer to civilizations. The name doesn't come from the
languages.

Besides Hindi is relatively new. It did not even exist as a language when the
number system was invented.

