The Decimal and Islamic Mathematics

Category: Featured, Life & Society, Nature & Science Topics: Al Khwarizmi, Hindus, Islam, Mathematics Views: 20640

The Decimal System

Muslim mathematicians were the first people to write numbers the way we do, and, although we are the heirs of the Greeks in geometry, part of our legacy from the Muslim world is our arithmetic. This is true even if it was Hindu mathematicians in India, probably a few centuries before the rise of Islamic civilization, who began using a numeration system with these two characteristics:

  1. The numbers from one to nine are represented by nine digits, all easily made by one or two strokes.
  2. The right-most digit of a numeral counts the number of units, and a unit in any place is ten of that to its right. Thus the digit in the second place counts the number of tens, that in the third place the number of hundreds (which is ten tens), and so on. A special mark, the zero, is used to indicate that a given place is empty.

These two properties describe our present system of writing whole numbers, and we may summarize the above by saying the Hindus were the first people to use a cipherized, decimal, positional system, "Cipherized" means that the first nine numbers are represented by nine ciphers, or digits, instead of accumulating strokes as the Egyptians and Babylonians did, and "decimal" means that it is base 10. However, the Hindus did not extend this system to represent parts of the unit by decimal fractions, and since it was the Muslims who first did so, they were the first people to represent numbers as we do. Quite properly, therefore, we call the system "Hindu-Arabic".

As to when the Hindus first began writing whole numbers according to this system, the available evidence shows that the system was not used by the great Indian astronomer Aryabhata (born in A.D. 476), but it was in use by the time of his pupil, Bhaskara I, around the year A.D. 520. (See Van der Waerden and Folkerts for more details.)

News of the discovery spread, for, about 150 years later, Severus Sebokht, a bishop of the Nestorian Church ( one of the several Christian faiths existing in the East at the time), wrote from his residence in Keneshra on the upper Euphrates river as follows:

I will not say anything now of the science of the Hindus, who are not even Syrians, of their subtle discoveries in this science of astronomy, which are even more ingenious then those of the Greeks and Babylonians, and of the fluent method of their calculation, which surpasses words. I want to say only that it is done with nine signs. If those who believe that they have arrived at the limit of science because they speak Greek ad known these things they would perhaps be convinced, even if a bit late, that there are others who know something, not only Greeks but also men of a different language.

It seems, then, that Christian scholars in the Middle East, writing only a few years after the great series of Arab conquests had begun, knew of Hindu numerals through their study of Hindu astronomy. The interest of Christian scholars in astronomy and calculation was, in the main, due to their need to be able to calculate the date of Easter, a problem that stimulated much of the Christian interest in the exact sciences during the early Middle Ages. It is not a trivial problem, because it requires the calculation of the date of the first new moon following the spring equinox. Even the great nineteenth-century mathematician and astronomer C.F. Gauss was not able to solve the problem completely, so it is no wonder that Severus Sebokht was delighted to find in Hindu sources a method of arithmetic that would make calculation easier.

We can perhaps explain the reference to the "nine signs" rather then the ten as follows: the zero (represented by a small circle) was not regarded as one of the digits of the system but simply a mark put in a place when it is empty, i.e. when no digit goes there. The idea that zero represents a number, just as any other digit does, is a modern notion, foreign to medieval though.

With this evidence that the Hindu system of numeration had spread so far by the year A.D. 662, it may be surprising to learn that the earliest Arabic work we know of explaining the Hindu system is one written early in the ninth century whose title may be translated as The Book of Addition and Subtraction According to the Hindu Calculation. The author was Muhammad ibn Musa al-Khwarizmi who, since the was born around the year A.D. 780, probably wrote his book after A.D. 800.

We mentioned in Chapter 1 that al-Khwarizmi, who was one of the earliest important Islamic scientists, came from Central Asia and was not an Arab. This was not unusual, for, by and large, in Islamic civilization it was not a man's place (or people) of origin, his native language, or (within limits) his religion that mattered, but his learning and his achievements in his chosen profession.

The question arises, however, where al-Khwarizmi learned of the Hindu arithmetic, given that his home was in a region far from where Bishop Sebokht learned of Hindu numerals 150 years earlier. In the absence of printed books and modern methods of communication, the penetration of a discovery into a given region by no means implied its spread to adjacent regions. Thus al-Khwarizmi may have learned of Hindu numeration not in his native Kharizm but in Baghdad, where, around 780, the visit of a delegation of scholars from Sind to the court of the Caliph al-Mansur led to the translation of Sanskrit astronomical works. Extant writings of al-Khwarizmi on astronomy show he was much influenced by Hindu methods, and it may be that it was from his study of Hindu astronomy that he learned of Hindu numerals.

Whatever the line of transmission to al-Khwarizmi was, his work helped spread Hindu numeration both in the Islamic world and in the Latin West. Although this work has not survived in the Arabic original (doubtless because it was superseded by superior treatises  later on), we possess a Latin translation, made in the twelfth century  A.D. From the introduction to this we learn that the work treated all the arithmetic operations and not only addition and subtraction as the title might suggest. Evidently al-Khwarizmi's usage is parallel to ours when we speak of a child who is studying arithmetic as "learning his sums".

This article is excerpted from the book "Episodes in the Mathematics of Medieval Islam" by J. L. Berggren. 

  Category: Featured, Life & Society, Nature & Science
  Topics: Al Khwarizmi, Hindus, Islam, Mathematics
Views: 20640

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Older Comments:
Why are you in the west. This stupid question always ask to the eastern immigrants, especially to Muslims. Let us analyze it. First, Muslims do not need anybody, we need them. Why? They control all the waterways. They have gas, petrol, uranium and other matchless resources. They have the greatest resource, that is human resource. Why did they come here then? Because we went to their countries, we bombed their countries, we enforced dictators and thugs upon them. We destroyed their sanitation systems, bombed their hospitals, sanctioned them for years. That is the reason Muslims are forced to scattered all over the world. The world need Muslim consumers, Muslim manpower, Muslim resources, Muslim-controlled-waterways. Let us talk intellectually, instead of child-like-ranting.

Romesh Chander has some deep misunderstandings of the Islamic Civilisation. I am a history student. I am still a devout Christian. First of all, Islamic Civilization took 600 years to collapse. 600 yars does not include that 500 years of golden period. None other civilization dare to match the acheivements of Muslims. If you read Indian history, the golden period of India evolves and revolves around Muslims. Mughal Empire was the golden period of India with absolutely no arugment. Hindu civilization in comparison is nothing but mythology and scattered small kingdoms. Surprisingly, Mughal Empire started in the middle of the Muslim decline. In other words Muslims were still on the top of the world during their decline. By the way, it is just not Muslims, every civilization boasts its past. Soviet Union took 10 years to collapse. America will not last another 10 years as a single unit. Persian Empire collpased in 4 years. Romans lost their power hardly in few years. Only Islamic Civilization took 600 years to collapse. Most surprisingly, Romans, Persians, Soveits, Spanish and now Americans have lost their superpower status to Muslim resistance. This is amazing.

Abid: I have restrictions on the topics and form of the answers I can give. If you can find an on topic board and post your question there I may be able to help you. If not, maybe someone on the board can.

To Romesh: There is not just one reason why people want to immigrate to the west.. there could be so many different reasons, it's impossible for me to explain them all. But the most common reason could be because of better job or business opportunities. And certainly the prime reason is not for going after the decadent lifestyle. Majority of the musilms immigrant to the west do still care about the afterlife and are very religious even though the are living in the middle of an unislamic society. I didn't mean that it's all bad in the west, that you cannot totally live there. It's still liveable for the muslims (in fact the whole corner of the world is liveable for the muslims) with a condition that you have a strong IMAN (faith). Not that I am saying all muslims are saints (before you manipulate my words), there could be isolated cases where you find people who claimed themselves to be muslim but are not practicing Islam. Then, remember not to judge Islam by the muslims (while Islam is perfect, the muslims are not).

You are wrong to say "If given the chance, just about all muslims will migrate to the west", (see here you are being an accusive Mr FaultFinder and making a quick judgement on the muslims again) because doesn't matter how many musilms have migrated to the west, there are still far more muslims who doesn't want to leave their homeland for the west. In fact there are many muslims and new muslims who move away from the west and live in Islamic countries for the Islamic environment. If you want prove you can read this article :

On a last note Mr. Romesh, i think it would be better of you if you become a Mr. TruthSeeker instead.

Note to Al-Bruni:
You write "Well, for the past 500 yrs, the west may have been in the lead in scientific field however they are so spritually sick and morally bankrupt that their advancement fails to give them true happiness in their lives (as evident by the rise in suicide rate in Japan and America the two most advanced countries) and in some cases it only causes so many destruction to the world in general.".

Yes, true. But then explain to me. Why do so many muslims want to immigrate to the decadent, spirtually sick and morally bankrupt western (i.e. scientifically advanced) countries?
If given the chance, just about all muslims will migrate to the west.

May be most people don't care about AFTERLIFE and want some enjoyment of decadence, moral bankruptcy and spirtual sickness in THIS life!!.

Assalamualaikum wrbt and greetings to all.

I am yet to see Romesh reply. Why ? Has he got nothing left to say ?

There were times when I thought that behinds the lines written by e mail writers like him, you can see his trend of thought or his level of intellectualism. What a question he asked when he demanded to know what is the contribution of Muslim over the past 500 years ?

I am a professional belonging to a certain discipline and I know when a question is intellectually stimulating or when it comes from a mere elementary attitude.

Note to Kris in Malaysia:

What you describe in your post regarding scientific contributions of muslims was more than 500 years ago. You write about Koran -- well that was over 1200 years ago, was'nt it.

So, my charge that muslims have contributed little or nothing towards science over the last 500 years or so still stands.

As usual, I find that muslims don't like to do any kind of self-analysis. They love living in the past, not in the present or future.

To Romesh: For your kind information, it is not the character of a true muslim to brag because it is against the teaching of Islam to brag or have pride in one's heart (the Prophet SAW says: "Anyone who possesses a speck of pride (kibr) in his heart will not enter Paradise"), thus when muslims talk about the past glory of Islam in science, it is not to brag and dwell in the past but as a lesson and reminder to the current generation that it is possible to be spiritual and advanced in science at the same time. Everybody knows and it's common sense that one cannot live in the past but we still need the past to be our teacher in order live rightfully in the present and the future. The way you write your comment sounds like that you belittling the muslims, but instead you are only belittling yourself as it shows your hatred, your prejudgement and bad conclusion for the muslims.

Well, for the past 500 yrs, the west may have been in the lead in scientific field however they are so spritually sick and morally bankrupt that their advancement fails to give them true happiness in their lives (as evident by the rise in suicide rate in Japan and America the two most advanced countries) and in some cases it only causes so many destruction to the world in general. In islam it is utterly useless to be scientifically advanced but without having regards for the afterlife. So articles like this serves as reminder that it is possible for muslims to make a difference to the ailing world by moving on forward and excel in science and be God-fearing at the same time.. given the right environment that is. Because, here I would like you know about reliable reports from Iraq in which a total 530 Iraqi scientist have been killed by Mossad with the collobration from American occupation forces... now how's that for being so modern by your beloved so-called advanced master.

Note to Nizam:

If you read my post carefully, I was talking about Islam and muslim scientists, not about a particluar country (like Egypt or Pakistan). So, in your note, you should not be talking about Indians; rather you should be talking about Hindus and their scientific contributions.

Yes, there have been very few scientific contributions by Hindus over the last 500 years. But, at least Hindus don't brag about their 2000 thousand nyears old contributions; however, muslims always brag about their 1400 year old contributions.

Now, one must look at the facts. India (i.e. Hindus) was ruled by Islam for over 1000 years. During that period, muslims did not contribute anything; and neither COULD Hindus (they never were the bosses).

Yes, Muslim and Hindus have fantastic brains. And that is shown by their recent contributions when they study in the west and move to the west.

Unfortunately, the situation was such that muslims, in muslim countries, could not use their brains for scientific contributions for the last 500 years. Why not?

Instead of muslims bragging about their past glory, they should start looking at the reasons for their non-contributions for over 500 years. You cannot live in the past; only in the present and the future.

I find, no effort is being expended for doing any kind of analysis by anybody in the islamic countries. Yes, Saudi Arabia is a very scientifically developed country. But, then, all of that was purchased from the west; nothing was invented there.

Assalalaikum wrbt and greetings to all.

Romesh wrote " What is the contribution of Muslim scientist over the past 500 years " ?

Well, the Algebra still form the integral part of Mathematics, there is no way of studying mathematics without it, the knowledge of astronomy is still relevant to the technologies of today, the compass was discovered by Muslims and is now being used by Romesh western associates to set sail their ships to annihilate Muslim countries, and it was the Muslims surgeons that performed the first medical surgery when the whole Dark Europe still using the most spartan methods in teatment of ailing and sick people.

It was and is , in the Quran that describes the biological growth of life, from the sperm, then developed into flesh, then clothed with bones and blood, and finally in which ALLAH breathe life into the soul of humans. This discovery of the development of the life in the womb was only discovered less than 200 years ago by your ( Romesh ) much pride western scientist but it was and has always been in the Quran ever since Islam was perfected ( note that I don't use the word revealed, because all faith brought by the previous Messengers of GOD are also Islam ).

It's just that your arrogance and ignorance, above all, in not being able to see these.

It was and has always been in the Quran which explains that the origin of this world was initially from gaseous matters, ( the term the Quran use is Dhukan ) and you and your western scientist only discovered these less than 500 years ago. But those Muslims who read the Quran and understands it's meaning will know and appreciate the truth.

Please Romesh, do not make a fool out of yourselves. Save your pride, before it's too late.

Dear Ramesh..I agree with you that not many muslim scientists from Islamic countries have contributed to science. This is a shame for the muslims and islamic countries. Also, not many indian scientist contributed in science like western countries did in last 500 years, but there are many indians around the globe, specially in western country have shown their talents by innovative research and bagged nobel prize. So far only one hindu scientist (C.V. Raman) have received nobel back in 1930 who carried out his entire research in his home soil! What does it show? It shows that not that indians are talented enough, rather opportunity is not there! Well, our position as indian is not very far from that of middle east! All in all, East did not contribute much in science in last 500 yrs, unfortunately, most of the muslims countries are in that part!

Charles Jack, your comments are profound. In case you see this post, can you please email me at [email protected]

I have some questions that you may help me with.


As it mentioned in article, when we're talking about Muslim mathematicians and scientists, their origin or their native language is not so important, but it could be mentioned that al-Khwarizmi was from Iran instead of saying: "he came from Central Asia." It seems that somehow the emphasis is on the 'Arabic' instead on "Islamic".
We could see that in other parts of this article; e.g., in the way that Nasir ad-Din Tusi's problem in Arabic (another Iranian Muslim scientist who has some articles and books in Arabic, like many other Muslim scientists from non-Arabic countries) has been introduced.
Anyway, we proud of all Muslim scientists...!

There is no God but the creator and nothing else worthy of worship, you shall love your neighbor as yourself and you shall garden God's garden. Such extrapolations from secondary revelation as "the world is flat" have been used by the west to weed out those that would follow truth. The Spanish Inquisition ferreting out any that held views contrary to their "truth" and the expulsion of the learned Muslims and Jews was in all likelihood triggered by the knowledge injected into the west challenging the official paradigm. The inquisition, torturing and murdering those that followed truth instead to the official dogma is not unlike the US rounding up, torturing and murdering those that would not follow their official version of "truth, and law" in Iraq, Lebanon and Palestine the military, cia and such playing the role of the inquisition with secret chambers of torture and inquiry.
What little schism there existed between the religious and scientific people in the west can be celebrated for we have seen the damage they can create when people like Bush use religion to justify the use of science (engineering the engines of war) to impose their twisted views on others. We can hope that the Muslims would not repeat the evils imposed on them on others as the Jews have repeated the evils imposed on them on their neighbors. If that were to occur it would be necessary to create a schism between the learned and the religious and that would be a loss for everybody.
Knowledge, can be a dangerous thing, and if it is not used to worship God, to love ones neighbor as oneself or to garden God's Garden then a veil will be imposed between those that are knowledgeable and those that do not qualify. And that is easy for God. 17:45-46

One can ask, in what universe created by The Creator would the symbols for "one" and "two" not have the relationship "one plus one equals two". Once you understand this you might note that The Creator, The Just, would not create a universe in which one plus one equals three. There are universal truths that the Creator follows and if we are wise, we study these and better understand The Creator and creation. (3:189-191) Indeed there are problems of optimization that are important to us today. While we might wish that there is a ridge that we can walk to ever higher levels, in most cases to break out of a local optimum one must be willing to travel the valleys from one local optimum to another. To complain about this would be to ask God to make one plus one equal three. The world could be made with strict adherence and those that break the laws punished immediately but God has willed mercy as a constraint (6:8).
Still it is instructive to study such universal truths as mathematics, the mother of language, teaches and in doing so better understand the guides The Creator follows. One view of this is to note that the Greek word for "word" as in "God only needs to say a word and a thing exists" also means "dialog" as in the Socratic method of teaching.
Primary revelation, creation, is open to all and if it is pointed out to you (41:53) your understanding of this dialog from The Creator to you may be no deeper than your understanding of creation. A lack of understanding has created some strange extrapolations from secondary revelation like "the world is flat", "the world is in at the center of the universe" and "the heavenly bodies are fixed on crystalline spheres", hence metaphoric use of the "seven heavens". This can even restrict revelation to use these falsehoods as the revelation must be in terms understood by the subjects of the revelation. It would be no use to inspire Mohammad in concepts incomprehensible.

Great. Now can anybody tell me what is the contribution of ISLAM/Muslims to Mathematics, Chemistry, Physics, Astronomy, Medicine, Semiconductors, Metallurgy over the last 500 years or so? I cannot find any. Scientific journals are devoid of any contributions from muslim countries.

If there has been any contribution by muslims, it was muslims educated in non-muslim countries and research work done by them in non-muslim countries.

It is foolhardy to live in the past glory; the world has changed. Even the weapons (and vehicles) used to fight against Russia (in Afghan war) and now US in Iraq are made in non-muslim countries.