1. Ancient Arab/Indian Mathematics The regions from which the "Arab mathematicians" came was centered on Iran/Iraq but varied with military conquest during the period. At its greatest extent it stretched to the west through Turkey and North Africa to include most of Spain, and to the east as far as the borders of China. We are considering a time frame of about 100 AD to 1200 AD. The regions from which the “Indian Mathematics” are pretty much what is modern day India. The Ancient Indian Civilization begin somewhere around 1000 BC.

2. Ancient Arab/Indian Mathematics The background to the mathematical developments which began in Baghdad around 800 is not well understood. Certainly there was an important influence which came from the Hindu mathematicians of Ancient India whose earlier development of the decimal system and numerals was important. This period begins under the Caliph Harun al-Rashid, the fifth Caliph of the Abbasid dynasty, whose reign began in 786. He encouraged scholarship and the first translations of Greek texts into Arabic, such as Euclid's Elements by al-Hajjaj, were made during al-Rashid's reign.Euclid The next Caliph, al-Ma'mun, encouraged learning even more strongly than his father al-Rashid, and he set up the Bayt al- Hikma or “House of Wisdom” in Baghdad which became the center for both the work of translating and of research. Al-Kindi (born 801) and the three Banu Musa brothers worked there, as did the famous translator Hunayn ibn Ishaq.Bayt al- Hikma Al-KindiBanu Musa brothersHunayn ibn Ishaq Caliph Harun al-Rashid, 763 – 809

3. Al'Khwarizmi was an Islamic mathematician who wrote on Hindu-Arabic numerals and was among the first to use zero as a place holder in positional base notation. The word algorithm derives from his name. His algebra treatise Hisab al-jabr w'al- muqabala gives us the word algebra and can be considered as the first book to be written on algebra. Hunayn ibn Ishaq was an Islamic mathematician who is most important as a translator, making Greek works available to the Islamic mathematicians. 808 - 873 about 790 - about 850 Ancient Arab/Indian Mathematics

4. The first sign that the Indian numerals were moving west comes from a source which predates the rise of the Arab nations. In 662 AD Severus Sebokht, a Nestorian bishop who lived in Keneshra on the Euphrates river, wrote:- Ancient Arab/Indian Mathematics I will omit all discussion of the science of the Indians,..., of their subtle discoveries in astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians, and of their valuable methods of calculation which surpass description. I wish only to say that this computation is done by means of nine signs.

5. 969 A.D. 1082 A.D. 1256 A.D. We can see that these numbers are very similar to what we use today. Ancient Arab/Indian Mathematics

6. One of the commonest questions which the readers of this archive ask is: Who discovered zero? The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as an empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name. (Our name "zero" derives ultimately from the Arabic sifr which also gives us the word "cipher".) Ancient Arab/Indian Mathematics

7. One might think that once a place-value number system came into existence then the 0 as an empty place indicator is a necessary idea, yet the Babylonians had a place- value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. Now there were exceptions to what we have just stated. The exceptions were the mathematicians who were involved in recording astronomical data. Here we find the first use of the symbol which we recognize today as the notation for zero, for Greek astronomers began to use the symbol O. The scene now moves to India where it is fair to say the numerals and number system was born which have evolved into the highly sophisticated ones we use today. Of course that is not to say that the Indian system did not owe something to earlier systems and many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers Ancient Arab/Indian Mathematics

8. What is certain is that by around 650AD the use of zero as a number came into Indian mathematics. The Indians also used a place-value system and zero was used to denote an empty place. We now come to considering the first appearance of zero as a number. Let us first note that it is not in any sense a natural candidate for a number. From early times numbers are words which refer to collections of objects. Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects. Ancient Arab/Indian Mathematics

9. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases. Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. Many believe this is because the Greek mathematical achievements were based on geometry. Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division. In three important books the Indian mathematicians Brahmagupta, Mahavira and Bhaskara tried to answer these questions.BrahmaguptaMahaviraBhaskara

10. Ancient Arab/Indian Mathematics BrahmaguptaBrahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century. He explained that given a number then if you subtract it from itself you obtain zero. He gave the following rules for addition which involve zero:- BrahmaguptaBrahmagupta then says that any number when multiplied by zero is zero but struggles when it comes to division:- 598 - 670 Brahmagupta was the foremost Indian mathematician of his time. He made advances in astronomy and most importantly in number systems including algorithms for square roots and the solution of quadratic equations. The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.

11. Ancient Arab/Indian Mathematics Mahavira was an Indian mathematician who extended the mathematics of Brahmagupta. about 800 - about 870 about 1114 – about 1185 Bhaskara II or Bhaskaracharya was an Indian mathematician and astronomer who extended Brahmagupta's work on number systems A number remains unchanged when divided by zero. BhaskaraBhaskara wrote over 500 years after Brahmagupta. Despite the passage of time he is still struggling to explain division by zero. He writes:-Brahmagupta A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.

12. Ancient Arab/Indian Mathematics A type of Indian Math called Jaina mathematics recognized five different types of infinity [2]:-2... infinite in one direction, infinite in two directions, infinite in area, infinite everywhere and perpetually infinite. 8 Modern mathematics claims that any number divided by 0 is undefined 2 588 = 1013 065324 433836 171511 818326 096474 890383 898005 918563 696288 002277 756507 034036 354527 929615 978746 851512 277392 062160 962106 733983 191180 520452 956027 069051 297354 415786 421338 721071 661056 The Jaina cosmology contained a time period of 2 588 years. Note that 2 588 is a very large number!

13. Of course there are still signs of the problems caused by zero. Recently many people throughout the world celebrated the new millennium on 1 January 2000. Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified. Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the third millennium and the 21 st century begin on 1 January 2001. Zero is still causing problems! Ancient Arab/Indian Mathematics