![]() We, of course, now say division by zero is not defined. Unfortunately he defines zero divided by zero as zero and gives a term for a number divided by zero without saying what the result would be. He defines zero as that which results from subtracting a number from itself and gives the correct rules for addition, subtraction and multiplication with zero. In chapter eighteen he deals with negative numbers and with zero, not as a placeholder but as a number. Chapter 12 is devoted to arithmetic and introduces the basic arithmetical operations. Although an astronomical work Brahmagupta devotes several chapters to mathematics. Writing scientific works in verse in ancient cultures was probably in order to make them easier to memorise in predominantly oral societies. He wrote his Brāhma-sphuṭa-siddhānta a treatise on astronomy written in verse, with 24 chapters and 1008 verses, in 628 CE. There are three principle figures, who played a central role in the transmission of the place value decimal number system and the first of these is the Indian astronomer Brahmagupta (c.598–c.668 CE), who lived most of his life in Bhillamala (modern Bhinmal) in North-western India. The Chinese also had a place value decimal number system but whether the Chinese influenced the Indians, the Indians the Chinese or both developed their systems independently is also not known. This poses the question whether the Indians got the idea of a place value system from the Babylonians but it is simply not known. The Babylonian system even had a placeholder zero in its later versions. The Babylonians developed a place value number system as early as the beginning of the second millennium BCE but it was a sexagesimal or base sixty number system rather than a decimal base ten one. It also throws up some important and unanswered questions. ![]() ![]() The history of the early development of the place value decimal number system is long, complicated and full of holes and I shan’t be dealing with it here. In what follows I shall sketch the path that this number system took from India to medieval Europe, a path that has several twist and turns. The numerals are usually referred to as Arabic numerals or more correctly as Hindu-Arabic numerals because we Europeans inherited them and the entire system of how to use them from the Islamic Empire in the High Middle Ages, which in turn had inherited them from India in the Early Middle Ages, where they originated. This is an incredibly powerful and efficient method of writing numbers and the algorithms that it uses also make it a very efficient system for conducting calculations. The value of the symbol changes according to its position – place – within the number that we write. It consists of just ten symbols (numerals) – 1,2,3,4,5,6,7,8,9,0 – with which we can express any number, of any size that we may require. In every day life we all do our calculations, whether for the taxman, our purchases, paying the household bills or in some academic discipline, using the place value decimal number system.
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