History of Numbers 1c. I can distinguish between an additive and positional system, and convert between Roman and Hindu-Arabic numbers. Roman Numerals The numeric system represented by Roman numerals originated in ancient Rome (753 BC–476 AD) and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. By the 11th century, the more eficient Hindu–Arabic numerals had been introduced into Europe by way of Arab traders. Roman numerals, however, remained in commo use well into the 14th and 15th centuries, even in accounting and other business records (where the actual calculations would have been made using an abacus). Roman numerals are still used today, in certain contexts. See: Modern Uses of Roman Numerals Numbers in this system are represented by combinations of letters from the Latin alphabet. Roman numerals, as used today, are based on seven symbols:

The numbers 1 to 10 are expressed in Roman numerals as: I, II, III, IV, V, VI, VII, VIII, IX, X.

This an additive system. Numbers are formed by combining symbols and adding together their values. For example, III is three (three ones) and XIII is thirteen (a ten plus three ones). Because each symbol (I, V, X ...) has a ixed value rather than representing multiples of ten, one hundred and so on (according to the numeral's position) there is no need for “place holding” zeros, as in numbers like 207 or 1066. Using Roman numerals, those numbers are written as CCVII (two hundreds, plus a ive and two ones) and MLXVI (a thousand plus a ifty plus a ten, a ive and a one). Symbols are placed from left to right in order of value, starting with the largest. The value of the number is usually determined by just adding the symbols; however, in a few speciic cases, to avoid four characters being repeated in succession (such as IIII or XXXX), subtractive notation is used, as in this table:

In summary: • I placed before V or X indicates one less, so four is IV (one less than ive) and nine is IX (one less than ten) • X placed before L or C indicates ten less, so forty is XL (ten less than ifty) and ninety is XC (ten less than a hundred) • C placed before D or M indicates a hundred less, so four hundred is CD (a hundred less than ive hundred) and nine hundred is CM (a hundred less than a thousand) 1 PRACTICE. In Canvas, work through these two sets of online practice problems in sequence: 1. Read Roman numerals 2. Convert to Roman numerals

The Hindu-Arabic Number System Our own number system, composed of the ten symbols {0,1,2,3,4,5,6,7,8,9} is called the Hindu- Arabic system. It is a base-ten (decimal) system, because place values increase by powers of ten.

This is now a positional system, which means that the position of a symbol indicates the value of that symbol within the number. For example, the position of the symbol 3 in the number 435,681 gives it a value much greater than the symbol 8 in that same number: the 3 represents 30,000, and the 8 represents 80. And the overall value of the number is much greater than if we were to simply add up its digits. The development of these ten symbols and their use in a positional system comes to us primarily from India.

But it originated as an additive system. It was not until the ifteenth century (the 1400’s) that the symbols we are familiar with today irst took form in Europe. However, the history of these numbers and their development goes back hundreds of years. When we look for the origins of our Hindu-Arabic numbers, we have to go back around 2,200 years, to the third century BCE in India. This earliest form is called the Brahmi numerals. They had separate symbols for the numbers 1 through 9. But, like with the Roman numerals, there were also distinct symbols for 10, 100, 1000 … and also for 20, 30, 40 … and others for 200, 300, 400 … 900. Because there were all these symbols for higher value numbers, we know this was an additive number system.

The Brahmi symbols for 1 - 9 are shown below. These numerals were used for around 500 years, until the fourth century CE (the 300’s), with variations through time and geographic location. For example, in the irst century CE, one set of Brahmi numerals took on the following form:

From the fourth century on, you can actually trace several different paths in the evolution of the Brahmi numerals. One of those paths led to our current numeral system, and went through what are called the Gupta numerals. The Gupta numerals were prominent during a time ruled by the Gupta dynasty and were spread throughout that empire as they conquered lands during the fourth through sixth centuries (up until about 600 CE). They have the following form:

2 How the numbers got to their Gupta form is unclear. The numerals may have come from the initial letters of the names for the numbers. This is not uncommon: the Greek numerals developed in this manner. The Gupta numerals eventually evolved into another form of numerals called the Nagari numerals, and these continued to evolve until the eleventh century, at which time they looked like this:

Note that with the Nagari numerals, the symbol for a "zero" has appeared. By this point, only ten symbols were needed to build any number - they no longer had to use separate symbols for 10, 100, 1000… 20, 30, 40 … 200, 300, 400 etc. Because they had a symbol for zero and only ten symbols altogether, it seems these numerals were now part of a positional number system. In fact, the oldest dated Indian document which contains a number written in a positional system is from around 600 CE. So we know that by at least the year 600, the Hindu numerals were no longer additive and had become positional. At irst the zero only functioned as a placeholder. But by the eleventh century, zero as an idea was more fully developed (this will be covered later). The Mayans in the Americas had a symbol for zero as well, as we shall see later.

The Nagari numerals were adopted by the Arabs, most likely in the eighth century during Islamic incursions into the northern part of India. It is believed that the Arabs were instrumental in spreading them to other parts of the world, including Spain. The diagram below shows various forms of these Hindu-Arabic numerals as they evolved to become the numbers most of us use today:

For a short introduction to the development of the Hindu-Arabic numerals, watch the video in Canvas. 3