# Why learning numbers was so hard in medieval Europe

In his article “Old-Fashioning versus Newfangled: Reading and Writing Numbers, 1200-1500,” math historian John Crossley explains that even by the end of the Middle Ages many writers had a lot of difficulty understanding how numbers worked, and preferred using the older system of Roman numerals.

When one was using Roman numerals they would know that the various characters had a fixed amount. If they saw a V it would be five, X would be ten, and M that would mean one thousand. Crossley writes:

with minor exceptions, Roman numerals do not change their meanings when they change their place. On the other hand Hindu-Arabic numerals do change their meaning when they change their place. Consider this question, what does ‘3’ mean? When we encounter 3 in 437 or in 3,145,872, it means two different things. It is not “just a 3!” In the first it means “thirty,” in the second “three million.”

A more extreme example is provided by the occurrences of 3 in 1,234,537, where it has two different meanings! This illustrates the distinctive feature of the use of Hindu-Arabic in representing numbers: their place notation. This is independent of the form of the numerals 0,1,…9, since, on the one hand, other symbols could be used instead of these digits and, on the other, a different place notation could be used.

Thus our system writes the numeral beginning with the largest number first: “123” means “one hundred and twenty three.” Ironically, the smallest number comes first in written Arabic because the direction of Arabic writing is opposed to the Hindu orientation, which has been retained in the numerals.

This concept of place notation proved to be very difficult for medieval Europeans to understand, especially with how they traditionally calculated sums. Combined with the fact that the symbols for numbers were also brand new for Europeans, it is not surprising that the process of changing over to the new system was slow.

Crossley examined 1398 manuscripts created between the years 1200 and 1500 to see how much use of the Hindu-Arabic numerals, and found that throughout this period Roman numerals were still largely preferred. For the 13th century, only 7% of manuscripts had the new numbers, rising to 17% for the 14th century and 47% for the 15th century. He also found that in many instances where writers were mixing the two systems, sometimes within the same number – for example, one sometimes found M (for 1000) followed by Arabic numerals.

The impetus for changing to the Hindu-Arabic numbers in medieval Europe seems to have come from businessmen. Crossley writes

There was also a clear distinction in the domains in which the two kinds of numerals were used. Roman numerals were used in academia where universities taught about abstract properties: square numbers, triangular numbers, etc. Hindu-Arabic numerals were used for the practical world of commerce. This occurred in special, so-called abacus schools where merchants and their employees were taught the new Hindu-Arabic numerals. Such schools were prevalent in Italy. Since they were intimately involved with sometimes quite complicated calculations, the commercial used ultimately led to the development of algebra. It was not until the sixteenth century that the two domains came together. At that time academia at last embraced the study of methods of calculation, in particular algebra, while retaining its theoretical concern with abstract properties of numbers.

The article “Old-fashioned versus newfangled: Reading and writing numbers, 1200-1500” appears in the journal Studies in Medieval and Renaissance History, Third Series, Volume 10 (2013). John Crossley is professor emeritus at Monash University in Australia – click here to visit his website.