Numbers: Their Tales, Types, and Treasures.

Chapter 1: Numbers and Counting

1.12.THE NUMERAL SYSTEM IN WRITING

Once culture developed among early settlers, the society became more complex, goods were produced, and division of labor began. Yet resources were unequally distributed, and therefore trade and exchange between communities became necessary. The need arose to communicate what could be offered in what quantity or how much of a certain good was demanded in exchange. Notched tally sticks or bags with pebbles and bones to quantify number soon became impractical and number words came into use. With the development of writing, number symbols were also invented, and the spoken numeral system was translated into a written form.

We have seen that a numeral system is obtained by hierarchical grouping. In the example of the previous section, every item was represented by a pebble, every 10 pebbles by a stick, every 10 sticks by a bone. This is the foundation of a base-10 numeral system. From here it is still a long way to a symbolic representation of numbers in writing.

A systematic method to write arbitrary numbers typically uses arrangements of basic symbols that we call digits. We are used to the ten digits—0, 1, 2, 3, 4, 5, 6, 7, 8, 9—that also serve as symbols for the first natural numbers. In order to express larger numbers, we use combinations of the basic symbols. This can be done in quite different ways. Let us see how this problem was resolved in our own culture. (In chapter 3 we will describe some other historically interesting methods of symbolizing numbers.)

Currently, we still use two different systems of writing numbers. One is the system familiar from everyday use and the other, although rarely seen, is the Roman system. In spoken language, the Roman numerals are not fundamentally different from the numerals in English. The number forty-nine would be quadrāgintā novem in Latin, which would literally translate into forty nine. In writing, however, these two systems take completely different approaches to symbolizing numbers—just compare 49 with its equivalent XLIX in the Roman system! It is worthwhile to take some time and describe these different methods in more detail.

Consider, for example, the numeral 1776. It is written by putting the digits one-seven-seven-six in a row. It is a very ingenious notational trick that allows us to write a relatively large number in such a compact form, using only a few symbols. We all know immediately that 1776 actually means one thousand seven hundred seventy-six, or

1 × thousand + 7 × hundred + 7 × ten + 6 × one,

or in the language of the shepherd from the previous section: one stone, seven bones, seven sticks, and six pebbles.

In our way of symbolizing numbers, every digit in 1776 has a meaning that is given not only by its numerical value but also by the place where it appears. The digit 7 even appears twice, but each time its meaning is quite different. Reading the number from left to right, the first 7 is seven hundreds, and the second 7 means seven tens. The actual value of every digit depends on its position. The rightmost digit always counts the ones, the next counts the tens, and so on. Each digit contributes with a certain value to the final numerical meaning of 1776. Because the value of a digit depends on the place where it is written, our numeral system is called a place-value system. The most important consequence of the place-value system is that we do not need special symbols for ten, hundred, thousand, and so on.

In order to write a number like

1 × thousand + 7 × hundred + 6 × one,

we need a special symbol that denotes the absence of a position. We cannot simply omit the place describing the tens, because 176 would be something completely different. And leaving a gap, as in 17 6, is bound to create confusion. Therefore, using the symbol 0 as a place holder, we write 1706 to indicate that there are no tens. Without that symbol, it would be very difficult to distinguish between 176, 1076, 1706, and 1760.

In our place-value system, the numerical value of a numeral is determined by two operations:

1. multiplication of every digit with its place-value (one, ten, hundred,…)

2. addition of the results from step 1.

It is quite different from the ancient Roman system, which uses addition (and sometimes subtraction). In the Roman system, each power of ten has a separate symbol: 10 is written as X, 100 is written as C, 1000 is written as M. With additional symbols for 5=V, 50=L, and 500=D, the base-10 numeral 1776 would be written as

Therefore, in order to find the numerical value of a Roman numeral, we just have to perform addition, no multiplication. There are some exceptions to this rule: As a shortcut, one writes IV (V – I = 5 – 1 = 4) instead of IIII, and similarly in other cases, where the symbol with a smaller value is written first to indicate subtraction instead of addition: IX = X – I = 9, and so on. Note that we need no placeholder symbol to write a Roman numeral; 1706 would simply be MDCCVI.

Today, we still encounter Roman numerals occasionally; for example when denoting the year of construction on the cornerstone of a building, or sometimes the year of production at the end of a movie.