Algebra Know-It-ALL

Roman Numerals

The toothpick numeration system just described bears a resemblance to another system that

was actually used in much of the world until a few centuries ago: the Roman numeration

system, more often called Roman numerals.

Basic symbols

In Roman numerals, a quantity of one is represented by a capital letter I. A quantity of five

is represented by a capital V. A quantity of ten is denoted as a capital X, fifty is a capital L, a

hundred is a capital C, five hundred is a capital D, and a thousand is usually represented by a

capital M. (Sometimes K is used instead.)

So far, this looks like a refinement of the toothpick numeration scheme. But there are

some subtle differences. You don’t always write the symbols in straightforward order from left

to right, as you lay down the sticks in the toothpick system. There are exceptions, intended

to save symbols.

Arranging the symbols

The people who designed the Roman system did not like to put down more than three identi-

cal symbols in a row. Instead of putting four identical symbols one after another, the writer

would jump up to the next higher symbol and then put the next lower one to its left, indicat-

ing that the smaller quantity should be taken away from the larger.

For example, instead of IIII (four ones) to represent four, you would write IV (five with

one taken away). Instead of XXXX (four tens) to represent forty, you’d write XL (fifty with

ten taken away). Instead of MDXXXX to represent one thousand nine hundred, you’d write

MCM (a thousand and then another thousand with a hundred taken away).

What about zero?

By now you must be thinking, “No wonder people got away from Roman numerals, let alone

hash marks. They’re confusing!” But that’s not the only trouble with the Roman numeral

system or the toothpick numeral system we made up earlier. There’s a more serious issue.

Neither of these schemes give you any way to express the quantity zero. This might not seem

important at first thought. Why make a big fuss over a symbol that represents nothing?

Sometimes the best way to see why something is important is to try to get along with-

out it. When you start adding and subtracting, and especially when you start multiplying

6 Counting Methods

Figure 1- 3 Two different ways of expressing seven hundred

seventy-seven in toothpick numerals. The top

method is preferred because it is more “elegant.”