![]()
Part I
Techniques and Code
![]()
Chapter 1
Computer Number Systems
Chapter Summary
One of the fundamental applications of a computer system is the processing and storing of numeric data, sometimes called “number crunching.” In order to perform more efficient digital operations on numeric data, mathematicians have devised systems and structures that differ from those used traditionally. This chapter presents the background material necessary for understanding and using the number systems and numeric data storage structures employed in digital computers. The material includes low- and high-level integer conversion routines.
1.0 Counting
The origin and the fundamental application of a number system is counting. Imagine a primitive hunter using his or her fingers to show other members of the tribe how many mammoth were spotted during a scouting trip. By this simple scheme the hunter is able to transmit a unique type of information. The information the hunter wishes to convey does not relate to the species, size, or color of the animals, but to their numbers. This is possible because our minds are able to isolate the notion of “oneness” from other properties of objects.
The most primitive method of counting consists of using a common object to represent the degrees of oneness. The hunter uses one finger to represent each mammoth. Alternatively, the hunter could have resorted to pebbles, sticks, lines on the ground, or scratches on the cave wall to show how many units there were of the object, independently of any other characteristic or attribute.
1.0.1 The Tally System
The first number system probably started from notches on a stick or scratches on a cave wall. In the simplest form, sometimes called a tally system, each scratch, notch, or line represents an object. The method is so simple and intuitive that we continue resorting to it, even after mastering the intricacies of the calculi. Tallying is based on a one-to-one correspondence between objects and their representation. The hunter counts the mammoth hiding in the ravine by making notches on a stick, at the rate of one notch for each mammoth. In order to share this information with other members of the tribe, the hunter can later transfer the notches on a stick to lines or scratches on the cave wall. Tallying requires no knowledge of quantity and no elaborate symbols. Had there been 12 mammoth in the ravine the cave wall would have appeared as follows:
The next step in the evolution of the tally system probably consisted of grouping the marks to help visualize the number of objects. Since we have five fingers on each hand, the 12 mammoth may have been grouped as follows:
A primitive mathematical genius added one final sophistication to the tally system. By drawing one tally line diagonally the visualization is further improved, as in this familiar style:
1.0.2 Roman Numerals
By observing the early Roman numerals we can see how a simple graphical tally evolved into a symbolic numeric representation. The first five digits were encoded with the symbols
The Roman symbol V is conceivably a simplification of the tally encoding using a diagonal line to complete the grouping. Table 1.1 shows the decimal value of the Roman symbols.
Table 1.1
Symbols in the Roman Numeration System
ROMAN | DECIMAL |
I | 1 |
V | 5 |
X | 10 |
L | 50 |
C | 100 |
D | 500 |
M | 1000 |
The Roman numeral system is based on an add-subtract rule whereby the elements of a number, read left-to-right, are interpreted as follows:
1. If the value of the numeral to the right is equal to or larger than the numeral to its left, its value is added to the previous total.
2. If the value of the numeral to the right is larger, its value is subtracted from the previous total. For example:
The decimal 1994 is represented in Roman numerals as follows:
The uncertainty in the positional value of each digit, the absence of a symbol for zero, and the fact that some numbers require either one or two symbols (I, IV, V, IX, and X) complicates the rules of arithmetic using Roman numerals.
1.1 The Origins of Our Number System
The one element of our civilization which has transcended all cultural and social differences, is a system of numbers. While mankind is yet to agree on the most desirable political order, on generally acceptable rules of moral behavior, or on a universal language, the Hindu-Arabic numerals have been adopted by practically all the nations and cultures of the world. There must be something extraordinary about this number system for it to achieve such general acceptance.
By the year 800 A.D. the Arabs were using a ten-symbol positional system of numbers which included the special symbol for 0. This system (later called the Hindu-Arabic numerals) was introduced into Europe during the 8th century, probably through Spain. Pope Sylvester II, who had studied the Hindu-Arabic numbers in Spain, was the first European scholar known to adopt and teach them. The Latin title of the first book on the subject of “Indian numbers” is Liber Algorismi de Numero Indorum. The author is the Arab mathematician al-Khuwarizmi.
In spite of the evident advantages of this number system its adoption in Europe took place only after considerable debate and ...
