3. Binary number system

Base two has several other names, including the binary positional numeration system and the dyadic system. Many civilizations have used the binary system in some form, including inhabitants of Australia, Polynesia, South America, and Africa. Ancient Egyptian arithmetic depended on the binary system. Records of Chinese mathematics trace the binary system back to the fifth century and possibly earlier. The Chinese were probably the first to appreciate the simplicity of noting integers as sums of powers of 2, with each coefficient being 0 or 1. For example, the number 10 would be written as 1010:

10= 1 x 23 + 0 x 22 + 1 x 21 + 0 x 20

Users of the binary system face something of a trade-off. The two-digit system has a basic purity that makes it suitable for solving problems of modern technology. However, the process of writing out binary numbers and using them in mathematical computation is long and cumbersome, making it impractical to use binary numbers for everyday calculations. There are no shortcuts for converting a number from the commonly used denary scale (base ten) to the binary scale.

Over the years, several prominent mathematicians have recognized the potential of the binary system. Francis Bacon (1561-1626) invented a "bilateral alphabet code," a binary system that used the symbols A and B rather than 0 and 1. In his philosophical work, The Advancement of Learning , Bacon used his binary system to develop ciphers and codes. These studies laid the foundation for what was to become word processing in the late twentieth century. The American Standard Code for Information Interchange (ASCII), adopted in 1966, accomplishes the same purpose as Bacon's alphabet code. Bacon's discoveries were all the more remarkable because at the time Bacon was writing, Europeans had no information about the Chinese work on binary systems.

A German mathematician, Gottfried Wilhelm von Leibniz (1646-1716), learned of the binary system from Jesuit missionaries who had lived in China. Leibniz was quick to recognize the advantages of the binary system over the denary system, but he is also well known for his attempts to transfer binary thinking to theology. He speculated that the creation of the universe may have been based on a binary scale, where "God, represented by the number 1, created the Universe out of nothing, represented by 0." This widely quoted analogy rests on an error, in that it is not strictly correct to equate nothing with zero.

The English mathematician and logician George Boole (1815-1864) developed a system of Boolean logic that could be used to analyze any statement that could be broken down into binary form (for example, true/false, yes/no, male/female). Boole's work was ignored by mathematicians for 50 years, until a graduate student at the Massachusetts Institute of Technology realized that Boolean algebra could be applied to problems of electronic circuits. Boolean logic is one of the building blocks of computer science, and computer users apply binary principles every time they conduct an electronic search.

The binary system works well for computers because the mechanical and electronic relays recognize only two states of operation, such as on/off or closed/open.

Operational characters 1 and 0 stand for 1 = on = closed circuit = true 0 = off = open circuit = false

The telegraph system, which relies on binary code, demonstrates the ease with which binary numbers can be translated into electrical impulses. The binary system works well with electronic machines and can also aid in encrypting messages. Calculating machines using base two convert decimal numbers to binary form, then take the process back again, from binary to decimal.

The binary system, once dismissed as primitive, is thus central to the development of computer science and many forms of electronics. Many important tools of communication, including the typewriter, cathode ray tube, telegraph, and transistor, could not have been developed without the work of Bacon and Boole. Contemporary applications of binary numerals include statistical investigations and probability studies. Mathematicians and everyday citizens use the binary system to explain strategy, prove mathematical theorems, and solve puzzles.