Binary Number System
The binary number system is a method of representing numbers using positional notation and powers of 2. The decimal number system is similar, but employs powers of 10 rather than of 2. Consider, for example, the decimal number 451. The numerals 4, 5, and 1 are each understood--because of their notational positions--to be multiplied by specific powers of 10; that is, 451 = (4 x 102) + (5 x 101) + (1 x 100). Numbers expressed in the binary number system are interpreted similarly, except that the numerals 0 and 1 are used (rather than 0 through 9) and powers of 2 are used rather than powers of 10. The value of the binary number 101 can therefore be read off as (1 x 22) + (0 x 21) + (1 x 20). Adding these three terms reveals that binary 101 equals decimal 5.
The first known description of binary numbers was provided in 1679 by the mathematician Gottfried Wilhelm von Leibniz, who also constructed one of the earliest mechanical calculators.
Leibniz wrote: "Despite its length [i.e., the lengthiness of numbers written out in binary rather than in decimal notation], the binary system, in other words counting with 0 and 1, is scientifically the most fundamental system, and leads to new discoveries. When numbers are reduced to 0 and 1, a beautiful order prevails everywhere." This "beautiful order" indeed prevails everywhere today, as all digital computers employ binary numbers. All data in a digital computer--not only numbers as such but programs, documents, images, and sounds--are stored as patterns of on/off device-states that may be interpreted as binary digits.
As Leibniz noted over three hundred years ago, the binary number system usually requires many more digit positions to represent a given number than the more familiar decimal system. However, the binary digits 0 and 1 can be conveniently assigned to the on/off conditions of certain electronic devices. Devices that flip between two states (0 and 1, off and on) rather than many (0-9, say) are not only simpler to build but are more resistant to electronic noise than many-state devices; analogously, a disk lying on a table (2 stable states, i.e., the disk has two sides) is less likely to be flipped over by a random blow than is a cube (6 stable states, i.e., six sides). Moreover, though a number like decimal 1,537 appears more compact than its binary equivalent 11000000001--and is, in the sense that it uses less ink on this page--computers do not run on ink; they store numbers in electronic devices. And since 10 device states are required per digit of a base 10 number (to represent the numerals 0 through 9), storing any 4-digit decimal number requires a device capable of taking on 10 x 10 x 10 x 10 = 104 = 10,000 states. To represent the binary equivalent of decimal 1,537--11000000001--however, a device capable of taking on only 211 = 2,048 states is required. A larger 4-digit decimal number would require more binary digits, and a small one fewer binary digits, but binary numbers allow the absolute minimum of hardware to be used in the representation of any given number.
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