History of the Binary System

The Binary System of numeration is the simplest of all positional number systems. The base - or the radix - of the binary system is 2, which means that only two digits - 0 and 1 - may appear in a binary representation of any number. The binary system is of great help in the Nim-like games:Plainim, Nimble, Turning Turtles, Scoring, Northcott's game, etc. More importantly, the binary system underlies modern technology of electronic digital computers. Computer memory comprises small elements that may only be in two states - off/on - that are associated with digits 0 and 1. Such an element is said to represent one bit - binary digit.

The first electronic computer - ENIAC which stood for Electronic Numerical Integrator And Calculator - was built in 1946 at the University of Pennsylvania, but the invention of the binary system dates almost 3 centuries back. Gottfried Wilhelm Leibniz (1646-1716), the co-inventor ofCalculus, published his invention in 1701 in the paper Essay d'une nouvelle science des nombresthat was submitted to the Paris Academy to mark his election to the Academy. However the actual discovery occurred more than 20 years earlier.

According to the Oxford Encyclopedic Dictionary (see Earliest Known Uses of Some of the Words of Mathematics), an entry BINARY ARITHMETIC first appeared in English in 1796 in A Mathematical and Philosophical Dictionary.

Binary numbers are written with only two symbols - 0 and 1. For example, a = 1101. Since symbols 0 and 1 are also a part of the decimal system and in fact of a positional system with any base, there's an ambiguity as to what 1101 actually stands for. To avoid confusion, the base is often written explicitly, like in a = (1101)₂ or b = (1101)₁₀. In the decimal system, 1101 is interpreted as1 thousand 1 hundred 1, which is just a sum of powers of 10 with coefficients that are the digits of the number. More accurately,

(1101)₁₀ = 1·10³ + 1·10² + 0·10 + 1

To represent numbers, the decimal system uses the powers of 10, whereas the binary system uses in a similar manner the powers of 2.

(1101)₂ = 1·2³ + 1·2² + 0·2 + 1

The numbers are different. In fact,

(1101)₂ = 8 + 4 + 1 = 13    ( = (13)₁₀.)

There are several problems with using more than one number system at the same time. Should we read (1101)₂ as 1 thousand 1 hundred 1 in binary? Or, after some mental calculations, just 13 without mentioning the base? The latter possibility is overtaxing and unreasonable: why to use a system other than the decimal in writing while depending on the decimal in speech? The former is inappropriate altogether for etymological reasons. We might say thousand to indicate a 1 in the fourth position from the right regardless of the base of the system in use, but this would conflict with the etymology of the word thousand, and the same is true of the word hundred. Both are related to the base 10 and no other.

In Words of Mathematics we find the following entries:

hundred (numeral): a native English compound. The first element, hund, actually means "ten." It comes from dekt-tom, an extension of the more basic Indo-European root dekm "ten." The second element is from the Old English rad "number", so that hundred means literally the "tens-number" in the sense that it is ten times ten.

and

thousand (numeral): actually an English compound, thus-hund. The first component is related to English thumb and thigh, and means "swollen, large." The Indo-European root is teu- "to swell." Related borrowings from Latin are tumor and tumulus. The second component is the root found in hundred (q.v.), which is based on the Indo-European root dekm-"ten." The literal meaning of thousand is "a swollen or big hundred" because it is ten times a hundred.