8CHAPTER 1. NUMERATION SYSTEMSNow, why would anyone use binary numeration? The decimal system, with its ten ciphers, makesa lot of sense, being that we have ten ﬁngers on which to count between our two hands. (It is inter-esting that some ancient central American cultures used numeration systems with a base of twenty.Presumably, they used both ﬁngers and toes to count!!). But the primary reason that the binarynumeration system is used in modern electronic computers is because of the ease of representing twocipher states (0 and 1) electronically. With relatively simple circuitry, we can perform mathematicaloperations on binary numbers by representing each bit of the numbers by a circuit which is either on(current) or oﬀ (no current). Just like the abacus with each rod representing another decimal digit,we simply add more circuits to give us more bits to symbolize larger numbers. Binary numerationalso lends itself well to the storage and retrieval of numerical information: on magnetic tape (spotsof iron oxide on the tape either being magnetized for a binary ”1” or demagnetized for a binary ”0”),optical disks (a laser-burned pit in the aluminum foil representing a binary ”1” and an unburnedspot representing a binary ”0”), or a variety of other media types.Before we go on to learning exactly how all this is done in digital circuitry, we need to becomemore familiar with binary and other associated systems of numeration.1.3Decimal versus binary numerationLet’s count from zero to twenty using four diﬀerent kinds of numeration systems: hash marks,Roman numerals, decimal, and binary:System:Hash MarksRomanDecimalBinary-----------------------------------Zeron/an/a00One|I11Two||II210Three|||III311Four||||IV4100Five/|||/V5101Six/|||/ |VI6110Seven/|||/ ||VII7111Eight/|||/ |||VIII81000Nine/|||/ ||||IX91001Ten/|||/ /|||/X101010Eleven/|||/ /|||/ |XI111011Twelve/|||/ /|||/ ||XII121100Thirteen/|||/ /|||/ |||XIII131101Fourteen/|||/ /|||/ ||||XIV141110Fifteen/|||/ /|||/ /|||/XV151111Sixteen/|||/ /|||/ /|||/ |XVI1610000Seventeen/|||/ /|||/ /|||/ ||XVII1710001Eighteen/|||/ /|||/ /|||/ |||XVIII1810010Nineteen/|||/ /|||/ /|||/ ||||XIX1910011Twenty/|||/ /|||/ /|||/ /|||/XX2010100Neither hash marks nor the Roman system are very practical for symbolizing large numbers.