10CHAPTER 1. NUMERATION SYSTEMS.bits =101.011.-------weight =421111(in decimal///notation)248410+ 110+ 0.2510+ 0.12510= 5.375101.4Octal and hexadecimal numerationBecause binary numeration requires so many bits to represent relatively small numbers comparedto the economy of the decimal system, analyzing the numerical states inside of digital electroniccircuitry can be a tedious task. Computer programmers who design sequences of number codesinstructing a computer what to do would have a very diﬃcult task if they were forced to work withnothing but long strings of 1’s and 0’s, the ”native language” of any digital circuit. To make it easierfor human engineers, technicians, and programmers to ”speak” this language of the digital world,other systems of place-weighted numeration have been made which are very easy to convert to andfrom binary.One of those numeration systems is called octal, because it is a place-weighted system with abase of eight. Valid ciphers include the symbols 0, 1, 2, 3, 4, 5, 6, and 7. Each place weight diﬀersfrom the one next to it by a factor of eight.Another system is called hexadecimal, because it is a place-weighted system with a base of sixteen.Valid ciphers include the normal decimal symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, plus six alphabeticalcharacters A, B, C, D, E, and F, to make a total of sixteen. As you might have guessed already,each place weight diﬀers from the one before it by a factor of sixteen.Let’s count again from zero to twenty using decimal, binary, octal, and hexadecimal to contrastthese systems of numeration:NumberDecimalBinaryOctalHexadecimal------------------------------------Zero0000One1111Two21022Three31133Four410044Five510155Six611066Seven711177Eight81000108Nine91001119Ten10101012AEleven11101113BTwelve12110014C