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# 3.4

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### 3.4

2. 2. e.g. If we count in eights (called the OCTAL system) the column headings go up in8’s. 512 64 8 1So 117 (in denary) is 1 lot of 64, leaving another 53.53 is 6 lots of 8 with 5 left over. Fitting this in the columns gives 512 64 8 1 0 1 6 5So 117 in denary is 165 in octal.Why bother with octal?Octal and binary are related. If we take the three digits of the octal number 165 andturn each one into binary using three bits each we get 1 = 001 6 = 110 5 = 101Put them together and we get 001110101 which is the binary value of 117 which wegot earlier.The value of this relationship is not important now, but it is the reason why octal is inthe syllabus.Another system is called HEXADECIMAL (counting in 16’s). This sounds verydifficult, but it needn’t be, just use the same principles. 256 16 1So 117 (in denary) is 7 lots of 16 (112) plus an extra 5. Fitting this in the columnsgives 256 16 1 0 7 5Notice that 7 in binary is 0111 and that 5 is 0101, put them together and we get01110101 which is the binary value of 117 again. So binary, octal and hexadecimalare all related in some way.There is a problem with counting in 16’s instead of the other systems. We needsymbols going further than 0 to 9 (only 10 symbols and we need 16!).We could invent 6 more symbols but we would have to learn them, so we use 6 thatwe already know, the letters A to F. In hexadecimal A stands for 10, B stands for 11and so on to F stands for 15.So a hexadecimal number BD stands for 11 lots of 16 and 13 units = 176 + 13 = 189 ( in denary)Note: B = 11, which in binary = 1011 D = 13, which in binary = 1101 Put them together to get 10111101 = the binary value of 189.Binary Coded DecimalSome numbers are not proper numbers because they don’t behave like numbers. Abarcode for chocolate looks like a number, and a barcode for sponge cake looks like anumber, but if the barcodes are added together the result is not the barcode forchocolate cake. The arithmetic does not give a sensible answer. Values like this thatlook like numbers but do not behave like them are often stored in binary coded 4.4 - 2