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Numerical Systems We are used to using the decimal system in our daily lives. Digital technology requires an understanding of several other number systems the most important being the Binary system. The decimal system has its origin in the fact that man has ten fingers (or digits) hence the decimal system has ten digits (0 through 9). The binary system and Boolean Algebra have their origin in that simple switches or valves have two highly distinguishable states. They are OFF or ON. The binary system has two digits. They are 0 and 1. ! Octal and Hexadecimal are two additional number systems frequently used when dealing with digital systems. The octal system has eight digits (07) and the hexadecimal system has sixteen (09, A, B, C, D, E, F) These numbering system are particularly useful in dealing with large binary numbers, for example, A9B3 (HEX) is far easier to remember than 1010 1001 1011 0011(Binary) or 76751(Octal) is easier to remember than 111 110 111 101 001(Binary) . Decimal Binary Octal Hexadecimal 0 0000 0 0 1 0001 1 1 2 0010 2 2 3 0011 3 3 4 0100 4 4 5 0101 5 5 6 0110 6 6 7 0111 7 7 8 1000 10 8 9 1001 11 9 10 1010 12 A 11 1011 13 B 12 1100 14 C 13 1101 15 D 14 1110 16 E 15 1111 17 F 16 1 0000 20 10
Adding Binary Numbers The rules of addition of binary numbers are essentially the same as the rules for decimal addition only simpler.
BINARY DECIMAL REMARKS 0 + 0 = 0 0 + 0 = 0 SAME 0 + 1 = 1 0 + 1 = 1 SAME 1 + 0 = 1 1 + 0 = 1 SAME 1 + 1 = 10 1 + 1 = 2 In binary we generated a zero and a carry. We see that the rules are exactly the same for decimal and binary addition until we reach one and one equals two. In binary we cannot generate the digit "2" since there are only two digits in Binary "0" and "1". Therefore when you add one to highest binary digit "1" you generate a carry and a result of zero, just as when you add one to the highest decimal digit "9" you generate a carry and a result of zero. The four rules for Binary addition 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1 and 1 + 1 = 0 and a carry are fundamental to implementing binary addition electronically. 0000 1100 0110 0011 0011
1111 0011 0011
Add the above binary numbers. and record your results on paper and then check your answer against answers at bottom of page.
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