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Numerical Representations Analog quantities that vary over a continuous range of values. Digital quantities that are measured in discrete steps, two state logic, on or off, (HIGH, 2-5 V) (LOW, 0 - .8V) +5V Logic HIGH +2V Undefined +0.8V Logic LOW 0V Digital Systems 1) Digital systems are easier to design because they utilize switching circuits. 2) Information storage is easy. 3) Accuracy and precision is greater because you can add additional switching circuits into the circuit. Analog circuit quantities are limited by the circuit components and supply. 4) Operation can be programmed. 5) Digital circuits are less affected by noise. 6) More digital circuitry can be fabricated on IC chips. Need an analog-digital-converter (ADC) to convert an analog quantity to a digital quantity to be processed. Also need a digital-analog-converter (DAC) to convert the digital signal back to an analog signal. NUMBERING SYSTEMS Decimal numbering system consists of 10 different symbols: 0,1,2,3,4,5,6,7,8,9. The decimal system is a Base 10 system because it consists of 10 different symbols. All numbers in the decimal system are made up of combinations of those 10 symbols to make up an infinite numbering system. Binary numbering system consists of 2 different symbols: 0,1. All numbers in the Binary system (Base 2) must be formed by using only those two digits. (LOW = 0, HIGH = 1) Representing Binary Quantities Two state logic. Digital Circuits 0 - .8 V Binary 0 2V - 5V Binary 1 Between .8V and 2V, not a value represented by a digital system. TTL (transistor-transistor logic) uses bipolar transistors CMOS (complementary metal-oxide semiconductor) uses enhancement MOSFET's Page 1 of 5 Octal numbering system consists of 8 different symbols: 0,1,2,3,4,5,6,7. The Octal system is a Base 8 system. Hexadecimal numbering system consists of 16 different symbols: The 10 decimal symbols and 6 others. The six others are the letters A - F. The hexadecimal numbering system consists of the following: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. The hexadecimal numbering system is a Base 16 system. Numbering Systems Chart Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 Octal 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Page 2 of 5 Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F Conversions DECIMAL Binary, Octal or Hexadecimal PROGRESSIVE DIVISION by the base you are converting to. Ex: 5710 Binary Decimal Remainder 57 2 = 28.5 .5 x 2 = 1 LSB 28 2 = 14 0 14 2 = 7 0 7 2 = 3.5 .5 x 2 = 1 3 2 = 1.5 .5 x 2 = 1 12=0 1 MSB Ans. 1110012 Ex: 5710 Octal Decimal Remainder 57 8 = 7.125 .125 x 8 = 1 78=0 7 LSB MSB Ans. 718 Ex: 5710 Hexadecimal Decimal Remainder 57 16 = 3.5625 .5625 x 16 = 38=0 9 3 LSB MSB Ans. 3916 Ex: 78310 Hexadecimal 783 16 = 48.9375 48 16 = 3 38=0 .9375 x 16 = Decimal Remainder 15 0 3 Hexadecimal Conversion F LSB 0 3 MSB Ans. 30F16 Page 3 of 5 BINARY, OCTAL or HEXADECIMAL Decimal POSITIONAL NOTATION using the base you are converting from Ex: 1110012 Decimal 32 16 8 4 2 1 25 24 23 22 21 20 1 1 1 0 0 1 5 4 3 2 1 0 1110012 (1 x 2 ) + ( 1 x 2 ) + (1 x 2 ) + ( 0 x 2 ) + (0 x 2 ) + (1 x 2 ) 1110012 32 16 8 1 57 Ans. 5710 Ex: 718 Decimal 8 1 81 80 7 1 1 0 718 (7 x 8 ) + (1 x 8 ) 718 56 + 1 57 Ans. 5710 Ex: 3916 Decimal 16 1 161 160 3 9 3916 (3 x 161) + (9 x 16) 3916 48 + 9 57 Ans. 5710 Page 4 of 5 BINARY Octal Group the BINARY number into 3 bit groups starting at the LSB and working towards the MSB (* Add zeros to the left of the far left group to make the last group into 3 bits if necessary) Write down the single digit OCTAL equivalent for each group Ex: 1110012 Octal Binary Octal 111 7 001 1 OCTAL Binary Write down the 3 bit BINARY equivalent for each OCTAL digit Ex: 718 Binary Octal Binary 7 111 1 001 BINARY Hexadecimal Group the BINARY number into 4 bit groups starting at the LSB and working towards the MSB (* Add zeros to the left of the far left group to make the last group into 4 bits if necessary) Write down the single digit HEXADECIMAL equivalent for each group Ex: 1110012 Hexadecimal Binary Hexadecimal 0011 3 1001 9 Hexadecimal Binary Write down the 4 bit BINARY equivalent for each HEXADECIMAL digit Ex: 3916 Binary Hexadecimal Binary 3 0011 9 1001 There is no direct conversion between OCTAL and HEXADECIMAL. You should change to BINARY, then use the conversion procedure from BINARY to either OCTAL or HEXADECIMAL. Page 5 of 5