Number Systems TableDecimal012345678910111213141516Binary0110111001011101111000100110101011110011011110111110000Octal01234567101112131415161720Hexadecimal0123456789ABCDEF10Red 1 and 2 = “Carry”A,B,C,D,E,F = extra hex digitsImportant number conversions to remember:(10)10 = (1010)2 = (A)16(11)10 = (1011)2 = (B)16Juxapositional NotationN = number=(an-1 an-2 ... a2 a1 a0 . a-1 a-2 a-3 ... a-m)rInteger |Fraction radix pointai = digitsr = radix0 ≤ ai <r, always holds.Example: decimal number(353.12)r=10 = 3 hundreds5 tens3 ones1 tenth2 hundrethsExample: binary number(1010.01)r=2 = ? How do we expand this number?Clue: look at polynomial representation.Polynomial Representation(353.12)r=10 = 3 x 102 +5 x 101 +3 x 100 +1 x 10-1 +2 x 10-2(1010.01)r=2 = 1 x 23 +0 x 22 +1 x 21 +0 x 20 +0 x 2-1 +1 x 2-2General Polynomial:N = number=n − 1 Σ i = − m a i ri = an-1 rn-1 + an-2 rn-2 + ... + a2 r2 + a1 r1 + a0 r0 + a-1 r-1 + a-2 r-2 + a-3 r-3 + ... + a-m r-mN =NI + NFConversion to Base 10N α to N 10, by polynomial subsitution:(101101)2 = ?10= 1 x 25 +0 x 24 +1 x 23 +1 x 22 +0 x 21 +1 x 20=(45)10(247.1)8 = ?10= 2 x 82 +4 x 81 +7 x 80 +1 x 8-1=(167.125)10(1AB)16 = ?10= 1 x 162 +A x 161 +B x 160(Recall: A = 10, B = 11)=(427)10Conversion to Base αN10 toN , by radix divide/multiply:α Example: to N2 Example: to N82k ConversionsOctal: 2 8 = 23, in groups of k = 3Hex: 2 16 = 24, in groups of k = 4Example: Convert (010101100)2 to base 8 and 16Example: Convert (110.110)2 to base 8 and 16Adding and Subtracting Binary #sExamples Adding:Examples Subtracting:2’s Complement Examples:Simplification of Circuits using Boolean AlgebraTwo-Level Standard FormsNMOS TransistorsGDSG = GateS = SourceD = Drain(S and D are interchangeable)The NMOS transistor may be configured to operate in one of three different states, as determined by the voltage at the gate terminal VG:VG = 0 VoltsDS1. Off State (Nonconducting)SDOpen SwitchVG = 5 VoltsDS2. On State (Conducting)SDClose SwitchVG = VD = 5 VoltsDS3. Resistive State (Resistive)SDResistorNMOS Based Logic Gates1. NOT Gate: 2. NAND Gate:3. NOR Gate:x x’+3 V xx’+3 V xyxyx•yxy+3 V x yx + yx•yx + yMintermsKarnaugh Maps02130264137504128151393715112614102 Variable Map:3 Variable Map:4 Variable Map:02641375041281513937151126141002641375041281513937151126141002641375026413750412815139371511261410041281513937151126141004128151393715112614105 Variable Map:16 20 28 2417 21 29 2519 23 31 2718 22 30