Table Of Content• Conversion between number systems:
– Radix-r to decimal.
– Decimal to binary.
– Decimal to Radix-r
– Binary to Octal
– Binary to Hex
• Binary arithmetic operations.
• Negative number representations.
• Switching Algebra Axioms & Theorems.
• Proof of identities:
– Using logic expression algebraic manipulation.
– Using Truth Table (perfect induction).
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#1 Midterm Review Winter 2001 1-22-2002
• Standard Representations of Logic Functions:
– Truth Table.
– Canonical Sum Representation:
• Full sum of minterms expression, or using SS notation.
– Canonical Product Representation:
• Full product of maxterms expression, or using PP notation.
•• CCoommbbiinnaattiioonnaall CCiirrccuuiitt AAnnaallyyssiiss// Synthesis.
• Combinational Circuit Minimization using K-maps:
– Sum of Products (SOP) Minimization using K-maps:
• Prime implicants, distinguished 1-cells, essential prime implicants
• Minimization with Don’t care Input Combinations.
– Product of Sums (POS) Minimization using K-maps:
• Prime implicates, distinguished 0-cells, essential prime implicates
• Detecting/Eliminating Static Hazards Using K-maps.
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Positional Number Systems
• A number system consists of an order set of symbols (digits) with relations
defined for +,-,*, /
• The radix (or base) of the number system is the total number of digits
allowed in the the number system.
–
Example, for the decimal number system:
• Radix, r = 10, Digits allowed = 0,1, 2, 3, 4, 5, 6, 7, 8, 9
• In positional number systems, a number is represented by a string of digits,
where each digit position has an associated weight.
• The value of a number is the weighted sum of the digits.
• The general representation of an unsigned number D with whole and
fraction portions number in a number system with radix r:
D = d d ….. d d .d d …. D
r p-1 p-2 1 0 -1 -2 -n
• The number above has p digits to the left of the radix point and n fraction
digits to the right.
• A digit in position i has as associated weight ri
• The value of the number is the sum of the digits multiplied by the associated
weight ri : D = (cid:229) p- 1 · i
d r
=-
i n i
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Positional Number Systems
Number: D = d d ….. d d .d d …. D
r p-1 p-2 1 0 -1 -2 -n
(cid:229) -
Value: D = p 1 · i
d r
i=- n i
• For example in the decimal number system:
5185.68 = 5x103 + 1x102 + 8x101 + 5x100 + 6 x 10-1 + 8 x 10-2
10
= 5x1000 + 1x100 + 8x10 + 5 x 1 + 6x.1 + 8x.01
• For the binary number system with radix = 2, digits 0, 1
D = d ·· 2p-1 ….. d ·· 21 + d . 20 + d ·· 2-1 + d ·· 2-2 …..
2 p-1 1 0 -1 -2
• For Example:
10011 = 1 ·· 16 + 0 ·· 8 + 0 ·· 4 + 1 ·· 2 + 1 ·· 1 = 19
2 10
| |
MSB LSB (least significant bit)
(most significant bit)
101.001 = 1x4 + 0x2 + 1x1 + 0x.5 + 0x.25 + 1x.125 = 5.125
2 10
Binary Point
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Name
Radix
Set of Digits Example
of Radix
Decimal r=10 {0,1,2,3,4,5,6,7,8,9} 255
10
Binary r=2 {0,1} 11111111
2
Octal r= 8 {0,1,2,3,4,5,6,7} 377
8
Hexadecimal r=16 {0,1,2,3,4,5,6,7,8,9,A, B, C, D, E, F} FF
16
Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Hex 0 1 2 3 4 5 6 7 8 9 A B C D E F
Binary
0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111
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RRaaddiixx--rr ttoo DDeecciimmaall CCoonnvveerrssiioonn
• The decimal value of a number in any radix r is found by converting
each digit to its radix 10 equivalent and expanding the value using
radix arithmetic: D = (cid:229) p- 1 · i
d r
• Examples: i=- n i
1101.101 = 1··23 + 1··22 + 1··20 + 1··2-1 + 1··2-3
2
= 8 + 4 + 1 + 0.5 + 0.125 = 13.625
10
572.6 = 5··82 + 7··81 + 2··80 + 6··8-1
8
= 320 + 56 + 16 + 0.75 = 392.75
10
2A.8 = 2··161 + 10··160 + 8··16-1
16
= 32 + 10 + 0.5 = 42.5
10
132.3 = 1··42 + 3··41 + 2··40 + 3··4-1
4
= 16 + 12 + 2 + 0.75 = 30.75
10
341.24 = 3··52 + 4··51 + 1··50 + 2··5-1 + 4··5-2
5
= 75 + 20 + 1 + 0.4 + 0.16 = 96.56
10
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• Separate the decimal number into whole and fraction portions.
• To convert the whole number portion to binary, use successive
division by 2 until the quotient is 0. The remainders form the
answer, with the first remainder as the least significant bit (LSB) and
the last as the most significant bit (MSB).
• Example: Convert 179 to binary:
10
179 / 2 = 89 remainder 1 (LSB)
/ 2 = 44 remainder 1
/ 2 = 22 remainder 0
/ 2 = 11 remainder 0
/ 2 = 5 remainder 1
/ 2 = 2 remainder 1
/ 2 = 1 remainder 0
/ 2 = 0 remainder 1 (MSB)
179 = 10110011
10 2
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• To convert decimal fractions to binary, repeated multiplication by 2 is
used, until the fractional product is 0 (or until the desired number of
binary places). The whole digits of the multiplication results produce
the answer, with the first as the MSB, and the last as the LSB.
• Example: Convert 0.3125 to binary
10
Result Digit
.3125 ·· 2 = 0.625 0 (MSB)
.625 ·· 2 = 1.25 1
.25 ·· 2 = 0.50 0
.5 ·· 2 = 1.0 1 (LSB)
0.3125 = .0101
10 2
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DDeecciimmaall ttoo RRaaddiixx--rr CCoonnvveerrssiioonn
• Separate the decimal number into whole and fraction portions.
• To convert the whole number portion to binary, use successive
division by r until the quotient is 0. The remainders form the
answer, with the first remainder as the least significant digit (LSD)
and the last as the most significant digit (MSD).
• To convert decimal fractions to radix-r, repeated multiplication by r
is used, until the fractional product is 0 (or until the desired number
of binary places). The whole digits of the multiplication results
produce the answer, with the first as the MSD, and the last as the
LSD.
• Example: Convert 467 to octal
10
467 / 8 = 58 remainder 3 (LSD)
/ 8 = 7 remainder 2
/ 8 = 0 remainder 7 (MSD)
467 = 723
10 8
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BBiinnaarryy ttoo OOccttaall CCoonnvveerrssiioonn
• Separate the whole binary number portion into groups of
3 beginning at the binary point and working to the left.
Add leading zeroes as necessary.
• Separate the fraction binary number portion into groups of
3 beginning at the binary point and working to the right.
Add trailing zeroes as necessary.
• Convert each group of 3 to the equivalent octal digit.
• Example:
3564.875 = 110 111 101 100.111
10 2
= (6 ·· 83) + (7 ·· 82) + (5 ·· 81)+(4 ·· 80)+(7 ·· 8-1)
= 6754.7
8
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#10 Midterm Review Winter 2001 1-22-2002
Description:Jan 22, 2002 Binary to Octal. – Binary to Hex. • Binary arithmetic operations. • Negative number
representations. • Switching Algebra Axioms & Theorems.
