Download 1-1Numerical Representations - ENGN1000

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
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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
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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
12=0
1
MSB
Ans. 1110012
Ex: 5710  Octal
Decimal
Remainder
57  8 = 7.125
.125 x 8 = 1
78=0
7
LSB
MSB
Ans. 718
Ex: 5710  Hexadecimal
Decimal
Remainder
57  16 = 3.5625 .5625 x 16 =
38=0
9
3
LSB
MSB
Ans. 3916
Ex: 78310  Hexadecimal
783  16 = 48.9375
48  16 = 3
38=0
.9375 x 16 =
Decimal
Remainder
15
0
3
Hexadecimal
Conversion
F
LSB
0
3
MSB
Ans. 30F16
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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
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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.
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