Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers?...
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Transcript of Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers?...
![Page 1: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/1.jpg)
Number Systems
![Page 2: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/2.jpg)
Common Number Systems
System Base Symbols
Used by humans?
Used in computers?
Decimal 10 0, 1, … 9 Yes No
Binary 2 0, 1 No Yes
Octal 8 0, 1, … 7 No No
Hexa-decimal
16 0, 1, … 9,
A, B, … F
No No
![Page 3: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/3.jpg)
Quantities/Counting (1 of 3)
Decimal Binary Octal
Hexa-decimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7p. 33
![Page 4: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/4.jpg)
Quantities/Counting (2 of 3)
Decimal Binary Octal
Hexa-decimal
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
![Page 5: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/5.jpg)
Quantities/Counting (3 of 3)
Decimal Binary Octal
Hexa-decimal
16 10000 20 10
17 10001 21 11
18 10010 22 12
19 10011 23 13
20 10100 24 14
21 10101 25 15
22 10110 26 16
23 10111 27 17 Etc.
![Page 6: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/6.jpg)
Conversion Among BasesThe possibilities:
Hexadecimal
Decimal Octal
Binary
pp. 40-46
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Quick Example
2510 = 110012 = 318 = 1916
Base
![Page 8: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/8.jpg)
Decimal to Decimal (just for fun)
Hexadecimal
Decimal Octal
Binary
Next slide…
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12510 => 5 x 100 = 52 x 101 = 201 x 102 = 100
125
Base
Weight
![Page 10: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/10.jpg)
Binary to Decimal
Hexadecimal
Decimal Octal
Binary
![Page 11: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/11.jpg)
Binary to DecimalTechnique
Multiply each bit by 2n, where n is the “weight” of the bit
The weight is the position of the bit, starting from 0 on the right
Add the results
![Page 12: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/12.jpg)
Example
1010112 => 1 x 20 = 11 x 21 = 20 x 22 = 01 x 23 = 80 x 24 = 01 x 25 = 32
4310
Bit “0”
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Octal to Decimal
Hexadecimal
Decimal Octal
Binary
![Page 14: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/14.jpg)
Octal to DecimalTechnique
Multiply each bit by 8n, where n is the “weight” of the bit
The weight is the position of the bit, starting from 0 on the right
Add the results
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Example
7248 => 4 x 80 = 42 x 81 = 167 x 82 = 448
46810
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Hexadecimal to Decimal
Hexadecimal
Decimal Octal
Binary
![Page 17: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/17.jpg)
Hexadecimal to DecimalTechnique
Multiply each bit by 16n, where n is the “weight” of the bit
The weight is the position of the bit, starting from 0 on the right
Add the results
![Page 18: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/18.jpg)
Example
ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560
274810
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Decimal to Binary
Hexadecimal
Decimal Octal
Binary
![Page 20: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/20.jpg)
Decimal to BinaryTechnique
Divide by two, keep track of the remainderFirst remainder is bit 0 (LSB, least-significant
bit)Second remainder is bit 1Etc.
![Page 21: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/21.jpg)
Example12510 = ?2
2 125 62 12 31 02 15 12 7 12 3 12 1 12 0 1
12510 = 11111012
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Octal to Binary
Hexadecimal
Decimal Octal
Binary
![Page 23: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/23.jpg)
Octal to BinaryTechnique
Convert each octal digit to a 3-bit equivalent binary representation
![Page 24: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/24.jpg)
Example7058 = ?2
7 0 5
111 000 101
7058 = 1110001012
![Page 25: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/25.jpg)
Hexadecimal to Binary
Hexadecimal
Decimal Octal
Binary
![Page 26: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/26.jpg)
Hexadecimal to BinaryTechnique
Convert each hexadecimal digit to a 4-bit equivalent binary representation
![Page 27: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/27.jpg)
Example10AF16 = ?2
1 0 A F
0001 0000 1010 1111
10AF16 = 00010000101011112
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Decimal to Octal
Hexadecimal
Decimal Octal
Binary
![Page 29: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/29.jpg)
Decimal to OctalTechnique
Divide by 8Keep track of the remainder
![Page 30: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/30.jpg)
Example123410 = ?8
8 1234 154 28 19 28 2 38 0 2
123410 = 23228
![Page 31: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/31.jpg)
Decimal to Hexadecimal
Hexadecimal
Decimal Octal
Binary
![Page 32: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/32.jpg)
Decimal to HexadecimalTechnique
Divide by 16Keep track of the remainder
![Page 33: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/33.jpg)
Example123410 = ?16
123410 = 4D216
16 1234 77 216 4 13 = D16 0 4
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Binary to Octal
Hexadecimal
Decimal Octal
Binary
![Page 35: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/35.jpg)
Binary to OctalTechnique
Group bits in threes, starting on rightConvert to octal digits
![Page 36: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/36.jpg)
Example10110101112 = ?8
1 011 010 111
1 3 2 7
10110101112 = 13278
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Binary to Hexadecimal
Hexadecimal
Decimal Octal
Binary
![Page 38: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/38.jpg)
Binary to HexadecimalTechnique
Group bits in fours, starting on rightConvert to hexadecimal digits
![Page 39: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/39.jpg)
Example10101110112 = ?16
10 1011 1011
2 B B
10101110112 = 2BB16
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Octal to Hexadecimal
Hexadecimal
Decimal Octal
Binary
![Page 41: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/41.jpg)
Octal to HexadecimalTechnique
Use binary as an intermediary
![Page 42: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/42.jpg)
Example10768 = ?16
1 0 7 6
001 000 111 110
2 3 E
10768 = 23E16
![Page 43: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/43.jpg)
Hexadecimal to Octal
Hexadecimal
Decimal Octal
Binary
![Page 44: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/44.jpg)
Hexadecimal to OctalTechnique
Use binary as an intermediary
![Page 45: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/45.jpg)
Example1F0C16 = ?8
1 F 0 C
0001 1111 0000 1100
1 7 4 1 4
1F0C16 = 174148
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Exercise – Convert ...
Don’t use a calculator!
Decimal Binary Octal
Hexa-decimal
33
1110101
703
1AF
Skip answer Answer
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Exercise – Convert …
Decimal Binary Octal
Hexa-decimal
33 100001 41 21
117 1110101 165 75
451 111000011 703 1C3
431 110101111 657 1AF
Answer
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Common Powers (1 of 2)Base 10
Power Preface Symbol
10-12 pico p
10-9 nano n
10-6 micro
10-3 milli m
103 kilo k
106 mega M
109 giga G
1012 tera T
Value
.000000000001
.000000001
.000001
.001
1000
1000000
1000000000
1000000000000
![Page 49: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/49.jpg)
Common Powers (2 of 2)Base 2
Power Preface Symbol
210 kilo k
220 mega M
230 Giga G
Value
1024
1048576
1073741824
• What is the value of “k”, “M”, and “G”?• In computing, particularly w.r.t. memory, the base-2 interpretation generally applies
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Example
/ 230 =
In the lab…1. Double click on My Computer2. Right click on C:3. Click on Properties
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Exercise – Free SpaceDetermine the “free space” on all drives on
a machine in the lab
Drive
Free space
Bytes GB
A:
C:
D:
E:
etc.
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Review – multiplying powersFor common bases, add powers
26 210 = 216 = 65,536
or…
26 210 = 64 210 = 64k
ab ac = ab+c
![Page 53: Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, … 9YesNo Binary20, 1NoYes Octal80, 1, … 7No.](https://reader035.fdocuments.in/reader035/viewer/2022062217/5697bfe11a28abf838cb3eb2/html5/thumbnails/53.jpg)
Binary Addition (1 of 2)Two 1-bit values
pp. 36-38
A B A + B
0 0 0
0 1 1
1 0 1
1 1 10“two”
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Binary Addition (2 of 2)Two n-bit values
Add individual bitsPropagate carriesE.g.,
10101 21+ 11001 + 25 101110 46
11
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Multiplication (1 of 3)Decimal (just for fun)
pp. 39
35x 105 175 000 35 3675
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Multiplication (2 of 3)Binary, two 1-bit values
A B A B
0 0 0
0 1 0
1 0 0
1 1 1
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Multiplication (3 of 3)Binary, two n-bit values
As with decimal valuesE.g.,
1110 x 1011 1110 1110 0000 111010011010
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FractionsDecimal to decimal (just for fun)
pp. 46-50
3.14 => 4 x 10-2 = 0.041 x 10-1 = 0.1
3 x 100 = 3 3.14
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FractionsBinary to decimal
pp. 46-50
10.1011 => 1 x 2-4 = 0.06251 x 2-3 = 0.1250 x 2-2 = 0.01 x 2-1 = 0.50 x 20 = 0.01 x 21 = 2.0 2.6875
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FractionsDecimal to binary
p. 50
3.14579
.14579x 20.29158x 20.58316x 21.16632x 20.33264x 20.66528x 21.33056
etc.11.001001...
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Exercise – Convert ...
Don’t use a calculator!
Decimal Binary Octal
Hexa-decimal
29.8
101.1101
3.07
C.82
Skip answer Answer
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Exercise – Convert …
Decimal Binary Octal
Hexa-decimal
29.8 11101.110011… 35.63… 1D.CC…
5.8125 101.1101 5.64 5.D
3.109375 11.000111 3.07 3.1C
12.5078125 1100.10000010 14.404 C.82
Answer
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http://www.ascii-code.com/
ascii-code
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Binary Logic
Truth Tables, Boolean Expressions, and Logic Gates
x y z
0 0 0
0 1 0
1 0 0
1 1 1
x y z
0 0 0
0 1 1
1 0 1
1 1 1
x z
0 1
1 0
AND OR NOT
xy z x
y z
z = x • y = x y z = x + y z = x = x’
x z
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Switching Circuits
AND OR
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Binary LogicLogic gates
Example of binary signals
0
1
2
3Logic 1
Logic 0
Un-define
Figure 1.3 Example of binary signals
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Binary LogicLogic gates
Graphic Symbols and Input-Output Signals for Logic gates:
Fig. 1.4 Symbols for digital logic circuits
Fig. 1.5 Input-Output signals for gates
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Binary LogicLogic gates
Graphic Symbols and Input-Output Signals for Logic gates:
Fig. 1.6 Gates with multiple inputs