1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers?...
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Transcript of 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers?...
![Page 1: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/1.jpg)
1. Number Systems
![Page 2: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/2.jpg)
Common Number Systems
System Base SymbolsUsed 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: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/3.jpg)
Quantities/Counting (1 of 3)
Decimal Binary OctalHexa-
decimal
0 0 0 01 1 1 12 10 2 23 11 3 34 100 4 45 101 5 56 110 6 67 111 7 7
![Page 4: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/4.jpg)
Quantities/Counting (2 of 3)
Decimal Binary OctalHexa-
decimal
8 1000 10 89 1001 11 9
10 1010 12 A11 1011 13 B12 1100 14 C13 1101 15 D14 1110 16 E15 1111 17 F
![Page 5: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/5.jpg)
Quantities/Counting (3 of 3)
Decimal Binary OctalHexa-
decimal
16 10000 20 1017 10001 21 1118 10010 22 1219 10011 23 1320 10100 24 1421 10101 25 1522 10110 26 1623 10111 27 17 Etc.
![Page 6: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/6.jpg)
Conversion Among Bases
• The possibilities:
Hexadecimal
Decimal Octal
Binary
![Page 7: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/7.jpg)
Quick Example
2510 = 110012 = 318 = 1916
Base
![Page 8: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/8.jpg)
Decimal to Decimal (just for fun)
Hexadecimal
Decimal Octal
Binary
Next slide…
![Page 9: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/9.jpg)
12510 => 5 x 100 = 52 x 101 = 201 x 102 = 100
125
Base
Weight
![Page 10: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/10.jpg)
Binary to Decimal
Hexadecimal
Decimal Octal
Binary
![Page 11: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/11.jpg)
Binary to Decimal
• Technique– 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: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/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”
![Page 13: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/13.jpg)
Hexadecimal to Decimal
Hexadecimal
Decimal Octal
Binary
![Page 14: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/14.jpg)
Hexadecimal to Decimal
• Technique– 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 15: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/15.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
![Page 16: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/16.jpg)
Decimal to Binary
Hexadecimal
Decimal Octal
Binary
![Page 17: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/17.jpg)
Decimal to Binary
• Technique– Divide by two, keep track of the remainder– First remainder is bit 0 (LSB, least-significant
bit)– Second remainder is bit 1– Etc.
![Page 18: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/18.jpg)
Example
12510 = ?22 125 62 12 31 02 15 12 7 12 3 12 1 12 0 1
12510 = 11111012
![Page 19: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/19.jpg)
Hexadecimal to Binary
Hexadecimal
Decimal Octal
Binary
![Page 20: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/20.jpg)
Hexadecimal to Binary
• Technique– Convert each hexadecimal digit to a 4-bit
equivalent binary representation
![Page 21: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/21.jpg)
Decimal to Hexadecimal
Hexadecimal
Decimal Octal
Binary
![Page 22: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/22.jpg)
Decimal to Hexadecimal
• Technique– Divide by 16– Keep track of the remainder
![Page 23: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/23.jpg)
Example
123410 = ?16
123410 = 4D216
16 1234 77 216 4 13 = D16 0 4
![Page 24: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/24.jpg)
Binary to Hexadecimal
Hexadecimal
Decimal Octal
Binary
![Page 25: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/25.jpg)
Binary to Hexadecimal
• Technique– Group bits in fours, starting on right– Convert to hexadecimal digits
![Page 26: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/26.jpg)
Example
10101110112 = ?16
10 1011 1011
2 B B
10101110112 = 2BB16
![Page 27: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/27.jpg)
Exercise – Convert ...
Don’t use a calculator!
Decimal BinaryHexa-
decimal
331110101
1AF
![Page 28: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/28.jpg)
Exercise – Convert …
Decimal BinaryHexa-
decimal
33 100001 21117 1110101 75451 111000011 1C3431 110101111 1AF
Answer
![Page 29: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/29.jpg)
Binary Addition (1 of 2)
• Two 1-bit values
A B A + B0 0 00 1 11 0 11 1 10
“two”
![Page 30: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/30.jpg)
Binary Addition (2 of 2)
• Two n-bit values– Add individual bits– Propagate carries– E.g.,
10101 21+ 11001 + 25 101110 46
11
![Page 31: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/31.jpg)
Multiplication (1 of 3)
• Decimal (just for fun)
pp. 39
35x 105 175 000 35 3675
![Page 32: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/32.jpg)
Multiplication (2 of 3)
• Binary, two 1-bit values
A B A B0 0 00 1 01 0 01 1 1
![Page 33: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/33.jpg)
Multiplication (3 of 3)
• Binary, two n-bit values– As with decimal values– E.g.,
1110 x 1011 1110 1110 0000 111010011010
![Page 34: 1. Number Systems. Common Number Systems SystemBaseSymbols Used by humans? Used in computers? Decimal100, 1, 9YesNo Binary20, 1NoYes Octal80, 1, 7No.](https://reader033.fdocuments.us/reader033/viewer/2022060206/5a4d1bc57f8b9ab0599d456d/html5/thumbnails/34.jpg)
Thank you