CT

1
CT

• Seeram Chapter 2A
• Digital Arithmetic

2
The Bit

• Fundamental unit of computer storage
• Only 2 allowable values
• 0
• 1
• Computers do all operations with 0s
  1sBUTComputers group bits together

3
Popular Bit Groupings

• Bit (binary digit)
• Smallest binary unit has value 0 or 1 only
• Byte
• 8 bits
• 28 256 unique values
• Word
• 16 bits
• 216 65536 unique values

4
Special Binary Digit Grouping Terms

• Nibble
• 4 binary bits (0101)
• Byte
• 8 binary bits (1000 1011)
• Word
• 16 binary bits (1100 0100 1100 0101)
• Double Word
• 32 binary bits(1110 0100 0000 1011 0101 0101
  1110 0101)

5
of values which can be represented by 1 bit
2 unique combinations / values
1
2
6
of values which can be represented by 2 bits
1
2
4 unique combinations / values
3
4
7
of values which can be represented by 3 bits
5
1
6
2
7
3
8
4
8 unique combinations / values
8
of values which can be represented by bits
of possible Values
Possible Values
Bits
1 2 3 . . . 8
0, 1 00, 01, 10, 11 000, 001, 010, 011, 100, 101,
110, 111 . . . 00000000, 00000001, ... 11111111
2 1 2 2 2 4 2 3 8 . . . 2 8 256
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Base 10
485

• 485 means
• 5 ones
• 8 tens
• 4 hundreds

• Digit position
• right digit
• 1s or 100 s
• Next digit to left
• 10s or 101 s
• Next digit to left
• 100s or 102 s

10
Base 10 vs. Base 2
Base 10
Base 2

• Digit position
• right digit
• 1s or 20 s
• Next digit to left
• 2s or 21 s
• Next digit to left
• 4s or 22 s
• Next digit to left
• 8s or 23 s

• Digit position
• right digit
• 1s or 100 s
• Next digit to left
• 10s or 101 s
• Next digit to left
• 100s or 102 s
• Next digit to left
• 1000s or 103 s

11
Base 10 vs. Base 2
1011
Base 10
Base 2

• 1011 means
• 1 one
• 1 tens
• 0 hundreds
• 1 thousand

• 1011 means
• 1 one
• 1 two
• 0 fours
• 1 eight

11
12
Base 10 vs. Base 2
Base 10
Base 2

• 0000 00001 10010 20011 30100 40101 50110 6
  0111 7

1000 81001 91010 101011 111100 121101 13111
0 141111 15
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Shortcut NomenclatureGrouping Binary digits

• Binary (base 2) s are awkward (101100111010)
• Binary digits often grouped by 3s or 4s
• Grouping by 3s
• octal
• Grouping by 4s
• hexadecimal

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Shortcut NomenclatureOctal - Grouping by 3s

• Each group of 3 digits assigned its binary value

Grouping by 3s
000 0001 1010 2011 3100 4101 5110
6111 7
101100111010 101 100 111 010 5 4
7 25472 Octal (Base 8)
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Shortcut NomenclatureOctal - Grouping by 3s

• Octal Value is actually base 8

Base 8

• 5472 means
• 2 1s
• 7 8s
• 4 64s
• 5 512s

Grouping by 3s
101100111010 101 100 111 010 5 4
7 25472 Octal (Base 8)
16
Try this one!

• Base 2 110001
• Octal ?
• Decimal ?

17
Shortcut NomenclatureHexadecimal - Grouping by
4s

• Each group of 3 digits assigned its binary value

Grouping by 4s
0000 00001 10010 20011 30100 40101 50110 6
0111 7
1000 81001 91010 A1011 B1100 C1101 D1110 E
1111 F
101100111010 1011 0011 1010 B 3
AB3A Hexadecimal(Base 16)
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Shortcut NomenclatureHexadecimal - Grouping by
4s

• Hexadecimal is actually base 16

Base 16

• B3A means
• A or 10 1s
• 3 16s
• B or 11 256s

Grouping by 4s
101100111010 1011 0011 1010 B 3
AB3A Hexadecimal(Base 16)
19
Try this one!

• Base 2 10010001
• Hexadecimal ?
• Decimal ?

20
Counting Bytes

• bytes different from of unique values for a
  byte
• Kilobyte
• 210 or 1024 bytes
• sometimes rounded to 1000 bytes
• Megabyte
• 213 or 1,048,576 bytes or 1024 kilobytes
• sometimes rounded to 1,000,000 bytes or 1,000
  kilobytes

21
How Computers Handle Text

• Each character assigned a
• Assignment standards
• ASCII
• American Standard Code for Information
  Interchange
• EBCDIC
• Extended binary coded decimal interchange code

22
ASCII Codes
23
How Computers Handle TextAn Example George
Letter Value Binary Hexadecimal

• G 71 0100 0111 47e 101 0110 0101 65o
  111 0110 1111 6Fr 114 0111 0010 72g
  103 0110 0111 67e 101 0110 0101 65

ASCII