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Chapter 1
Digital Computers
and Information
Digital Computer Basics
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Digital values
represented as
voltage values, e.g.
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Logic 1 is 4.0-5.0 V
Logic 0 is 0.0-1.0 V
Why digital? Why
binary?
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Simplest
Cheapest
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A Generic Computer
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Memory = RAM,
caches
I/O = keyboard,
terminal, hard disk
CPU
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Datapath= performs
basic instructions
CPU = supervisor, flow
MMU = memory
management
FPU = floating point
specialist
Number Systems
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Radix r number represented as:
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An-1 ... A1 A0 . A-1 A-2 ... A-m
Value is An-1rn-1 + An-2rn-2 + ... A1r1 +
A0r0 + A-1r-1 + ... A–mr-m
Digit range: 0 - (r-1)
An-1 is the “most significant digit”
A-m is the “least significant digit”
Notation: (6342)7 – use parens and subscript
to indicate number base, in this case 7
4 common bases: 2 (binary), 8 (octal), 10
(decimal), 16 (hex)
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Base R Wishlist
We want to be able to “handle” radix R
numbers, including:
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Decimal shortcut - Convert to decimal a
number in any radix
Power of 2 shortcut - Power of 2 conversions
between binary, octal, and hex numbers
General radix conversion - convert a number
in radix R1 to radix R2
We also want to be able to perform the
standard math operations
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Convert to Decimal
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We use decimal (r=10), so conversion from
other bases to decimal is common...
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Ex: Convert (43.2)5 to decimal
Trick: Just expand each digit. We know how to
manipulate decimal numbers which makes this
process “easy”.
Ans: 4 * 5 + 3 * 1 + 2 * .2 = 23.4
Note 1: Conversion from decimal is handled
by standard technique, no shortcut
Note 2: We are lucky. We can easily
convert to decimal to check our work!
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Convert to Decimal, cont
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There is another way (shortcut?) called
Horner’s rule (not in the text except as a
homework problem
W = an-1
for i=n-2 to 0 by –1
w = w*r + ai
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Ex: Convert (2504)6 to decimal
2 * 6 + 5 = 17
17 * 6 + 0 = 102
102 * 6 + 4 = 616
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This is actually more efficient as
well
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Power of 2 Conversions
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The trick(s)
Base 8 (octal) is Base
2 “cubed”
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24 = 16
So, 4 bits equals
one hex digit
Convert to Binary
Ex: (743.2)8
 Expand each octal digit
into its 3-bit value
 Ans:
(111 100 010. 010)2
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23 = 8
So, 3 bits equals
one octal digit
Base 16 (hex) is Base 2
to the 4th power
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Convert from Binary
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Ex: (10010101)2 to
hex
Trick: Collapse each 4
bit group into its hex
value
Ans: (95)16
General Conversion, cont.
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Don’t forget fractions
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Handle fractional parts separately
Similar process as integer part, except multiply
and use whole numbers, not remainders
Ex. (0.625)10 to binary
0.625 * 2 = 1.25, use 1
0.25 * 2 = 0.5, use 0
0.5 * 2 = 1.0, use 1
Answer is whole numbers in order: (101)2
Again, you can check work in decimal.
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General Number Conversion
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Trick: To convert to radix R, successively
divide your number by R and the
remainders are your converted value.
Ex: Convert (169)10 to octal
169 / 8 = 21 R1
21 / 8 = 2 R5
2 / 8 = 0 R2
Answer is reverse of remainders (251)8
Check our work:
(251)8 = 2*8*8 + 5*8 + 1 = 169
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Arithmetic Operations
We want to add, subtract and multiply
numbers of radix R
 Good news: Same basic rules as decimal
math
 Bad news: Your decimal brain may rebel at
the non-decimal carries, borrows, etc.
 More good news: Convert to decimal to
verify your work
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Adding
Let’s add:
(3A)16 + (69)16
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Watch the carries!
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Carry value is now
worth 16 (the base)
Double check with
decimal values
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Subtracting
Let’s subtract:
(69)16 - (3A)16
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Watch the
borrows!
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Borrow value is now
worth 16 (the base)
Double check with
decimal values
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Multiplying
Let’s multiply:
(3A)16 * (69)16
 Labor-intensive, error-prone with multiple
steps
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Binary-coded Decimals
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BCD = binary-coded
decimal
“Human” binary
coding of decimal
numbers
Convert number
one digit at a time:
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BCD addition
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Ex:
17 = 0001 0111
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Add each digit
separately
Carry is “special”
If sum for any digit >=
10, add 6 (0110) to sum
to get proper BCD
result
Ex. 8 + 5
= 1000 + 0101
= 1101 (Add 0110!)
= 0001 0011
Alphanumeric codes
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ASCII – encoding for alphanumeric
data
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7 bits long
Covers a-z, A-Z, 0-9, !@#$ and many
other special characters
See table on page 21 for complete
mapping
8th bit often added for parity check
Only for English-language characters
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Chapter 1 Summary
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Most important:
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Generic computer & information
Understand radix R number systems
Radix conversion (general and shortcuts)
Radix math operations
BCD encoding and addition
ASCII code basics
CSC 480 – Winter 2002