Transcript Lectures/Lect 2 - Arith op and codes

Arithmetic Operators
Decimal Codes
9/15/09 - L2
Copyright 2009 - Joanne DeGroat, ECE, OSU
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Class 2 outline
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Number Ranges
Arithmetic Operators
Decimal Codes
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Material from sections 1-3 and 1-4 of text
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Copyright 2009 - Joanne DeGroat, ECE, OSU
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Number Ranges
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What do you mean by number range?
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On paper you have as many digits as you can fit.
In a computer or digital system, it is built with a
limit to the number of bits that can be in a
number.
So the maximum number that can be represented
is limited by the number of bits
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m bits allows representation of 2m numbers,
Ranged 0 through 2m -1
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Numbers ranges with binary
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With 1 bit – 21 = 2 numbers
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With 2 bits – 22 = 4 numbers
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0 and 1
0, 1, 2, 3 (00, 01, 10, 11)
Or in fixed point 0, .5, 1, 1.5
Or in fractional 0, .25, .5, .75
With 3 bits – 23 = 8 numbers (Octal)
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0,1,2,3,4,5,6,7 (000,001,010,011,100,101,110,111)
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Other number ranges
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With 4 bits – Hex – 24 = 16 numbers
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With 8 bits – 256 – commonly referred to as a
byte
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0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
0 through (28-1), or 0 through 255
With 16 bits
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0 through (216-1), or 0 through 65,535 if the 16
bits represent and unsigned number
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Arithmetic Operators
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Arithmetic in any base is according to the
same rules
Just must be careful to only use the r digits of
the radix.
Rules for binary addition are the same as for
base 10.
9/15/09 - L2
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Addition
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Basic – consider the bit by bit possibilities
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Augend
Addend
Sum
0
0
0
0
1
1
1
0
1
1
1
10
1
0
0
1
1 1 1
0 1 1
1 0 1
10 10 11
And including a carry
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Carry
Augend
Addend
Sum
0
0
0
0
0
0
1
1
0
1
0
1
0
1
1
10
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Result carry
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For those instances where the result was 10 or
11, the msb 1 of this addition are a carry into
the next bit position.
Muti-bit example
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Carries
Augend
Addend
Sum
01100
00101
10001
12
5
17
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Result carry
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For those instances where the result was 10 or
11, the msb 1 of this addition are a carry into
the next bit position.
Muti-bit example
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Carries
Augend
Addend
Sum
11000
01100
00101
10001
12
5
17
Copyright 2009 - Joanne DeGroat, ECE, OSU
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Binary Subtraction
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Just like subtraction in any other base
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Minuend
Subtrahand
Difference
10110
- 10010
00100
And when a borrow is needed. Note that the
borrow gives us 2 in the current bit position.
9/15/09 - L2
Copyright 2009 - Joanne DeGroat, ECE, OSU
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Binary subtraction (cont)
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And more ripple -
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Binary Multiplication
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Just like base 10, it is a series of adds
5
x 3
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x 15
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Operations in other bases
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Operations in octal and hex follow the same
rules.
Book provides and example of octal
multiplication
9/15/09 - L2
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Conversion from Decimal
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To binary (the most common)
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Have seen one method so far
A General Procedure
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Step 1 – Separate into integer and fractional parts
if there is a binary point
Repeat: Divide by radix, r, until remainder is <r
The final remainder is the msb.
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Example of conversion
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Convert 5410 to binary
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Another example
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Convert 4210 to binary
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Book also show fractional conversion
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Decimal codes
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We are cavemen at heart and like base 10.
Computer electronics are such that it likes
base 2.
IS THERE A COMPROMISE?
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Binary Coded Decimal (BCD)
Group the binary into groups of 4 bits. Each
group can represent 16 distinct symbols. Use
only the 10 digits.
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BCD example
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56 + 42
Encode as BCD digits
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56 = 0101 0110
42 = 0100 0010
Simply do the binary addition of each BCD
digit
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98 = 1001 1000
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Another example
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46 + 68
Encode as BCD digits
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46 = 0100 0110
68 = 0110 1000
Simply do the binary addition of each BCD
digit
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?? = 1010 1110 but both are out of the
useable BCD representation range
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What to do?
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Starting with least significant digit add 6
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Had the bcd representation 1010 1110
Add 6 or 0110
to the 1110
Getting 10100 and have BCD digit 0100 and a
carry to the next BCD position
WHY 6?
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Add in the carry
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Continue by adding the carry to the next digit
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Had the bcd representation 1010 1110
Add the carry and get 1011 so must again add 6
Or add 0110
to the 1011
Getting 10001 and have BCD digit 0001 and a
carry to the next BCD position
Final answer is 0001 0001 0100 or 114 √
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Copyright 2009 - Joanne DeGroat, ECE, OSU
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Class 2 wrapup
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Covered sections 1-3 and 1-4
Problems for hand in
1-12 and 1-13
Problems for practice
1-19
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Reading for next class: sections 1-5 thru 1-7
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9/15/09 - L2
Copyright 2009 - Joanne DeGroat, ECE, OSU
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