CMPE12 – Notes chapter 3
Integers
(Textbook Chapter 2)
Positional Number System
(again)
•  Select a number as the base, b, ex:
•  Define an alphabet of b-1 symbols plus a
symbol for zero to represent all numbers…
•  Use an ordered sequence of 2 or more
digits to represent numbers…
•  The represented number is the sum of all
digits, each multiplied by b to the power of
the digit’s position p
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Base conversion
Three cases:
I.  From any base b to base 10
II.  From base 10 to any base b
III. From any base b to any other base c
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I. From base b to base 10
Use sum of products:
n-1
Value = i=0
Σ (di b i)
•  Ex: 123410 = 123410
•  Ex: 12345 = (625+250+75+4)10 = 95910
•  Ex: 1001012 = (32+4+1)10 = 3710
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II. From base 10 to base b
•  Use successive divisions. Divide by the base
and the remainder is the least significant
digit that is not yet
i
N
q
r
found
Trivial example:
436710 = ?10
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0 4367 436
7
1
2
436
43
43
4
6
3
3
4
0
4
3-5
From base 10 to base b
•  A nontrivial example. Ex: 12310 to base 3
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i
N
q
r
0
1
2
3
4
5
6
7
123
41
13
4
1
41
13
4
1
0
0
2
1
1
1
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III. From base b to base c
•  The easiest way is to convert from base b to
base 10 first, and then from 10 to c
•  Or, in some cases, it’s easier to use base 2
as the intermediate base (we’ll see them soon)
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Positional Addition
Examples:
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Positional multiplication
1135
*425
---------------------------
Not as hard as it looks.
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Numbers for computers
•  There are many ways to “represent” a number
•  Representation does not affect computation result
•  Representation affects difficulty of computing results
•  Computers need a representation that works with
(fast) electronic circuits
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Binary Number System
• Base (radix): 2
• Alphabet: 0, 1
• Binary Digits, or bits
• 10012 =
• 110002 =
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Octal Number System
•  Base (radix): 8
•  Alphabet: 0 – 7
•  3458 = 3*64+4*8+5 = 22910
•  10018 = 51310
•  In C, octal numbers are represented with a
leading 0. Ex:
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Hexadecimal Number System
•  Base (radix): 16
•  Alphabet: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C,
D, E, F
Hex
Decimal
A
10
B
11
C
12
D
13
E
14
F
15
•  In C: leading “0x” (e. g. 0xa3)
•  In LC-3: leading “x” (for example, “x3000”)
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Examples of converting
hex numbers to decimal
A316 =
3E816 =
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Decimal -> Binary Conversion
•  Method 1:
• Divide decimal value by 2 until the value is 0
•  Ex:
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Decimal -> Binary Conversion
• Method 2:
• Know your powers of two and subtract
• Ex: 61
20
1
21
2
22
4
23
8
24
16
25
32
26
64
27
128
28
256
29
512
210
1024
• What is the biggest power of two that fits?
• What is the remainder?
• What fits?
• What is the remainder?
• …
• What is the binary representation?
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Binary -> Octal Conversion
• Group into triplets starting at least significant bit
• Add leading 0’s if needed (why not trailing?)
• Write 1 octal digit for each group of three binary
digits
•  Examples:
100 010 111 (binary)
(octal)
10 101 110 (binary)
(octal)
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Octal -> Binary Conversion
It is simple, just write down the 3-bit binary
code for each octal digit
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Octal
Binary
0
000
1
001
2
010
3
011
4
100
5
101
6
110
7
111
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Binary -> Hex Conversion
• Group into 4’s starting at least significant bit
• Add leading 0’s if needed
• Write 1 hex digit for each group
• Ex:
0x
0x
1001 1110 0111 00002 =
0001
1111 1010 00112 =
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Hex -> Binary Conversion
Again, simply write down the 4-bit binary code
for each hex digit
Ex:
0x
3
9
c
8 =
2
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Conversion Table
Decimal
Hexadecimal
Octal
Binary
0
0
0
0000
1
1
1
0001
2
2
2
0010
3
3
3
0011
4
4
4
0100
5
5
5
0101
6
6
6
0110
7
7
7
0111
8
8
10
1000
9
9
11
1001
10
A
12
1010
11
B
13
1011
12
C
14
1100
13
D
15
1101
14
E
16
1110
15
F
17
1111
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Hex and Octal can be used to
represent Binary!
Because A53216 is less error prone and is
easier to write than 10100101001100102 !
So when I am talking about data in a register
or a memory address as a base 16 number it
really represents a base 2 number.
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Largest number
What is M, the largest number that we can
represent on n digits
• In base 10?
• In base 2?
• In octal?
• In hex?
• In base 7?
• In base b?
How many different numbers can we represent with
n digits in base b?
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Binary addition examples
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Unsigned binary subtraction
Same principle as in decimal (or any base b): ripple
borrow. Ex:
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Unsigned binary multiplication
Same principle as in decimal (or any base b): sum of
partial products. Ex:
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What about Negative Numbers?
Until now we have been talking
about “unsigned” integers.
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Signed numbers: in “human”
•  Most humans precede number with “-” (e.g.,
-2000)
•  Accountants, however, use parentheses: (2000)
or color 2000
•  Called Sign-and-magnitude format
•  Example: -100010 in hex?
100010
= 3 x 162 + e x 161 + 8 x 160
= -3E816
If there is no sign, it is assumed positive
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Signed numbers: in binary
Problem: There is no voltage specifically for -
Four main choices:
• 
• 
• 
• 
Sign-and-magnitude
One’s complement
Two’s complement
Biased notation
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Sign-and-magnitude (without a
minus sign)
•  How does it work?
x
y
000
001
•  Advantages
–  Intuitive
–  Easy to invert the sign
•  Disadvantages:
–  Add and sub must compare sign
and magnitude of the operands
–  Two representations of zero
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010
011
100
101
110
111
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Sign-and-magnitude addition
and subtraction
•  Depends on sign of operands:
–  Same sign: add the magnitudes, keep the sign
–  Different: subtract smaller from larger, keep
sign of larger
•  Ex: A+B, with A = 6, B = -10, on 5 bits
Complicated and slow
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Sign-and-magnitude mapping
Number
represented
Number 000
encoded
001
010
011
100
101
110
111
Range on
n bits: from -2n-1-1
to 2n-1-1
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One’s complement
•  How does it work?
x
y
000
001
010
•  Advantages
–  Still intuitive…
–  Easy to invert the sign
•  Disadvantages:
- Add and sub require a special
operation (end-around carry)
–  Two representations of zero
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011
100
101
110
111
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One’s complement addition and
subtraction
Add/subtract normally, add 1 if carry-out
•  Ex: A+B, with A = 3, B = 2, on 4 bits.
•  Ex: A+B, with A = -3, B = -2.
•  Ex: A+B, with A = -3, B = 2.
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One’s complement mapping
Number
represented
Number 000
encoded
001
010
011
100
101
110
111
Range on
n bits:
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Two’s complement
•  How does it work?
x
y
000
001
•  Advantages
–  Add/sub are simple (and easy in hw)
–  Subtraction only requires one adder
–  Single zero
•  Disadvantages:
–  Asymmetric range
–  Not intuitive at first for humans.
010
011
100
101
110
111
This is what today’s computers use!
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Finding the Additive Inverse in
2’s complement notation
•  Often called “taking the 2’s Complement”
•  Process is used to
–  Convert an unsigned integer to a negative 2’s complement
number.
–  Find the absolute value of a negative 2’s complement
number
–  First step to convert to signed decimal from a negative 2’s
complement integer.
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Finding the Additive Inverse in
2’s complement notation
•  Method 1: “flip the bits and add one”
•  Method 2: “keep all the leftmost zeros and
the first one, flip all other bits”
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Finding the Additive Inverse in
2’s complement notation
•  Method 2: “keep all the leftmost zeros and
the first one, flip all other bits”
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More Examples
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Converting from Two’s
Complement to Decimal
•  If Positive?
•  If Negative?
1. Find its additive inverse (sometimes
called “taking the two’s complement of”).
2. Convert to Decimal
3. Insert negative sign.
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Examples
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Addition and subtraction in
two’s complement
Addition: normal binary addition
•  Ex: A+B, with A = 2, B = 1, on 3 bits
Subtraction: convert to adding the negative (two’s
complement)
•  Ex: A-B, with A = 1, B = 2, on 3 bits
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More Examples
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Overflow
•  Numbers are stored in some form of memory (or
registers) and the memory is organized with limits
on the number of bits available to represent the
number.
•  This leads to the issue of what to do when you
perform an operation and your answer is outside of
the range that can be represented by the number
of bits available.
•  In a computer this is called overflow.
•  Overflow depends on the representation’s precision
(number of bits).
( Next slide for example)
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Overflow Continued
•  Ex: unsigned binary integer addition to be stored
in a 32 bit register: A32616+900016
All computer systems have a procedure to
deal with overflow.
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2’s Complement: Overflow
•  Overflow detection is easy! When adding two operands of
the same sign and the sign of the sum must be the same!
Ex: 0100 + 0101 with 4 bits of precision:
•  Ex: 1010 + 1001 with 4 bits:
•  NOTE: in two’s complement, there can’t be overflow with
operands of different sign. Why?
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Representation is too small for
the register
Unsigned Integers: how do you put
1010 into a 16 bit register?
2’s Complement: Sign-Extension.
“Extend” the sign bit out to the end
of the register
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Examples
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Two’s complement mapping
-2
1
2
3
-4
Number 000
encoded
001
010
011
100
101
110
100
101
110
111
000
001
010
011
-3
-2
-1
0
1
2
3
Number
represented
0
-4
-3
-1
111
Range on
n bits:-2n-1 < x < 2n-1-1
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Integer data types
•  Byte: ordered set of 8 bits
•  Nybble: ordered set of 4 bits
•  Word: ordered set of…
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Character Data type
7-bit ASCII (American Standard
Code for Information Interchange) table
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Exercises
due Friday January 28
• 
• 
• 
• 
Ex 2.1 to 2.8
Ex 2.9 to 2.15, 2.17
Ex 2.28, 2.30, 2.34
Ex 2.37, 2,38
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