CS61C L02 Number Representation (1) Garcia,Spring 2007? UCB
Lecturer SOE Dan Garcia
www.cs.berkeley.edu/~ddgarcia
inst.eecs.berkeley.edu/~cs61cCS61C,Machine Structures
Lecture #2 – Number Representation
2007-01-19 There is one handout today at the front and
back of the room!
Great book?
The Universal History
of Numbers
by Georges Ifrah
CS61C L02 Number Representation (2) Garcia,Spring 2007? UCB
Great DeCal courses I supervise (2 units)
UCBUGG
UC Berkeley Undergraduate Graphics Group
Thursdays 5:30-7:30pm in 310 Soda
Learn to create a short 3D animation
No prereqs (but they might have too many students,so admission not guaranteed)
http://ucbugg.berkeley.edu
MS-DOS X
Macintosh Software Developers for OS X
Thursdays 5-7pm in 320 Soda
Learn to program the Macintosh and write an
awesome GUI application
No prereqs (other than interest)
http://msdosx.berkeley.edu
CS61C L02 Number Representation (3) Garcia,Spring 2007? UCB
Review
Continued rapid improvement in computing
2X every 2.0 years in memory size;
every 1.5 years in processor speed;
every 1.0 year in disk capacity;
Moore’s Law enables processor
(2X transistors/chip ~1.5 yrs)
5 classic components of all computers
Control Datapath Memory Input Output
Processor
}
CS61C L02 Number Representation (4) Garcia,Spring 2007? UCB
My goal as an instructor
To make your experience in CS61C as enjoyable & informative as possible
Humor,enthusiasm,graphics &
technology-in-the-news in lecture
Fun,challenging projects & HW
Pro-student policies (exam clobbering)
To maintain Cal & EECS standards of
excellence
Your projects & exams will be just as
rigorous as every year,Overall,B- avg
To be an HKN,7.0” man
I know I speak fast when I get excited
about material,I’m told every semester,
Help me slow down when I go toooo fast.
Please give me feedback so I improve!
Why am I not 7.0 for you? I will listen!!
CS61C L02 Number Representation (5) Garcia,Spring 2007? UCB
Putting it all in perspective…
,If the automobile had followed the same development cycle as the computer,
a Rolls-Royce would today cost $100,get a million miles per gallon,
and explode once a year,
killing everyone inside.”
– Robert X,Cringely
CS61C L02 Number Representation (6) Garcia,Spring 2007? UCB
Decimal Numbers,Base 10
Digits,0,1,2,3,4,5,6,7,8,9
Example:
3271 =
(3x103) + (2x102) + (7x101) + (1x100)
CS61C L02 Number Representation (7) Garcia,Spring 2007? UCB
Numbers,positional notation
Number Base B? B symbols per digit:
Base 10 (Decimal),0,1,2,3,4,5,6,7,8,9
Base 2 (Binary),0,1
Number representation,
d31d30,.,d1d0 is a 32 digit number
value = d31? B31 + d30? B30 +,.,+ d1? B1 + d0? B0
Binary,0,1 (In binary digits called,bits”)
0b11010 = 1?24 + 1?23 + 0?22 + 1?21 + 0?20
= 16 + 8 + 2
= 26
Here 5 digit binary # turns into a 2 digit decimal #
Can we find a base that converts to binary easily?
#s often written
0b…
CS61C L02 Number Representation (8) Garcia,Spring 2007? UCB
Hexadecimal Numbers,Base 16
Hexadecimal,
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Normal digits + 6 more from the alphabet
In C,written as 0x… (e.g.,0xFAB5)
Conversion,Binary?Hex
1 hex digit represents 16 decimal values
4 binary digits represent 16 decimal values
1 hex digit replaces 4 binary digits
One hex digit is a,nibble”,Two is a,byte”
Example:
1010 1100 0011 (binary) = 0x_____?
CS61C L02 Number Representation (9) Garcia,Spring 2007? UCB
Decimal vs,Hexadecimal vs,Binary
Examples:
1010 1100 0011 (binary)
= 0xAC3
10111 (binary) = 0001 0111 (binary)
= 0x17
0x3F9 = 11 1111 1001 (binary)
How do we convert between
hex and Decimal?
00 0 000001 1 0001
02 2 001003 3 0011
04 4 0100
05 5 010106 6 0110
07 7 011108 8 1000
09 9 1001
10 A 101011 B 1011
12 C 110013 D 1101
14 E 1110
15 F 1111MEMORIZE!
1100
11 ( 011 (
11 1111 1001 (binary)
CS61C L02 Number Representation (10) Garcia,Spring 2007? UCB
Kilo,Mega,Giga,Tera,Peta,Exa,Zetta,Yotta
Common use prefixes (all SI,except K [= k in SI])
Confusing! Common usage of,kilobyte” means
1024 bytes,but the,correct” SI value is 1000 bytes
Hard Disk manufacturers & Telecommunications are the only computing groups that use SI factors,so
what is advertised as a 30 GB drive will actually only
hold about 28 x 230 bytes,and a 1 Mbit/s connection
transfers 106 bps.
Name Abbr Factor SI size
Kilo K 210 = 1,024 103 = 1,000
Mega M 220 = 1,048,576 106 = 1,000,000
Giga G 230 = 1,073,741,824 109 = 1,000,000,000
Tera T 240 = 1,099,511,627,776 1012 = 1,000,000,000,000
Peta P 250 = 1,125,899,906,842,624 1015 = 1,000,000,000,000,000
Exa E 260 = 1,152,921,504,606,846,976 1018 = 1,000,000,000,000,000,000
Zetta Z 270 = 1,180,591,620,717,411,303,424 1021 = 1,000,000,000,000,000,000,000
Yotta Y 280 = 1,208,925,819,614,629,174,706,176 1024 = 1,000,000,000,000,000,000,000,000
physics.nist.gov/cuu/Units/binary.html
CS61C L02 Number Representation (11) Garcia,Spring 2007? UCB
kibi,mebi,gibi,tebi,pebi,exbi,zebi,yobi
New IEC Standard Prefixes [only to exbi officially]
International Electrotechnical Commission (IEC) in 1999 introduced these to specify binary quantities.
Names come from shortened versions of the
original SI prefixes (same pronunciation) and bi is
short for,binary”,but pronounced,bee”,-(
Now SI prefixes only have their base-10 meaning
and never have a base-2 meaning.
Name Abbr Factor
kibi Ki 210 = 1,024
mebi Mi 220 = 1,048,576
gibi Gi 230 = 1,073,741,824
tebi Ti 240 = 1,099,511,627,776
pebi Pi 250 = 1,125,899,906,842,624
exbi Ei 260 = 1,152,921,504,606,846,976
zebi Zi 270 = 1,180,591,620,717,411,303,424
yobi Yi 280 = 1,208,925,819,614,629,174,706,176
en.wikipedia.org/wiki/Binary_prefix
As of this
writing,this
proposal has
yet to gain
widespread
use…
CS61C L02 Number Representation (12) Garcia,Spring 2007? UCB
What is 234? How many bits addresses (I.e.,what’s ceil log
2 = lg of) 2.5 TiB?
Answer! 2XY means…
X=0? ---
X=1? kibi ~103
X=2? mebi ~106
X=3? gibi ~109
X=4? tebi ~1012
X=5? pebi ~1015
X=6? exbi ~1018
X=7? zebi ~1021
X=8? yobi ~1024
The way to remember #s
Y=0? 1
Y=1? 2
Y=2? 4
Y=3? 8
Y=4? 16
Y=5? 32
Y=6? 64
Y=7? 128
Y=8? 256
Y=9? 512MEMORIZE!
CS61C L02 Number Representation (13) Garcia,Spring 2007? UCB
What to do with representations of numbers?
Just what we do with numbers!
Add them
Subtract them
Multiply them
Divide them
Compare them
Example,10 + 7 = 17
…so simple to add in binary that we can build circuits to do it!
subtraction just as you would in decimal
Comparison,How do you tell if X > Y?
1 0 1 0
+ 0 1 1 1
-------------------------
1 0 0 0 1
11
CS61C L02 Number Representation (14) Garcia,Spring 2007? UCB
Which base do we use?
Decimal,great for humans,especially when
doing arithmetic
Hex,if human looking at long strings of binary numbers,its much easier to convert
to hex and look 4 bits/symbol
Terrible for arithmetic on paper
Binary,what computers use;
you will learn how computers do +,-,*,/
To a computer,numbers always binary
Regardless of how number is written:
32ten == 3210 == 0x20 == 1000002 == 0b100000
Use subscripts,ten”,“hex”,“two” in book,
slides when might be confusing
CS61C L02 Number Representation (15) Garcia,Spring 2007? UCB
BIG IDEA,Bits can represent anything!!
Characters?
26 letters? 5 bits (25 = 32)
upper/lower case + punctuation
7 bits (in 8) (“ASCII”)
standard code to cover all the world’s
languages?8,16,32 bits (“Unicode”)
www.unicode.com
Logical values?
0? False,1? True
colors? Ex:
locations / addresses? commands?
MEMORIZE,N bits? at most 2N things
Red (00) Green (01) Blue (11)
CS61C L02 Number Representation (16) Garcia,Spring 2007? UCB
How to Represent Negative Numbers?
So far,unsigned numbers
Obvious solution,define leftmost bit to be sign!
0? +,1? –
Rest of bits can be numerical value of number
Representation called sign and magnitude(原码）
MIPS uses 32-bit integers,+1ten would be:
0000 0000 0000 0000 0000 0000 0000 0001
And –1ten in sign and magnitude would be:
1000 0000 0000 0000 0000 0000 0000 0001
CS61C L02 Number Representation (17) Garcia,Spring 2007? UCB
Shortcomings of sign and magnitude?
Arithmetic circuit complicated
Special steps depending whether signs are
the same or not
Also,two zeros
0x00000000 = +0ten
0x80000000 = –0ten
What would two 0s mean for programming?
Therefore sign and magnitude abandoned
CS61C L02 Number Representation (18) Garcia,Spring 2007? UCB
Administrivia
Upcoming lectures
Next three lectures,Introduction to C
Lab overcrowding
Remember,you can go to ANY discussion (none,or one that
doesn’t match with lab,or even more than one if you want)
Overcrowded labs - consider finishing at home and getting
checkoffs in lab,or bringing laptop to lab
HW
HW0 due in discussion next week
HW1 due this Wed @ 23:59 PST
HW2 due following Wed @ 23:59 PST
Reading
K&R Chapters 1-6 (lots,get started now!); 1st quiz due Sun!
Soda locks doors @ 6:30pm & on weekends
Look at class website,newsgroup often!
http://inst.eecs.berkeley.edu/~cs61c/
ucb.class.cs61c
CS61C L02 Number Representation (19) Garcia,Spring 2007? UCB
Another try,complement the bits
Example,710 = 001112 –710 = 110002
Called One’s Complement（反码）
Note,positive numbers have leading 0s,
negative numbers have leadings 1s.
00000 00001 01111...
111111111010000,..
What is -00000? Answer,11111
How many positive numbers in N bits?
How many negative numbers?
CS61C L02 Number Representation (20) Garcia,Spring 2007? UCB
Shortcomings of One’s complement?
Arithmetic still a somewhat complicated.
Still two zeros
0x00000000 = +0ten
0xFFFFFFFF = -0ten
Although used for awhile on some
computer products,one’s complement
was eventually abandoned because
another solution was better.
CS61C L02 Number Representation (21) Garcia,Spring 2007? UCB
Standard Negative Number Representation
What is result for unsigned numbers if tried to subtract large number from a small one?
Would try to borrow from string of leading 0s,so result would have a string of leading 1s
3 - 4?00…0011 –00…0100 = 11…1111
With no obvious better alternative,pick
representation that made the hardware simple
As with sign and magnitude,
leading 0s? positive,leading 1s? negative
000000...xxx is ≥ 0,111111...xxx is < 0
except 1…1111 is -1,not -0 (as in sign & mag.)
This representation is Two’s Complement
CS61C L02 Number Representation (22) Garcia,Spring 2007? UCB
2’s Complement Number,line”,N = 5
2N-1 non-negatives
2N-1 negatives
one zero
how many
positives?
00000 00001
00010
11111
11110
10000 0111110001
0 1 2-1-2
-15 -16 15
.
.
.
.
.
.
-3
11101
-411100
00000 00001 01111...
111111111010000,..
CS61C L02 Number Representation (23) Garcia,Spring 2007? UCB
Two’s Complement for N=32
0000,.,0000 0000 0000 0000two = 0ten0000,.,0000 0000 0000 0001
two = 1ten0000,.,0000 0000 0000 0010
two = 2ten.,,
0111,.,1111 1111 1111 1101two = 2,147,483,645ten
0111,.,1111 1111 1111 1110two = 2,147,483,646ten0111,.,1111 1111 1111 1111
two = 2,147,483,647ten1000,.,0000 0000 0000 0000
two = –2,147,483,648ten1000,.,0000 0000 0000 0001
two = –2,147,483,647ten1000,.,0000 0000 0000 0010
two = –2,147,483,646ten.,,
1111,.,1111 1111 1111 1101two = –3ten
1111,.,1111 1111 1111 1110two = –2ten
1111,.,1111 1111 1111 1111two = –1ten
One zero; 1st bit called sign bit
1,extra” negative:no positive 2,147,483,648ten
CS61C L02 Number Representation (24) Garcia,Spring 2007? UCB
Two’s Complement Formula
Can represent positive and negative numbers in terms of the bit value times a power of 2:
d31 x -(231) + d30 x 230 +,.,+ d2 x 22 + d1 x 21 + d0 x 20
Example,1101two
= 1x-(23) + 1x22 + 0x21 + 1x20
= -23 + 22 + 0 + 20
= -8 + 4 + 0 + 1
= -8 + 5
= -3ten
CS61C L02 Number Representation (25) Garcia,Spring 2007? UCB
Two’s Complement shortcut,Negation
Change every 0 to 1 and 1 to 0 (invert or complement),then add 1 to the result
Proof*,Sum of number and its (one’s) complement must be 111...111
two
However,111...111two= -1ten
Let x’?one’s complement representation of x
Then x + x’ = -1?x + x’ + 1 = 0? -x = x’ + 1
Example,-3 to +3 to -3x,1111 1111 1111 1111 1111 1111 1111 1101
twox’,0000 0000 0000 0000 0000 0000 0000 0010
two+1,0000 0000 0000 0000 0000 0000 0000 0011
two()’,1111 1111 1111 1111 1111 1111 1111 1100
two+1,1111 1111 1111 1111 1111 1111 1111 1101
two
You should be able to do this in your head…
*Check out www.cs.berkeley.edu/~dsw/twos_complement.html
CS61C L02 Number Representation (26) Garcia,Spring 2007? UCB
Two’s comp,shortcut,Sign extension
Convert 2’s complement number rep,using n bits to more than n bits
Simply replicate the most significant bit (sign bit) of smaller to fill new bits
2’s comp,positive number has infinite 0s
2’s comp,negative number has infinite 1s
Binary representation hides leading bits;
sign extension restores some of them
16-bit -4ten to 32-bit,
1111 1111 1111 1100two
1111 1111 1111 1111 1111 1111 1111 1100two
CS61C L02 Number Representation (27) Garcia,Spring 2007? UCB
What if too big?
Binary bit patterns above are simply
representatives of numbers,Strictly speaking they are called,numerals”.
Numbers really have an? number of digits
with almost all being same (00…0 or 11…1) except
for a few of the rightmost digits
Just don’t normally show leading digits
If result of add (or -,*,/ ) cannot be
represented by these rightmost HW bits,
overflow is said to have occurred.
00000 00001 00010 1111111110
unsigned
CS61C L02 Number Representation (28) Garcia,Spring 2007? UCB
Peer Instruction Question
X = 1111 1111 1111 1111 1111 1111 1111 1100two
Y = 0011 1011 1001 1010 1000 1010 0000 0000two
A,X > Y (if signed)
B,X > Y (if unsigned)
C,An encoding for Babylonians could have 2N
non-negative numbers w/N bits!
ABC
0,FFF
1,FFT
2,FTF
3,FTT
4,TFF
5,TFT
6,TTF
7,TTT
CS61C L02 Number Representation (29) Garcia,Spring 2007? UCB
Number summary...
We represent,things” in computers as
particular bit patterns,N bits? 2N
Decimal for human calculations,binary for computers,hex to write binary more easily
1’s complement - mostly abandoned
2’s complement universal in computing,cannot avoid,so learn
Overflow,numbers?; computers finite,errors!
00000 00001 01111...
111111111010000,..
00000 00001 01111...
111111111010000,..
Lecturer SOE Dan Garcia
www.cs.berkeley.edu/~ddgarcia
inst.eecs.berkeley.edu/~cs61cCS61C,Machine Structures
Lecture #2 – Number Representation
2007-01-19 There is one handout today at the front and
back of the room!
Great book?
The Universal History
of Numbers
by Georges Ifrah
CS61C L02 Number Representation (2) Garcia,Spring 2007? UCB
Great DeCal courses I supervise (2 units)
UCBUGG
UC Berkeley Undergraduate Graphics Group
Thursdays 5:30-7:30pm in 310 Soda
Learn to create a short 3D animation
No prereqs (but they might have too many students,so admission not guaranteed)
http://ucbugg.berkeley.edu
MS-DOS X
Macintosh Software Developers for OS X
Thursdays 5-7pm in 320 Soda
Learn to program the Macintosh and write an
awesome GUI application
No prereqs (other than interest)
http://msdosx.berkeley.edu
CS61C L02 Number Representation (3) Garcia,Spring 2007? UCB
Review
Continued rapid improvement in computing
2X every 2.0 years in memory size;
every 1.5 years in processor speed;
every 1.0 year in disk capacity;
Moore’s Law enables processor
(2X transistors/chip ~1.5 yrs)
5 classic components of all computers
Control Datapath Memory Input Output
Processor
}
CS61C L02 Number Representation (4) Garcia,Spring 2007? UCB
My goal as an instructor
To make your experience in CS61C as enjoyable & informative as possible
Humor,enthusiasm,graphics &
technology-in-the-news in lecture
Fun,challenging projects & HW
Pro-student policies (exam clobbering)
To maintain Cal & EECS standards of
excellence
Your projects & exams will be just as
rigorous as every year,Overall,B- avg
To be an HKN,7.0” man
I know I speak fast when I get excited
about material,I’m told every semester,
Help me slow down when I go toooo fast.
Please give me feedback so I improve!
Why am I not 7.0 for you? I will listen!!
CS61C L02 Number Representation (5) Garcia,Spring 2007? UCB
Putting it all in perspective…
,If the automobile had followed the same development cycle as the computer,
a Rolls-Royce would today cost $100,get a million miles per gallon,
and explode once a year,
killing everyone inside.”
– Robert X,Cringely
CS61C L02 Number Representation (6) Garcia,Spring 2007? UCB
Decimal Numbers,Base 10
Digits,0,1,2,3,4,5,6,7,8,9
Example:
3271 =
(3x103) + (2x102) + (7x101) + (1x100)
CS61C L02 Number Representation (7) Garcia,Spring 2007? UCB
Numbers,positional notation
Number Base B? B symbols per digit:
Base 10 (Decimal),0,1,2,3,4,5,6,7,8,9
Base 2 (Binary),0,1
Number representation,
d31d30,.,d1d0 is a 32 digit number
value = d31? B31 + d30? B30 +,.,+ d1? B1 + d0? B0
Binary,0,1 (In binary digits called,bits”)
0b11010 = 1?24 + 1?23 + 0?22 + 1?21 + 0?20
= 16 + 8 + 2
= 26
Here 5 digit binary # turns into a 2 digit decimal #
Can we find a base that converts to binary easily?
#s often written
0b…
CS61C L02 Number Representation (8) Garcia,Spring 2007? UCB
Hexadecimal Numbers,Base 16
Hexadecimal,
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Normal digits + 6 more from the alphabet
In C,written as 0x… (e.g.,0xFAB5)
Conversion,Binary?Hex
1 hex digit represents 16 decimal values
4 binary digits represent 16 decimal values
1 hex digit replaces 4 binary digits
One hex digit is a,nibble”,Two is a,byte”
Example:
1010 1100 0011 (binary) = 0x_____?
CS61C L02 Number Representation (9) Garcia,Spring 2007? UCB
Decimal vs,Hexadecimal vs,Binary
Examples:
1010 1100 0011 (binary)
= 0xAC3
10111 (binary) = 0001 0111 (binary)
= 0x17
0x3F9 = 11 1111 1001 (binary)
How do we convert between
hex and Decimal?
00 0 000001 1 0001
02 2 001003 3 0011
04 4 0100
05 5 010106 6 0110
07 7 011108 8 1000
09 9 1001
10 A 101011 B 1011
12 C 110013 D 1101
14 E 1110
15 F 1111MEMORIZE!
1100
11 ( 011 (
11 1111 1001 (binary)
CS61C L02 Number Representation (10) Garcia,Spring 2007? UCB
Kilo,Mega,Giga,Tera,Peta,Exa,Zetta,Yotta
Common use prefixes (all SI,except K [= k in SI])
Confusing! Common usage of,kilobyte” means
1024 bytes,but the,correct” SI value is 1000 bytes
Hard Disk manufacturers & Telecommunications are the only computing groups that use SI factors,so
what is advertised as a 30 GB drive will actually only
hold about 28 x 230 bytes,and a 1 Mbit/s connection
transfers 106 bps.
Name Abbr Factor SI size
Kilo K 210 = 1,024 103 = 1,000
Mega M 220 = 1,048,576 106 = 1,000,000
Giga G 230 = 1,073,741,824 109 = 1,000,000,000
Tera T 240 = 1,099,511,627,776 1012 = 1,000,000,000,000
Peta P 250 = 1,125,899,906,842,624 1015 = 1,000,000,000,000,000
Exa E 260 = 1,152,921,504,606,846,976 1018 = 1,000,000,000,000,000,000
Zetta Z 270 = 1,180,591,620,717,411,303,424 1021 = 1,000,000,000,000,000,000,000
Yotta Y 280 = 1,208,925,819,614,629,174,706,176 1024 = 1,000,000,000,000,000,000,000,000
physics.nist.gov/cuu/Units/binary.html
CS61C L02 Number Representation (11) Garcia,Spring 2007? UCB
kibi,mebi,gibi,tebi,pebi,exbi,zebi,yobi
New IEC Standard Prefixes [only to exbi officially]
International Electrotechnical Commission (IEC) in 1999 introduced these to specify binary quantities.
Names come from shortened versions of the
original SI prefixes (same pronunciation) and bi is
short for,binary”,but pronounced,bee”,-(
Now SI prefixes only have their base-10 meaning
and never have a base-2 meaning.
Name Abbr Factor
kibi Ki 210 = 1,024
mebi Mi 220 = 1,048,576
gibi Gi 230 = 1,073,741,824
tebi Ti 240 = 1,099,511,627,776
pebi Pi 250 = 1,125,899,906,842,624
exbi Ei 260 = 1,152,921,504,606,846,976
zebi Zi 270 = 1,180,591,620,717,411,303,424
yobi Yi 280 = 1,208,925,819,614,629,174,706,176
en.wikipedia.org/wiki/Binary_prefix
As of this
writing,this
proposal has
yet to gain
widespread
use…
CS61C L02 Number Representation (12) Garcia,Spring 2007? UCB
What is 234? How many bits addresses (I.e.,what’s ceil log
2 = lg of) 2.5 TiB?
Answer! 2XY means…
X=0? ---
X=1? kibi ~103
X=2? mebi ~106
X=3? gibi ~109
X=4? tebi ~1012
X=5? pebi ~1015
X=6? exbi ~1018
X=7? zebi ~1021
X=8? yobi ~1024
The way to remember #s
Y=0? 1
Y=1? 2
Y=2? 4
Y=3? 8
Y=4? 16
Y=5? 32
Y=6? 64
Y=7? 128
Y=8? 256
Y=9? 512MEMORIZE!
CS61C L02 Number Representation (13) Garcia,Spring 2007? UCB
What to do with representations of numbers?
Just what we do with numbers!
Add them
Subtract them
Multiply them
Divide them
Compare them
Example,10 + 7 = 17
…so simple to add in binary that we can build circuits to do it!
subtraction just as you would in decimal
Comparison,How do you tell if X > Y?
1 0 1 0
+ 0 1 1 1
-------------------------
1 0 0 0 1
11
CS61C L02 Number Representation (14) Garcia,Spring 2007? UCB
Which base do we use?
Decimal,great for humans,especially when
doing arithmetic
Hex,if human looking at long strings of binary numbers,its much easier to convert
to hex and look 4 bits/symbol
Terrible for arithmetic on paper
Binary,what computers use;
you will learn how computers do +,-,*,/
To a computer,numbers always binary
Regardless of how number is written:
32ten == 3210 == 0x20 == 1000002 == 0b100000
Use subscripts,ten”,“hex”,“two” in book,
slides when might be confusing
CS61C L02 Number Representation (15) Garcia,Spring 2007? UCB
BIG IDEA,Bits can represent anything!!
Characters?
26 letters? 5 bits (25 = 32)
upper/lower case + punctuation
7 bits (in 8) (“ASCII”)
standard code to cover all the world’s
languages?8,16,32 bits (“Unicode”)
www.unicode.com
Logical values?
0? False,1? True
colors? Ex:
locations / addresses? commands?
MEMORIZE,N bits? at most 2N things
Red (00) Green (01) Blue (11)
CS61C L02 Number Representation (16) Garcia,Spring 2007? UCB
How to Represent Negative Numbers?
So far,unsigned numbers
Obvious solution,define leftmost bit to be sign!
0? +,1? –
Rest of bits can be numerical value of number
Representation called sign and magnitude(原码）
MIPS uses 32-bit integers,+1ten would be:
0000 0000 0000 0000 0000 0000 0000 0001
And –1ten in sign and magnitude would be:
1000 0000 0000 0000 0000 0000 0000 0001
CS61C L02 Number Representation (17) Garcia,Spring 2007? UCB
Shortcomings of sign and magnitude?
Arithmetic circuit complicated
Special steps depending whether signs are
the same or not
Also,two zeros
0x00000000 = +0ten
0x80000000 = –0ten
What would two 0s mean for programming?
Therefore sign and magnitude abandoned
CS61C L02 Number Representation (18) Garcia,Spring 2007? UCB
Administrivia
Upcoming lectures
Next three lectures,Introduction to C
Lab overcrowding
Remember,you can go to ANY discussion (none,or one that
doesn’t match with lab,or even more than one if you want)
Overcrowded labs - consider finishing at home and getting
checkoffs in lab,or bringing laptop to lab
HW
HW0 due in discussion next week
HW1 due this Wed @ 23:59 PST
HW2 due following Wed @ 23:59 PST
Reading
K&R Chapters 1-6 (lots,get started now!); 1st quiz due Sun!
Soda locks doors @ 6:30pm & on weekends
Look at class website,newsgroup often!
http://inst.eecs.berkeley.edu/~cs61c/
ucb.class.cs61c
CS61C L02 Number Representation (19) Garcia,Spring 2007? UCB
Another try,complement the bits
Example,710 = 001112 –710 = 110002
Called One’s Complement（反码）
Note,positive numbers have leading 0s,
negative numbers have leadings 1s.
00000 00001 01111...
111111111010000,..
What is -00000? Answer,11111
How many positive numbers in N bits?
How many negative numbers?
CS61C L02 Number Representation (20) Garcia,Spring 2007? UCB
Shortcomings of One’s complement?
Arithmetic still a somewhat complicated.
Still two zeros
0x00000000 = +0ten
0xFFFFFFFF = -0ten
Although used for awhile on some
computer products,one’s complement
was eventually abandoned because
another solution was better.
CS61C L02 Number Representation (21) Garcia,Spring 2007? UCB
Standard Negative Number Representation
What is result for unsigned numbers if tried to subtract large number from a small one?
Would try to borrow from string of leading 0s,so result would have a string of leading 1s
3 - 4?00…0011 –00…0100 = 11…1111
With no obvious better alternative,pick
representation that made the hardware simple
As with sign and magnitude,
leading 0s? positive,leading 1s? negative
000000...xxx is ≥ 0,111111...xxx is < 0
except 1…1111 is -1,not -0 (as in sign & mag.)
This representation is Two’s Complement
CS61C L02 Number Representation (22) Garcia,Spring 2007? UCB
2’s Complement Number,line”,N = 5
2N-1 non-negatives
2N-1 negatives
one zero
how many
positives?
00000 00001
00010
11111
11110
10000 0111110001
0 1 2-1-2
-15 -16 15
.
.
.
.
.
.
-3
11101
-411100
00000 00001 01111...
111111111010000,..
CS61C L02 Number Representation (23) Garcia,Spring 2007? UCB
Two’s Complement for N=32
0000,.,0000 0000 0000 0000two = 0ten0000,.,0000 0000 0000 0001
two = 1ten0000,.,0000 0000 0000 0010
two = 2ten.,,
0111,.,1111 1111 1111 1101two = 2,147,483,645ten
0111,.,1111 1111 1111 1110two = 2,147,483,646ten0111,.,1111 1111 1111 1111
two = 2,147,483,647ten1000,.,0000 0000 0000 0000
two = –2,147,483,648ten1000,.,0000 0000 0000 0001
two = –2,147,483,647ten1000,.,0000 0000 0000 0010
two = –2,147,483,646ten.,,
1111,.,1111 1111 1111 1101two = –3ten
1111,.,1111 1111 1111 1110two = –2ten
1111,.,1111 1111 1111 1111two = –1ten
One zero; 1st bit called sign bit
1,extra” negative:no positive 2,147,483,648ten
CS61C L02 Number Representation (24) Garcia,Spring 2007? UCB
Two’s Complement Formula
Can represent positive and negative numbers in terms of the bit value times a power of 2:
d31 x -(231) + d30 x 230 +,.,+ d2 x 22 + d1 x 21 + d0 x 20
Example,1101two
= 1x-(23) + 1x22 + 0x21 + 1x20
= -23 + 22 + 0 + 20
= -8 + 4 + 0 + 1
= -8 + 5
= -3ten
CS61C L02 Number Representation (25) Garcia,Spring 2007? UCB
Two’s Complement shortcut,Negation
Change every 0 to 1 and 1 to 0 (invert or complement),then add 1 to the result
Proof*,Sum of number and its (one’s) complement must be 111...111
two
However,111...111two= -1ten
Let x’?one’s complement representation of x
Then x + x’ = -1?x + x’ + 1 = 0? -x = x’ + 1
Example,-3 to +3 to -3x,1111 1111 1111 1111 1111 1111 1111 1101
twox’,0000 0000 0000 0000 0000 0000 0000 0010
two+1,0000 0000 0000 0000 0000 0000 0000 0011
two()’,1111 1111 1111 1111 1111 1111 1111 1100
two+1,1111 1111 1111 1111 1111 1111 1111 1101
two
You should be able to do this in your head…
*Check out www.cs.berkeley.edu/~dsw/twos_complement.html
CS61C L02 Number Representation (26) Garcia,Spring 2007? UCB
Two’s comp,shortcut,Sign extension
Convert 2’s complement number rep,using n bits to more than n bits
Simply replicate the most significant bit (sign bit) of smaller to fill new bits
2’s comp,positive number has infinite 0s
2’s comp,negative number has infinite 1s
Binary representation hides leading bits;
sign extension restores some of them
16-bit -4ten to 32-bit,
1111 1111 1111 1100two
1111 1111 1111 1111 1111 1111 1111 1100two
CS61C L02 Number Representation (27) Garcia,Spring 2007? UCB
What if too big?
Binary bit patterns above are simply
representatives of numbers,Strictly speaking they are called,numerals”.
Numbers really have an? number of digits
with almost all being same (00…0 or 11…1) except
for a few of the rightmost digits
Just don’t normally show leading digits
If result of add (or -,*,/ ) cannot be
represented by these rightmost HW bits,
overflow is said to have occurred.
00000 00001 00010 1111111110
unsigned
CS61C L02 Number Representation (28) Garcia,Spring 2007? UCB
Peer Instruction Question
X = 1111 1111 1111 1111 1111 1111 1111 1100two
Y = 0011 1011 1001 1010 1000 1010 0000 0000two
A,X > Y (if signed)
B,X > Y (if unsigned)
C,An encoding for Babylonians could have 2N
non-negative numbers w/N bits!
ABC
0,FFF
1,FFT
2,FTF
3,FTT
4,TFF
5,TFT
6,TTF
7,TTT
CS61C L02 Number Representation (29) Garcia,Spring 2007? UCB
Number summary...
We represent,things” in computers as
particular bit patterns,N bits? 2N
Decimal for human calculations,binary for computers,hex to write binary more easily
1’s complement - mostly abandoned
2’s complement universal in computing,cannot avoid,so learn
Overflow,numbers?; computers finite,errors!
00000 00001 01111...
111111111010000,..
00000 00001 01111...
111111111010000,..
