Number Representations and Computer Arithmetic. CS 21a 9/23/02 Odds and Ends Slide 2 © Luis F. G. Sarmenta and John Paul Vergara, Ateneo de Manila University

Number Representations andComputer Arithmetic

Odds and EndsSlide 2

© Luis F. G. Sarmenta and John Paul Vergara, Ateneo de Manila University

CS 21a9/23/02

“Computing” with Computers

Back in the “old days” (WW II time), the word “computer” meant this

• “Computers” were used to compute things such as trajectory tables, etc.– redundant

“computers” for reliability and speed• Quite effective!

– new computers were tested against human computers



CS 21a9/23/02

Of course, we don’t do things that way anymore!

But HOW do (electronic) computers compute?

The trick is to use BINARY numbers use 1’s and 0’s corresponds on

current/voltage being ON or OFF

easy to implement resilient against noise

“Computing” with Computers



CS 21a9/23/02

Java types: int and double

int range:-2,147,483,648 to 2,147,483,647

double range: 4.94e-324 to 1.80e+308

These ranges depend on: Storage size Internal data representation



CS 21a9/23/02

Review: Decimal Numbers Integer Representation

number is sum of DIGIT * “place value”

3 7 9 2 + 0 5 3 1

Adding two decimal numbers add by “place value”, one digit at a time

2

100

d0

9

101

d1

7

102

d2

3

103

d3

0

104

d4

0

105

d5

0

106

d6

0

107

d7

379210= 3 103 + 7 102 + 9 101 + 2 100

= 3000 + 700 + 90 + 2

3792 + 531

???

Range0 to 10n - 1

1 “carry 1” because9+3 = 12

0 4 3 2 3



CS 21a9/23/02

Binary Numbers

(Unsigned) Binary Integer Representation “base” of place values is 2, not 10

0

20

b0

0

21

b1

1

22

b2

0

23

b3

0

24

b4

1

25

b5

1

26

b6

0

27

b7

011001002 = 26 + 25 + 22

= 64 + 32 + 4 = 10010

Range0 to 2n - 1

• Humans can naturally count up to 10 values,

• But computers can count only up to 2 values (OFF and ON, or 0 and 1)

aka “0b01100100”



CS 21a9/23/02

Converting from Binary to Decimal

20 = 121 = 222 = 423 = 824 = 1625 = 3226 = 6427 = 12828 = 25629 = 512210 = 1,024

or “1K”

VERY USEFUL trick for a CS/MIS person … Memorize powers of 2 from 20 up to 210

Lets you approximate any power of 2 “1 KB” is actually 1,024 bytes, not 1000 bytes “1 MB” is 1K*1KB = 220 bytes

= 1,048,576 bytes = approximately 1 million bytes

“1 GB” is 1K * 1MB = 230 bytes = (approx 1 billion) Example 1: what is 216?

26 * 210 = 64 * 1024 = 64 K = 65,536 Example 2:

The Pentium processor does integer math with 32-bit numbers. What’s the highest unsigned number it can handle (approximately)?

range = 2n-1 = 232 - 1232 = 22 * 230

= 4 * 1 G = approximately 4 billion = actually 4*1024*1024*1024 = 4,294,967,296 (minus 1)



CS 21a9/23/02

20 = 121 = 222 = 423 = 824 = 1625 = 3226 = 6427 = 12828 = 25629 = 512210 = 1,024

or “1K”

More practice

0b00000010

0b00110001

0b00001110

0b10110001

Converting from Binary to Decimal



CS 21a9/23/02

From Decimal to Binary

General rule: Divide and write remainderfrom right to left

Why this works remainder gives bit 0 odd numbers have a 1 in bit 0

Note that this rule works in converting decimal to any base

e.g., HEX numbers (more later)

3792 / 2 = 1896 rem 01896 / 2 = 948 rem 0 948 / 2 = 474 rem 0 474 / 2 = 237 rem 0 237 / 2 = 118 rem 1 118 / 2 = 59 rem 0 59 / 2 = 29 rem 1 29 / 2 = 14 rem 1 14 / 2 = 7 rem 0 7 / 2 = 3 rem 1 3 / 2 = 1 rem 1 1 / 2 = 0 rem 1

0b111011010000= 211+210+29+27+26+24

= 3792



CS 21a9/23/02

Binary Arithmetic

1 1 1 0 + 0 1 1 1

Adding two binary numbers same, but “1 + 1 = 10”

14 + 7

21

3 7 9 2 + 0 5 3 1

Adding two decimal numbers

3792 + 531

???

1 “carry 1” because9+3 = 12

0 4 3 2 3

1 1 1 “carry 1” because1+1 = 10

1 0 1 0 1



CS 21a9/23/02

Binary Arithmetic

A bits 1 1 1 0B bits + 0 1 1 1

In general: Add up to 3 bits at a

time per place value A and B “carry in”

Output 2 bits at a time sum bit for that place

value “carry out” bit

(becomes carry-in of next bit)

carry-in bits 1 1 1 0 0

sum bits 0 1 0 1carry-out bits 1 1 1 1 0



CS 21a9/23/02

So what about negative values?

In math, it is easy to represent negative values

Just put a negative sign (-) prefix In computers, we extend the binary notation

in order to support signed values We can use the following 3 methods:

Sign and magnitude (using the Most Significant Bit or MSB)

1’s complement 2’s complement



CS 21a9/23/02

Signed Integers Sign-and-Magnitude

MSB represents sign bit (0 for positive, 1 for negative)

has 2 “zeroes” 1’s complement

flip bits easy to do subtraction STILL has 2 “zeros”

2’s complement flip bits, then add 1 easy subtraction

(just negate, then add) has only 1 zero Another interpretation:

add place values as before, except that MSB is negative(i.e., MSB is place value is 2n-1)

210 0 010

-210 1 010

010 0 000

-010 1 000

210 0 010

-210 1 101

010 0 000

-010 1 111

210 0 010

-210 1 110

010 0 000

-010 0 000



CS 21a9/23/02

Binary Subtraction 1’s complement subtraction: flip the negative

0 1 0 1 +1 1 0 0

1 0 0 0 1

5 - 3 = 2

0101 00111100

flip

-3 in 1’s complement form

1 Add the carry overflow

0 0 1 1 +1 0 1 0

1 1 0 1

3 - 5 = -2

0011


01011010

flip

-2

0 0 1 0

2



CS 21a9/23/02

Binary Subtraction 2’s complement subtraction: flip then add 1

0 1 0 1 +1 1 0 1

1 0 0 1 0

5 - 3 = 2

0101 001111001101

flip+1


2ignoreoverflow

0 0 1 1 +1 0 1 1

1 1 1 0

3 - 5 = -2

0011


010110101011

flip+1

-200010010

flip+1

(flip+1 also gives positive of negative number)

2



CS 21a9/23/02

Range of binary numbers

Java (and most computer platforms today) use 2’s comp.

Hence the range for the different int types

byte (8 bits): ? short (16 bits): ? int (32 bits): -2,147,483,648 to

2,147,483,647 long (64 bits): ?



CS 21a9/23/02

Hexadecimal (Hex) Numbers Base 16 Each hex digit goes

from 0-9, then A-F Hex is a convenient

shortform for binary 4 bits = 1 hex digit

(aka nibble) 1 byte = 8 bits = 2 hex digits Another useful trick:

memorize binary of 0 to F Addition and conversion

to/from decimal is similar except use base 16 instead of 2

0

20

b0

0

21

b1

1

22

b2

0

23

b3

0

24

b4

1

25

b5

1

26

b6

0

27

b7

6416 = 6*1610 + 4 = 10010

aka “0x64”



CS 21a9/23/02

Some Exercises

What’s the range of an n-bit sign-mag integer? How about 1’s comp? 2’s comp?

Convert the ff signed 8-bit values to decimal 0b01010101, 0b1110111, 0x14, 0x41

Convert the ff to 16-bit binary and hex numbers: 413, 39, 1045, -3, -124, -134

Add the following pairs 0b00001011 + 0b00100100, 0x3F + 0x2F

If 0x7F and 0x32 are signed 8-bit integers, what’s wrong with adding them and storing the result in a byte?

Puzzle: How can I use my 10 fingers to count up to 1000?

Documents

Number Representations and Computer Arithmetic. CS 21a 9/23/02 Odds and Ends Slide 2 © Luis F. G. Sarmenta and John Paul Vergara, Ateneo de Manila University