BITS,BINARY AND LOGIC

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FLOATING POINT

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HEXADECIMAL

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0 0111 100

UNSIGNED INTEGER

CONVERTING LOGIC OPERATION EXITMAIN MENU

1’S COMPLIMENT

L PIF 0 0111 100

UNSIGNED INTEGER


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11 100

+

1’S COMPLIMENT

2’S COMPLIMENT

0 011UNSIGNED INTEGER


0 1 1 0

2023

22 21

Mostsignificant

Leastsignificant

n 2n

0 1

1 2

2 4

3 8

4 16

5 32

6 64

7 128

8 256

9 512

10 1024


1 1

=

2023 22 21 +++

4 2+

6=

0 0

0 1 1 0

202322 21

ONLY 1 (CIRCLE) WAS COUNT

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663

22

01

12 1

00110

READ FROM DOWN TO UP


1 010

0 100+SIGNED

INTEGER

ANSWER

1 100SIGNED INTEGER


10 010

+

1’S COMPLIMENT

2’S COMPLIMENT

1 101NEGATIVE INTEGER


1 0001 011

-1

ANSWER

1 1005


1 0111 011

OR

ANSWER

1 010

ORuseful for setting bitsOR with zero = no changeOR with one = 1

1 0001 000

AND

ANSWER

1 110

ANDuseful for clearing bitsAND with zero = 0AND with one = no change

• It is often convenient to write binary (base-2) numbersas hexadecimal (base-16) numbers instead.– fewer digits -- four bits per hex digit– less error prone -- easy to corrupt long string of 1’s

and 0’s

HEXADECIMAL NOTATION


Binary Hex Decimal

0000 0 0

0001 1 1

0010 2 2

0011 3 3

0100 4 4

0101 5 5

0110 6 6

0111 7 7

Binary Hex Decimal

1000 8 8

1001 9 9

1010 A 10

1011 B 11

1100 C 12

1101 D 13

1110 E 14

1111 F 15



Every four bits is a hex digit.– start grouping from right-hand side– Is refer to hexamdecimal table

1101 0001 1010 1111

FA1D



• Large values: 6.023 x 1023 -- requires 79 bits• Small values: 6.626 x 10-34 -- requires >110 bits

• Use equivalent of “scientific notation”: F x 2E

• Need to represent F (fraction), E (exponent), and sign.• IEEE 754 Floating-Point Standard (32-bits):

S Exponent Fraction

1b 8b 23b

0exponent,2fraction.0)1(

254exponent1,2fraction.1)1(126

127exponent

S

S

N

N

FLOATING POINT

00101000.101 (40.625)

+ 11111110.110 (-1.25)

00100111.011 (39.375)

2-1 = 0.52-2 = 0.252-3 = 0.125

How can we represent fractions?Use a “binary point” to separate positivefrom negative powers of two -- just like “decimal point.”2’s comp addition and subtraction still work.

if binary points are aligned


2-22

Single-precision IEEE floating point number:

1 01111110 10000000000000000000000

sign exponent fractionSign is 1 – number is negative.Exponent field is 01111110 = 126 (decimal).Fraction is 0.100000000000… = 0.5 (decimal).

Value = -1.5 x 2(126-127) = -1.5 x 2-1 = -0.75.

FLOATING POINT

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BITS,BINARY AND LOGIC