1. Information Representation
Computer use a binary systems
Why binary?
Electronic bi-stable environment
on/off, high/low voltage
Bit: each bit can be either 0 or 1
Reliability
With only 2 values, can be widely separated, therefore clearly differentiated
“drift” causes less error
Example:
-3 mv
0 mv 1
Digital v.s, Analog

2. Binary Representation in Computer System
All information of diverse type is represented within computers in the form of bit patterns.
e.g., text, numerical data, sound, and images
One important aspect of computer design is to decide how information is converted ultimately to a bit pattern
Writing software also frequently requires understanding how information is represented along with accuracies

3. Number Systems Decimal Number System Base is 10 or ‘D’ or ‘Dec’
Ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Each place is weighted by the power of 10
Example:
or 1234D
= 1 x x x x 100 =
$1,000
$100
$10 $1

4. Binary Number System Base is 2 or ‘b’ or ‘B’ or ‘Bin’
Two symbols: 0 and 1
Each place is weighted by the power of 2
Example:
10112 or 1011 B
= 1 x x x x 20 =
= 1110
11 in decimal number system is 1011 in binary number system

5. Conversion between Decimal and Binary
Conversion from decimal number system to binary system
Question: represent 3410 in the binary number system
Answer: using the divide-by-2 technique repeatedly
If we write the remainder from right to left :
3410  1 x x x x 22+ 1x x 20  34 8 17 2 4 1
Remainder
div-by-2

6. Converting decimal to binary
Example: 162:
So, =
162 / 2 = 81 rem 0
81 / 2 = 40 rem 1
40 / 2 = 20 rem 0
20 / 2 = 10 rem 0
10 / 2 = 5 rem 0
5 / 2 = 2 rem 1
2 / 2 = 1 rem 0
1 / 2 = 0 rem 1

7. Some Exercises
13D = (?) B
23D = (?) B
72D = (?) B
1101B

8. Conversion from binary number system to decimal system
Example: check if is 3410
using the :weights” appropriately
 1 x x x x x x 20  

9. Some Exercises Ex: 0101B ( ? ) D Ex: 1100B  ( ? ) D
Bit 4
23 = 8
Bit 3
22 = 4
Bit 2
21 = 2
Bit 1
20 = 1 1 4 + 1
= 5D
= 12D
= 92D

10. Binary Arithmetic on Integers
Addition
a b a + b
0 1
1 1 1 1
1 0
Carry bit
Example: find binary number of a + b
If a = 13D , b = 5D
If a = 15D, b = 10D 1 1 1 b
13D D
18D + b 1 1 0b 1 b b b +
15D
+ 10D
25D

11. Binary Arithmetic on Integers (cont.)
Multiplication
a b a x b
0 1
1 0
1 1 1
Example: if a = b , b = 101b , find a x b b
x b
 33D
 5D b
 165D

12. Hexadecimal Number System
Base = 16 or ‘H’ or ‘Hex’
16 symbols:
{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A(=10), B(=11), C(=12), D(=13), E(=14), F(=15)}
Hexadecimal to Decimal
(an-1an-2…a1a0)16 = (an-1 x 16n-1 + an-2 x 16n-2 + …+ a1 x a0 x 160 )D
Example: (1C7)16 = (1 x x x 160 )10 = ( )10 = (455)10
Decimal to Hexadecimal Repeated division by 16
Similar in principle to generating binary codes
Example: (829)10 = (? )16
Stop, since quotient = 0
Hence, (829)10 = (3313)16
Divide-by-16
Quotient
Remainder
829 / 16
51 / 16
3 / 16 51 3 13
(33D)16

13. Hexadecimal Conversions
Hexadecimal to Binary
Expand each hexadecimal digit to 4 binary bits.
Example: (E29)16 = (1110 | 0010 | 1001)2
Binary to Hexadecimal
Combine every 4 bits into one hexadecimal digit
Example: (0101 | 1111 | 1010 | 0110)2 = (5FA6)16

14. Octal Number System Octal Number System Base = 8 or ‘o’ or ‘Oct’
8 symbols: { 0, 1, 2, 3, 4, 5, 6, 7}
Octal to Decimal
(an-1an-2…a1a0)8 = (an-1 x 8n-1 + an-2 x 8n-2 + …+ a1 x 81 + a0 x 80 )10
Example: (127)8 = (1 x x x 80 )10 = ( )10 = (87)10
Decimal to Octal
Repeated division by 8 (similar in principle to generating binary codes)
Example: (213)10 = (? )8
Stop, since quotient = 0
Hence, (213)10 = (325)8
Divide-by -8
Quotient
Remainder
Octal digit
213 / 8
26 / 8
3 / 8 26 3 5 2
Lower digit = 5
Second digit =2
Third digit =3

15. Octal to Binary Binary to Octal
Expand each octal digit to 3 binary bits.
Example: (725)8 = (111 | 010 | 101)2
Binary to Octal
Combine every 3 bits into one octal digit
Example: (110 | 010 | 011)2 = (623)8

16. Mini Homework
1)  Convert the following binary numbers to decimal numbers:
(a)     0011 B
(b)     0101 B
(c)     B
(d)     B
2) Convert the following decimal numbers to binary:
(a)     21 D
(b)     731 D
(c)     1,023 D

17. 3) Convert the following binary numbers to hexadecimal numbers:
(a)     0011 B
(b)     0101 B
(c)     B
(d)     B
(a)     21 D
(b)     731 D
(c)     1,023 D
4.) Perform the following binary additions and subtractions. Show your work without using decimal numbers during conversion.
(a)     111 B B
(b)     1001 B + 11 B