A bit is the smallest unit of data that can represent either a 1 or 0. Eight bits form a byte, which can represent one character. Common units for data storage are bytes, kilobytes, megabytes and gigabytes. There are different number systems used to represent numeric values in computing, including binary, octal and hexadecimal. Binary numbers can be added, subtracted and multiplied using logic gates. Decimal numbers can be converted to their binary or octal equivalent through repeated division and obtaining remainders.

1. M. Raihan
Email: rianku11@gmail.com

2. Data Representation
Part-2

3. Bit and Byte
21-Aug-17
Bit:
A "bit", like an atom, the smallest unit of storage.
A bit stores just a 0 or 1.
"In the computer it's all 0's and 1's" ... Bits.
In a chip: electric charge = 0/1
In a hard drive: spots of North/South magnetism = 0/1
A bit is too small to be much use.
Group 8 bits together to make 1 byte.

4. Bit and Byte
21-Aug-17
Byte:
One byte = grouping of 8 bits.
e.g. 0 1 0 1 1 0 1 0
One byte can store one character, e.g. 'A' or 'x' or '$‘
Used to measure data storage.
 For example, a CD holds 700MB (megabytes) of data and a hard
drive may hold 250GB (gigabytes).
Bytes are often used when referring to the size of an electronic file,
such as music, photos and videos.

5. Decimal to Binary
21-Aug-17
Example: convert (41.6875)10 into a binary number.
Integer part: 41 and fractional part: 0.6875
For integer part: 41
(41)10 = (101001)2

6. Decimal to Binary (continue)
21-Aug-17
For fractional part: (0.6875)10
Number Base Product Fractional Part Integer Part
0.6875 × 2 = 1.3750 0.3750 1
0.3750 × 2 = 0.7500 0.7500 0
0.7500 × 2 = 1.5000 0.5000 1
0.5000 × 2 = 1.0000 0.0000 1
(0.6875)10 = (.1011)2

7. Decimal to Binary (continue)
21-Aug-17
Example: convert (41.6875)10 into a binary number.
(41.6875)10 = (101001.1011)2
(41)10 = (101001)2
(0.6875)10 = (1011)2

8. Decimal to Octal
21-Aug-17
Example: convert (153.513)10 into a octal number.
Integer part: 153
Base Number Remainder
8 153
8 19 1
8 2 3
0 2
(153)10 = (231)8

9. Decimal to Octal (continue)
21-Aug-17
Example: convert (153.513)10 into a octal number.
Integer part: 0.513
(0.513)10 = (.4065…)8
Number Base Product Fractional Part Integer Part
0.513 × 8 = 4.104 0.104 4
0.104 × 8 = 0.832 0.832 0
0.832 × 8 = 6.656 0.656 6
0.656 × 8 = 5.248 0.248 5
.
.
.
.
.
.
.
.
.
.
.
.

10. Decimal to Octal (continue)
21-Aug-17
Example: convert (153.513)10 into a octal number.
(153.513)10 = (231.4065….)8
(153)10 = (231)8
(0.513)10 = (.4065….)8

11. Binary Arithmetic
21-Aug-17
Addition Subtraction Multiplication Division
𝟎 + 𝟎 = 𝟎 𝟎 − 𝟎 = 𝟎 𝟎 × 𝟎 = 𝟎 𝟎 ÷ 𝟎 = 𝒏𝒐 𝒎𝒆𝒂𝒏𝒊𝒏𝒈
𝟎 + 𝟏 = 𝟏 𝟏 − 𝟎 = 𝟏 𝟏 × 𝟎 = 𝟎 𝟏 ÷ 𝟎 = 𝒏𝒐 𝒎𝒆𝒂𝒏𝒊𝒏𝒈
𝟏 + 𝟎 = 𝟏 𝟏 − 𝟏 = 𝟎 𝟎 × 𝟏 = 𝟎 𝟎 ÷ 𝟏 = 𝟎
𝟏 + 𝟏 = 𝟎 𝒄𝒂𝒓𝒓𝒚 𝟏 𝟎 − 𝟏 = 𝟏 𝒃𝒐𝒓𝒓𝒐𝒘 𝟏 𝒆𝒒𝒖𝒂𝒍 𝒕𝒐 𝟏𝟎 𝟏 × 𝟏 = 𝟏 𝟏 ÷ 𝟏 = 𝟏
𝟏 + 𝟏 + 𝟏 = 𝟏 𝒄𝒂𝒓𝒓𝒚 𝟏 - - -

12. Binary Addition
21-Aug-17
Example: (101)2 + (001)2
101
+ 001
110

13. Binary Subtraction
21-Aug-17
Example: (10110)2 - (01001)2
10110
- 01001
01101

14. Complements
21-Aug-17
1’s Complement: 1’s complement of 110101 is 001010.
2’s Complement:
Way 1:
First perform 1’s complement
Add 1 to this to get 2’s complement
(13)10 = 1101 --> 0010
0010+1 = 0011 which is 2’s complement for -13
Way 2:
Scan right to left and complement all bits appearing after the first
appearance of 1
Example : 1010 = 0110

15. Signed Integers
21-Aug-17
Three methods to represent signed integers,
especially negative integer numbers:
Signed Magnitude Representation
Signed 1’s Complement Representation
Signed 2’s Complement Representation

16. Signed Magnitude Representation
21-Aug-17
In binary system, the sign of a number is represented using
an extra bit, called sign bit, at the extreme left most position
of the number.
By convention, sign bit 0 is for positive integer and sign bit
1 is for negative integer.
The rest bits represent the magnitude of the number.

17. Signed Magnitude Representation
21-Aug-17
Example:
(+15)10 = (0,0001111)2 and
(–15)10 = (1,0001111)2
Problem: Zero has two forms,
(0,0000000)2 = (+0)10 and
(1,0000000)2 = (–0)10

18. Binary Coded Decimal (BCD)
21-Aug-17
Using straight binary coding: a decimal number is
represented by its equivalent binary number.
BCD is a weighted code, each bit is assigned a weight.
From left to right the weights are 8, 4, 2 and 1. This is
known as 8–4–2–1 BCD.
As the highest decimal digit 9 requires 4-bits in binary
(i.e., 1001); each decimal digit is represented using a 4-
bit binary equivalent.

19. BCD
21-Aug-17
To convert a decimal number
to its BCD equivalent, just the
4-bits coded value is used for
each decimal digit.
Example: (1972)10 equals
(0001 1001 0111 0010)BCD
Decimal Digit
BCD
8 4 2 1
0 0 0 0 0
1 0 0 0 1
2 0 0 1 0
3 0 0 1 1
4 0 1 0 0
5 0 1 0 1
6 0 1 1 0
7 0 1 1 1
8 1 0 0 0
9 1 0 0 1

20. BCD (continue)
21-Aug-17
To convert a BCD number to
its decimal equivalent, all bits
are to be divided into groups of
4 bits. Then corresponding
decimal is replaced.
Example: (10101.1)BCD
= (0001 0101 . 1000)BCD
= (15.8)10
Decimal Digit
BCD
8 4 2 1
0 0 0 0 0
1 0 0 0 1
2 0 0 1 0
3 0 0 1 1
4 0 1 0 0
5 0 1 0 1
6 0 1 1 0
7 0 1 1 1
8 1 0 0 0
9 1 0 0 1

21. Binary Coded Decimal (BCD)
21-Aug-17
Advantages:
BCD system saves the efforts of conversion. Conversion from
decimal numbers to binary number system requires more
computation.
There is no round off error in BCD representation, but there
may be in straight binary conversion.
BCD system is easier to user, as generally user provides input in
decimal form and expects results also in decimal form.

22. Binary Coded Decimal (BCD)
21-Aug-17
Disadvantages:
It takes more digits in BCD system than straight binary coding.
For example, (1023)10 requires 16 bits in BCD,
(0001000000100011)BCD; where binary coding takes 9 bits
(111111111)2 .
Performing arithmetic operations with straight binary coding is
easier than the BCD coding.
Circuits required for BCD arithmetic operations are more
complex than those of binary circuits.
BCD system requires more time in executing numeric
computation.

23. Thank You
21-Aug-17