The document discusses the binary number system which computers use to store and manipulate data. It explains that computers operate using transistors which have two states (1 and 0), so computers use the binary number system. It then provides details on how decimal numbers are converted to and from binary, including how fractions are handled. Basic binary operations like addition are demonstrated. Finally, it discusses how binary operations are implemented using logic gates that perform boolean operations like AND, OR, and NOT.

1. THE BINARY NUMBER
SYSTEM
CSE 1110: Introduction to Computer Systems
Course teacher: Minhajul Bashir
12/20/2022 MINHAJUL@CSE.UIU.AC.BD 1

2. WHY BINARY?
Major computer components (CPU, RAM etc.) are made of transistors
A transistor is an electronic switch
 Has two states – on (1) and off (0)
This is why a computer understands the language of 1’s and 0’s only
 Binary number system
Every instruction of a computer is encoded in binary
Every data is also encoded in binary
Computer stores and manipulates these data in binary format
12/20/2022 MINHAJUL@CSE.UIU.AC.BD 2

3. BINARY NUMBER SYSTEM
Two digits to represent every number – 0 (zero) and
1 (one)
 Known as bits
8 bits together – a byte
A number of bytes together – a word
 Usually 4 or 8 bytes, depending on the processor, computer
architecture etc.
More units – kilobytes, megabytes, gigabytes,
terabytes, …
12/20/2022 MINHAJUL@CSE.UIU.AC.BD 3
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000
9 1001
10 1010
11 1011
12 1100
13 1101
14 1110
15 1111
16 10000

4. COMPUTER STORES
DATA IN
ITS BINARY FORM
Let’s see how a number is stored in its binary format

5. PART 1: CONVERT A DECIMAL
INTEGER TO BINARY
1. Divide the number by 2
2. Store the quotient and remainder
3. Let the quotient be the new number
4. Repeat steps 1 to 3 until the number is 0
5. Write down the remainders from last to first. That will be the
binary format

6. EXAMPLE: CONVERT 154 10
TO BINARY
12/20/2022 MINHAJUL@CSE.UIU.AC.BD 6
154
2
77 0
2
Quotient Remainder
38 1
2
19 0
2
9 1
2
4 1
2
2 0
2
1 0
2
0 1
154 10 = 10011010 2

7. PART 2: CONVERT A PROPER
FRACTION TO BINARY
1. Multiply the number by 2
2. Separate the integer and the fraction parts
3. Let the fraction part be the new number
4. Repeat steps 1 to 3 until the number is 0
5. Write down the integer parts from first to last after the binary
point. This will be the binary representation

8. EXAMPLE: CONVERT 0.625 10
TO BINARY
12/20/2022 MINHAJUL@CSE.UIU.AC.BD 8
Integer Fraction
- .625
x2
1 .250
x2
0 .500
x2
1 .000
0.625 10 = 0.101 2

9. HOW TO CONVERT
IMPROPER FRACTIONS?
To convert 154.625 10 to binary –
 Divide the number into integer and fraction part, i.e. 154 10 and 0.625 10
 Convert them individually to binary
 Join them up
154.625 10 = 10011010.101 2
12/20/2022 MINHAJUL@CSE.UIU.AC.BD 9

10. REVERSE ACTION: BINARY
TO DECIMAL
Multiply each bit with the place value of its place value, and sum the
results
Place values –
𝟐𝟔
𝟐𝟓 𝟐𝟒 𝟐𝟑 𝟐𝟐 𝟐𝟏 𝟐𝟎
12/20/2022 MINHAJUL@CSE.UIU.AC.BD 10
𝟐−𝟏 𝟐−𝟐 𝟐−𝟑
𝟐−𝟒

11. REVERSE ACTION: BINARY
TO DECIMAL
10011010.101 2
= 1 × 27
+ 0 × 26
+ 0 × 25
+ 1 × 24
+ 1 × 23
+ 0 × 22
+ 1 × 21
+ 0 × 20
+ 1 ×
2−1
+ 0 × 2−2
+ 1 × 2−3
= 128 + 0 + 0 + 16 + 8 + 0 + 2 + 0 + 0.5 + 0 + 0.125
= 154.625 10
12/20/2022 MINHAJUL@CSE.UIU.AC.BD 11

12. COMPUTER
MANIPULATES DATA IN
ITS BINARY FORM
Let’s see a basic manipulation example

13. ADDING TWO BINARY
NUMBERS
0101 + 0111 = 1100
0101 + 1011 = 10000
0111 + 0011 = 1010
How is it done?
 Watch carefully …

14. ADDING TWO BINARY
NUMBERS
1101011
+101110
=====
==
10011001
12/20/2022 MINHAJUL@CSE.UIU.AC.BD 14
1
1
1
1
+
0
1
1
0
1
0
0
1
1
1
1
1
1
0
1
0
1
1
0
1
0
1

15. IMPLEMENTING THE
MANIPULATIONS IN
CIRCUITS: BOOLEAN
OPERATIONS
Every manipulation is a combination of three basic operations
 Called the Boolean operations, by the name of George Bool
Three Boolean operations
 Logical addition (OR)
 Logical multiplication (AND)
 Inversion (NOT)
These operations are implemented using logic gates
 Gates take one or more bits as inputs, and give one bit an output, according to the
operations

16. LOGIC GATES
12/20/2022 MINHAJUL@CSE.UIU.AC.BD 16

17. BOOLEAN OPERATIONS
OR
𝐴 𝐵 𝐴 + 𝐵
0 0 0
0 1 1
1 0 1
1 1 1
AND
𝐴 𝐵 𝐴𝐵
0 0 0
0 1 0
1 0 0
1 1 1
NOT
𝐴 𝐴′
0 1
1 0

18. WHAT WILL BE THE OUTPUT
OF THIS CIRCUIT?
12/20/2022 MINHAJUL@CSE.UIU.AC.BD 18
A = 1
B = 0
C = 1
(AB)’
(B+C)BC