Introduction to Computer Science
Dr. Nagy Ramadan
E-mail: Nagyrdo@yahoo.com
Lecture - 3
1
Outline

Part 1 - Conversions in Numbering Systems

Part 2 - Conversion from Decimal to Binary System

Part 3 - Conversion from Decimal to Ternary System

Part 4 - Conversion from Decimal to Octal System

Part 5 - Conversion from Decimal to Hexadecimal System

Part 6 - Conversion from Binary to Octal System and vice
versa

Part 7 - Conversion from Binary to Hexadecimal System
and vice versa
2
Conversions in Numbering Systems
In lecture (2), we have studied the conversion from any number
system to the decimal one, i.e.
In this lecture, we are going to study the conversions in the
reverse direction, i.e.
3
Conversion from Decimal to Binary System
Ex1: Convert the decimal number 174.390625 to its equivalent binary number
R
L
L
R
Therefore, the binary number (10101110.011001)2 is exact equivalent to the decimal number (174.390625)10
4
Ex2: Obtain the equivalent binary number of the decimal number 47.763 with a
precision of 7-binary digits
R
L
L
R
Therefore, the number (10111.1100001)2 is the approximate binary equivalent to the decimal number
(47.763)10 with a precision of 7-binary digits .
5
Conversion from Decimal to Ternary System
Ex1: Convert the decimal number 124.33 to its equivalent ternary number with
precision of 5 ternary digits
R
L
L
R
Therefore, the number (11121.02222)3 is the approximate ternary equivalent to the decimal number
(124.33)10 with a precision of 5-ternary digits
6
Ex2: Obtain the ternary equivalent of the decimal number
R
L
L
R
Therefore, the number (1110.001)3 is the exact ternary equivalent to the decimal number
7
Conversion from Decimal to Octal System
Ex1: Convert the decimal number 167.390625 to its equivalent octal number
R
L
L
R
Therefore, the number (247.31)8 is the exact octal equivalent to the decimal number (167.390625)10
8
Ex2: Find the equivalent octal number of the decimal number 95.236 with a
precision up to 4 octal digits.
R
L
L
R
Therefore, the number (137.1706)8 is the approximate octal equivalent to the decimal
number (95.236)10 with a precision of 4-octal digits
9
Conversion from Decimal to Hexadecimal System
Ex1: Convert the decimal number 247.390625 to its equivalent hexadecimal
number
R
L
L
R
Therefore, the number (F7.64)16 is the exact hexadecimal equivalent to the decimal
number (247.390625)10
10
Conversion from Binary to Octal System and vice versa
1. Indirect conversion
By using the decimal system as the intermediate number system, i.e.
BDO
ODB
2. Direct Conversion
Each 3-binary digits are replaced by one octal digit , and vice versa ,
using the following table:
Octal
0
1
2
3
4
5
6
7
Binary
000
001
010
011
100
101
110
111
11
Ex1: Convert the binary number 1001101.1011 to octal
First Solution (indirect method)
Convert the binary number to decimal, then convert the decimal to octal.
(1001101.1011)2 = 1*26 +0*25 + 0*24 +1*23 + 1*22 + 0*21 + 1*20 + 1*2-1 + 0*2-2 +
1*2-3 + 1*2-4 = 64 + 8 + 4 + 1 + 0.5 + 0.125 + 0.0625 = (77.6875)10
R
L
L
R
Thus, (1001101.1011)2 = (77.6875)10 = (115.54)8
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Second Solution (direct method)
Start from the binary point and move to the right and to the left
grouping each 3-binary digits as one octal digit. Adding 0’s to the
extreme right (or left) is allowed.
Thus, (1001101.1011)2 = (115.54)8
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Ex2: Convert the octal number 115.54 to binary
Solution (direct method)
1
1
001
001
5
.
101 .
5
101
4
100
Thus, (115.54)8 = (001001101.101100)2
or (115.54)8 = (1001101.1011)2
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Conversion from Binary to Hexadecimal System and vice versa
1. Indirect conversion
By using the decimal system as the intermediate number system, i.e.
BDH
HDB
2. Direct Conversion
Each 4-binary digits are replaced by one hexadecimal digit, and vice
versa, Using the following table:
Hexadecimal 0
Binary
0000
1
0001
2
0010
3
0011
4
0100
5
0101
6
0110
7
0111
8
1000
9
1001
A
B
C
D
E
F
1010
1011
1100
1101
1110
1111
15
Ex: Convert the hexadecimal number 3BC. 2E directly to its
equivalent binary number.
Solution
Thus, (3BC.2E)16 = (001110111100.00101110)2
16
Ex: Convert the binary number 111010.11011 directly to its equivalent
hexadecimal number
Solution
Start from the binary point and move to the right and to the left
grouping each 4- binary digits as one hexadecimal digit. Adding
0’s to the extreme right (or left) is allowed. Thus, we have:
(111010.11011)2
=(00111010.11011000)2
=(
3
A . D
8
)16
17
Questions
18