Miscellaneous examples of binary , quinary and octonary numbers (2)

2. BY: ASSOCIATE PROFESSOR NADEEM UDDIN
Miscellaneous
EXAMPLES OF Binary ,Quinary and Octonary numbers
Example-36:
Convert ( 324 )5 into binary number.
SOLUTION:
First step is to convert ( 324 )5 into decimal number
3 2 4 = 3 x 52
+ 2 x 51
+ 4 x 50
3 2 4 = 3 x 25 + 2 x 5 + 4 x 1
3 2 4 = 75 + 10 + 4
(32 4)5 = ( 89 )10
Now convert ( 89 )10 into binary number
(324 )5 = (1011001 )2
2 89
2 44 1
2 22 0
2 11 0
2 5 1
2 2 1
1 0

3. Example-37:
Convert ( 345 )8 into binary number.
SOLUTION:
First step is to convert ( 345 )8 into decimal number
( 345 )8 = 3 x 82
+ 4 x 81
+ 5 x 80
( 345 )8 = 3 x 64 + 4 x 8 + 5 x 1
( 345 )8 = 192 + 32 + 5
( 345 )8 = ( 229 )10
Now convert ( 229 )10 into binary number
( 345 )8 = (11100101 )2
Example-38:
Convert (1011001 )2 into Quinary number.
SOLUTION:
First step is to convert (1011001 )2 into decimal number
(1011001 )2 =1 x 26
+ 0 x 25
+ 1 x 24
+ 1 x 23
+ 0 x 22
+ 0 x 21
+ 1x20
(1011001 )2 =1 x 64 + 0 x 32 + 1 x 16 + 1 x 8 + 0 x 4 + 0 x 2 + 1 x 1
(1011001 )2 = 64 + 0 + 16 + 8 + 0 + 0 + 1
(1011001 )2 = (89)10
2 229
2 114 1
2 57 0
2 28 1
2 14 0
2 7 0
2 3 1
1 1

4. Now convert ( 89 )10 into Quinary number
(1011001 )2 = (324 )5
Example-39:
Convert (11100101 )2 into Octal number.
SOLUTION:
First step is to convert (11100101 )2 into decimal number
(11100101 )2 = 1 x 27
+1 x 26
+ 1 x 25
+ 0 x 24
+ 0 x 23
+ 1 x 22
+ 0 x 21
+ 1x20
(11100101 )2 =1 x 128 + 1x 64 + 1 x 32 + 0 x 16 + 0 x 8 + 1 x 4 + 0 x 2 + 1 x 1
(11100101 )2 = 128 + 64 + 32 + 0 + 0 + 4 + 0 + 1
(11100101 )2 = (229)10
Now convert ( 229 )10 into Octal number
(11100101 )2 = ( 345 )8
5 89
5 17 4
3 2
8 229
8 28 5
3 4

5. Example-40:
Convert (313 )8 into Quinary number.
SOLUTION:
First step is to convert (313 )8 into decimal number
(313 )8 = 3 x 82
+ 1 x 81
+ 3 x 80
(313 )8 =3 x 64 + 1 x 8 + 3 x 1
(313 )8 = 192 + 8 + 3
(313 )8 = (203)10
Now convert ( 203 )10 into Quinary number
(313 )8 = (1303 )5
Example-41:
Convert (432 )5 into Octal number.
SOLUTION:
First step is to convert (432 )5 into decimal number
(432 )5 = 4 x 52
+ 3 x 51
+2 x50
(432 )5 = 4 x 25 + 3 x 5 + 2 x 1
(432 )5 = 100 + 15 + 2
(432 )5 = (117 )10
5 203
5 40 3
5 8 0
1 3

6. Now convert (117 )10 into Octal number
(432)5 = (165 )8
TABLE TRICK-1
Octal
Number
Binary Number
I II III IV
0 0 0 0
1 0 0 1
2 0 1 0
3 0 1 1
4 1 O 0
5 1 O 1
6 1 1 0
7 1 1 1
Step.1 Put all octal numbers (0 to 7) in the I column.
Step.2 Put 4 zeros and 4 ones in the II column.
Step.3 Keep on putting 2 zeros and 2 ones in the III column.
Step.4 Keep on putting zero and one in the IV column.
DIRECT METHOD
Example-42:
Convert (345 )8 into binary number by using TABLE TRICK-1.
SOLUTION:
From Table
3 = 011
4 = 100
5 = 101
In raw one write the octal number right to left.
In raw two write the binary number right to left.
3 4 5
011 100 101
Therefore
(345 )8 = (11100101 )2
8 117
8 14 5
1 6

7. Example-43:
Convert (11100101 )2 into Octal number by using TABLE TRICK-1.
SOLUTION:
From Table
Break binary numbers into 3 digits from the right to left
Binary number 011 100 101
Octal number 3 4 5
Therefore
(11100101 )2 = (345 )8
Example-44:
Convert (123 )10 into binary number.
SOLUTION:
(123)10 = (1111011)2
Example-45:
Convert (123 )10 into binary number by Trick method.
SOLUTION:
Step.1In this method divide the number by 2 and ignore the remainder.
Step.2Write the divided number from right to left.
Step.3write 1 for odd number and 0 for even number.
1 3 7 15 30 61 123
1 1 1 1 0 1 1
Hence the answer is (123)10 = (1111011)2
2 123
2 61 1
2 30 1
2 15 0
2 7 1
2 3 1
1 1

8. Example-46 (a):
Convert (1111011)2 into decimal number
SOLUTION:
Binary
Number
1 1 1 1 0 1 1
Place
value
64 32 16 8 4 2 1
Decimal
Number
64 32 16 8 0 2 1 (123)10
In raw one write the binary numbers right to left.
In raw two write the place value of binary numbers right to left.
In raw three multiply raw one and raw two you can get the deimal number by
adding all the values in raw three.
Example-46 (b):
Convert (432)5 into decimal number
SOLUTION:
Penta
Number
4 3 2
Place
value
25 5 1
Decimal
Number
100 15 2 (117)10
In raw one write the penta numbers right to left.
In raw two write the place value of penta numbers right to left.
In raw three multiply raw one and raw two you can get the deimal number by
adding all the values in raw three.

9. Example-46 (c):
Convert (313)8 into decimal number
SOLUTION:
Octal
Number
3 1 3
Place
value
64 8 1
Decimal
Number
192 8 3 (203)10
In raw one write the Octal numbers right to left.
In raw two write the place value of octal numbers right to left.
In raw three multiply raw one and raw two you can get the deimal number by
adding all the values in raw three.