The document discusses different number systems including positional and non-positional systems. It describes the decimal, binary, octal, and hexadecimal number systems. Conversion between these number systems is also explained through long and short-cut methods. Steps are provided to convert between decimal, binary, octal, and hexadecimal numbers. Conversion of both integer and fractional numbers is covered.

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Number system:
A system to represents different quantities or values numerically is known as number system.
Number system is of two types:
1) Non-positional number system.
2) Positional number system.
Non-positional number system:
In non-positional number system some symbols are used to represent numbers for ex: I for 1, II
for 2 etc. These symbols are not capable of representing their position in the number and
therefore, it is very difficult to perform any arithmetic operation using non-positional number
system. So, to overcome this drawback positional number system is developed.
Positional number system:
In positional number system digits are used as symbol. These digits represent different values,
which depends upon their position they carry or hold in the number. Total number of
digits/symbols used by a number system under positional number system is known as its Base
or Radix.
Some of the positional number systems used in computers are as follow:
1) Decimal number system.
2) Binary number system.
3) Octal number system.
4) Hexadecimal number system.
Decimal number system:
In decimal number system following symbols are used as digits:
0,1,2,3,4,5,6,7,8,9.
Therefore Decimal number system has base 10 because it uses total 10 digits.
For ex: (7)10 , (117)10 here base 10 signifies that the number is decimal number.
Binary number system:
In binary number system following symbols are used as digits:
0 and 1.
Therefore binary number system has base 2 because it uses total 2 digits.
For ex: (101101)2, (0101011011)2 here base 2 signifies that the number is binary number.

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Octal number system:
In octal number system following symbols are used as digits:
0, 1, 2, 3, 4, 5, 6, 7.
Therefore octal number system has base 8 because it uses total 8 digits.
For ex: (7)8, (107)8 here base 8 signifies that the number is octal number.
Hexadecimal number system:
In hexadecimal number system following symbols are used as digits:
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F.
Where,
A is used to represent 10.
B is used to represent 11.
C is used to represent 12.
D is used to represent 13.
E is used to represent 14.
F is used to represent 15.
For ex: (DE)16, (27C)16 here base 16 signifies that the number is hexadecimal number.
Therefore hexadecimal number system has base 16 because it uses total 16 digits.
1.29 Conversion from one number system to another:
1) Decimal to binary conversion:
Steps of conversion:
1. Divide the decimal number by 2.
2. Record the remainder.
3. Divide the quotient of the previous divide by 2.

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4. Repeat step (2) and (3) until the quotient becomes zero.
5. And write the answer equal to the remainder in upward direction.
For ex: (7)10 = (?)2
2 7 1
2 3 1
2 1 1
0
Answer: (7)10 = (111)2.
For fractional numbers steps of conversion:
1. Repeat the same steps as given above for the number before decimal i.e. integer part.
Now for the fractional part:
2. Multiply the fractional part by 2.
3. From the answer of multiplication record the integer part separate.
4. For the fractional part of answer of previous multiplication repeat the previous step (2).
Repeat the step (2), (3), and (4) four to five times or as per asked in the question.
For ex: (28.32)10 = (?)2.
For integer part:
2 28 0
2 14 0
2 7 1
2 3 1
2 1 1
0
Remainder
QUOTIENT
Answerwill be
recordedinupward
direction.

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For fractional part:
.32 X 2 = 0.64 0
.64 X 2 = 1.28 1
.28 X 2 = 0.56 0
.56 X 2 = 1.12 1
Answer: (28.32)10 = (11100.0101)2
2) Decimal to octal conversion:
Steps of conversion:
1. Divide the decimal number by 8.
2. Record the remainder.
3. Divide the quotient of the previous divide by 8.
4. Repeat step (2) and (3) until the quotient becomes zero.
5. And write the answer equal to the remainder in upward direction.
For ex: (21)10 = (?)8
8 21 5
8 2 2
0
Answer: (21)10 = (25)8.
For fractional numbers steps of conversion:
1. Repeat the same steps as given above for the number before decimal i.e. integer part.
Now for the fractional part:
2. Multiply the fractional part by 8.
3. From the answer of multiplication record the integer part separate.
4. For the fractional part of answer of previous multiplication repeat the previous step (2).
Repeat the step (2), (3), and (4) four to five times or as per asked in the question.
Answerwill be
recordedin
downwarddirection.
Answerwill be
recordedinupward
direction.
Integerpart(from the answer of multiplication)

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For ex: (64.33)10 = (?)8.
For integer part:
8 64 0
8 8 0
8 1 1
0
For fractional part: integer part (from the answer of multiplication)
.33 X 8 = 2.64 2
.64 X 8 = 5.12 5
.12 X 8 = 0.96 0
.96 X 8 = 7.68 7
Answer: (64.33)10 = (100.2507)8
3) Decimal to Hexadecimal conversion:
Steps of conversion:
1. Divide the decimal number by 16.
2. Record the remainder.
3. Divide the quotient of the previous divide by 16.
4. Repeat step (2) and (3) until the quotient becomes zero.
5. And write the answer equal to the remainder in upward direction.
For ex: (43)10 = (?)16.
16 43 B (B represents 11)
16 2 2
0
Answer: (43)10 = (2B) 16.
For fractional number steps of conversion:
1. Repeat the same steps as given above for the number before decimal i.e. integer part.
Answerwill be
recordedinupward
direction.
Answerwill be
recordedin
downwarddirection.
Answerwill be
recordedinupward
direction.

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Now for the fractional part:
2. Multiply the fractional part by 8.
3. From the answer of multiplication record the integer part separate.
4. For the fractional part of answer of previous multiplication repeat the previous step (2).
Repeat the step (2), (3), and (4) four to five times or as per asked in the question.
For ex: (86.73)10 = (?)16.
For integer part:
16 86 6
16 5 5
0
For fractional part: integer part (from the answer of multiplication)
.73 X 16 = 11.68 11 (B)
.68 X 16 = 10.88 10 (A)
.88 X 16 = 14.08 14 (E)
.08 X 16 = 1.28 1
Answer: (86.73)10 = (56.BAE1)16.
4) Binary to Decimal conversion:
Steps for conversion:
1. Multiply the last digit of the binary number by 2^0 and store the result.
2. Increment the power by 1.
3. Take the previous digit.
4. Multiply the digit in step (3) by 2^ power in step (2) and store the result.
5. Repeat the step (2), (3) and (4) until you reach to the first digit of the decimal
number.
6. Add all the results to get the answer.
Answerwill be
recordedinupward
direction.
Answerwill be
recordedin
downwarddirection.

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For ex: (101)2 = (?)10.
1 0 1
X22
X21
X20
4 + 0 + 1 = (5)10.
Answer: (101)2 = (5)10.
For fractional number steps of conversion:
1. Repeat the same steps as given above for the integer part.
Now for the fractional part:
2. Multiply the first digit of the fractional part by 2^-1 and store the result.
3. Increment the power by -1.
4. Take the next digit.
5. Multiply the digit in the step (4) by 2^ power in step (3) and store the result.
6. Repeat the step (2), (3) and (4) until you reach to the last digit of the decimal number.
7. Add all the results to get the answer.
For ex: (101.101)2 = (?)10.
Right to left (before decimal) left to right (after decimal)
1 0 1 . 1 0 1
X22
X21
X20
. X2-1
X2-2
X2-3
4 + 0 + 1 . .50 + 0 + .125
(=5) (=.625)
=5.625
Answer: (101.101)2 = (5.625)10.
5) Octal to Decimal conversion:
Procedure of multiplicationby
differentpowersof 2is inright to
leftdirection.

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Steps for conversion:
1. Multiply the last digit of the binary number by 8^0 and store the result.
2. Increment the power by 1.
3. Take the previous digit.
4. Multiply the digit in step (3) by 8^ power in step (2) and store the result.
5. Repeat the step (2), (3) and (4) until you reach to the first digit of the decimal
number.
6. Add all the results to get the answer.
For ex: (32)8 = (?)10.
3 2
X81
X80
24 + 2 = 26
Answer: (32)8 = (26)10.
For fractional number steps of conversion:
1. Repeat the same steps as given above for the integer part.
Now for the fractional part:
2. Multiply the first digit of the fractional part by 8^-1 and store the result.
3. Increment the power by -1.
4. Take the next digit.
5. Multiply the digit in the step (4) by 8^ power in step (3) and store the result.
6. Repeat the step (2), (3) and (4) until you reach to the last digit of the decimal number.
7. Add all the results to get the answer.
For ex: (32.23)8 = (?)10.
3 2 . 2 3
X81
X80
. X8-1
X8-2
24 + 2 . .25 + .05 = 26.3
Answer: (32.23)8 = (26.3)10.
6) Hexadecimal to decimal conversion:
Steps for conversion:

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1. Multiply the last digit of the binary number by 16^0 and store the result.
2. Increment the power by 1.
3. Take the previous digit.
4. Multiply the digit in step (3) by 16^ power in step (2) and store the result.
5. Repeat the step (2), (3) and (4) until you reach to the first digit of the decimal
number.
6. Add all the results to get the answer.
For ex: (2F)16 = (?)10.
2 F
X161
X160
32 + 15 = 47.
Answer: (2F) 16 = (47)10.
For fractional number steps of conversion:
1. Repeat the same steps as given above for the integer part.
Now for the fractional part:
2. Multiply the first digit of the fractional part by 16^-1 and store the result.
3. Increment the power by -1.
4. Take the next digit.
5. Multiply the digit in the step (4) by 16^ power in step (3) and store the result.
6. Repeat the step (2), (3) and (4) until you reach to the last digit of the decimal number.
7. Add all the results to get the answer.
For ex: (23.45)16 = (?)10.
2 3 . 4 5
X161
X160
. X 16-1
X16-2
32 + 3 . .25 + 0.02 = 35.27.
Answer: (23.45)16 = (35.27)10.
7) Binary to Octal conversion:
Long method:

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Step1: convert Binary to Decimal. (Same as discussed before).
Step2: convert Decimal to Binary. (Same as discussed before).
For ex: (10110)2 = (?)8.
(Step1: convert Binary to Decimal.)
1 0 1 1 0
X24
X23
X22
X21
X20
16 + 0 + 4 + 2 + 0 =(22)10
(Step2: convert Decimal to Octal.)
8 22 6
8 2 2
0
= (26)8
Answer: (10110)2 = (26)8.
Short-cut method:
1. Make the group of three digits starting from the right side.
2. Take the first group.
3. Multiply the last digit of the group by 1, next digit by 2 and the third digit by 4 then, add
all the results and store the sum.
4. Take the next group.
5. Repeat the step (3.) and (4.) for each group.
6. Write all the sums together in the same sequence (as groups were made), as a number
to get the answer.
For ex: (10110)2 = (?)8
Starting from the right hand side first group will be 110 and in the second group only 2 digits
were left but we need 3 to proceed further. So we can add 0 at left hand side of the digit. It will
not make any difference in our answer. Similarly, as you know if you add 0 after decimal it
doesn’t make any difference.
Answerwill be
recordedinupward
direction.

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Group2: Group1:
0 1 0 1 1 0
X4 X2 X1 X4 X2 X1
=0 + 2 + 0 = 4 + 2 + 0
=2 =6
Answer: (10110)2 = (26)8
For fractional number steps of conversion:
Long method:
Step1: convert Binary to Decimal. (Same as discussed before).
Step2: convert Decimal to Octal. (Same as discussed before).
Short-cut method:
1. Repeat the same steps as given above for the integer part.
Now for the fractional part:
2. Make the group of three digits starting from the left side.
3. Take the first group.
4. Multiply the last digit of the group by 1, next digit by 2 and the third digit by 4 then, add
all the results and store the sum.
5. Take the next group.
6. Repeat the step (3.) and (4.) for each group.
7. Write all the sums together in the same sequence (as groups were made), as a number
to get the answer.
For ex: (111011.1010)2 = (?)8.
Group1: Group2: Group3: Group4:
1 1 1 0 1 1 . 1 0 1 0 0 0
X4 X2 X1 X4 X2 X1 X4 X2 X1 X4 X2 X1
=4 + 2 + 1 =0 + 2 + 1 . =4 + 0 + 1 =0 + 0 + 0
=7 =3 . =5 =0
Answer: (111011.1010)2 = (73.50)8.
An Extra
zero.

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Note: two extra zero’s were added at the end of group4 as only one digit was left and we need
three digits.