Number System Conversions, Binary, Octal, Hex, CBSE Class 11

Master number system conversions for CBSE Class 11 CS. Binary, octal, decimal, hexadecimal conversions with step-by-step examples and practice problems.

Number system conversions are one of the most scoring topics in CBSE Class 11 Computer Science. Once you understand the method, you can solve any conversion problem correctly. This guide covers all four number systems and every type of conversion with step-by-step examples.

The Four Number Systems

Number System	Base	Digits Used	Example
Binary	2	0, 1	1010
Octal	8	0-7	127
Decimal	10	0-9	255
Hexadecimal	16	0-9, A-F	1A3F

Hexadecimal Digit Values

Hex Digit	Decimal Value
A	10
B	11
C	12
D	13
E	14
F	15

Decimal to Binary Conversion

Method: Divide the decimal number by 2 repeatedly and note the remainders. Read the remainders from bottom to top.

Example: Convert 45 to binary

Division	Quotient	Remainder
45 / 2	22	1
22 / 2	11	0
11 / 2	5	1
5 / 2	2	1
2 / 2	1	0
1 / 2	0	1

Reading remainders from bottom to top: (45)₁₀ = (101101)₂

Decimal Fraction to Binary

Method: Multiply the fractional part by 2 repeatedly and note the integer parts. Read from top to bottom.

Example: Convert 0.625 to binary

Multiplication	Result	Integer Part
0.625 x 2	1.250	1
0.250 x 2	0.500	0
0.500 x 2	1.000	1

Reading from top to bottom: (0.625)₁₀ = (0.101)₂

Complete example: Convert 45.625 to binary

Integer part: 45 = 101101 Fractional part: 0.625 = 0.101

Result: (45.625)₁₀ = (101101.101)₂

Binary to Decimal Conversion

Method: Multiply each bit by its positional value (power of 2) and add all the results.

Example: Convert (101101)₂ to decimal

Position:    5    4    3    2    1    0
Bit:         1    0    1    1    0    1
Value:       2⁵   2⁴   2³   2²   2¹   2⁰
           = 32    0    8    4    0    1

Sum = 32 + 0 + 8 + 4 + 0 + 1 = 45

(101101)₂ = (45)₁₀

Binary Fraction to Decimal

Example: Convert (0.101)₂ to decimal

Position:       -1     -2     -3
Bit:             1      0      1
Value:          2⁻¹    2⁻²    2⁻³
             = 0.5    0.0    0.125

Sum = 0.5 + 0.0 + 0.125 = 0.625

(0.101)₂ = (0.625)₁₀

Decimal to Octal Conversion

Method: Divide by 8 repeatedly. Read remainders from bottom to top.

Example: Convert 345 to octal

Division	Quotient	Remainder
345 / 8	43	1
43 / 8	5	3
5 / 8	0	5

Reading from bottom to top: (345)₁₀ = (531)₈

Octal to Decimal Conversion

Method: Multiply each digit by its positional value (power of 8) and add.

Example: Convert (531)₈ to decimal

Position:    2      1      0
Digit:       5      3      1
Value:       8²     8¹     8⁰
           = 320    24     1

Sum = 320 + 24 + 1 = 345

(531)₈ = (345)₁₀

Decimal to Hexadecimal Conversion

Method: Divide by 16 repeatedly. Read remainders from bottom to top. Replace remainders 10-15 with A-F.

Example: Convert 748 to hexadecimal

Division	Quotient	Remainder
748 / 16	46	12 (C)
46 / 16	2	14 (E)
2 / 16	0	2

Reading from bottom to top: (748)₁₀ = (2EC)₁₆

Hexadecimal to Decimal Conversion

Method: Multiply each digit by its positional value (power of 16) and add.

Example: Convert (2EC)₁₆ to decimal

Position:    2        1        0
Digit:       2        E(14)    C(12)
Value:       16²      16¹      16⁰
           = 512      224      12

Sum = 512 + 224 + 12 = 748

(2EC)₁₆ = (748)₁₀

Binary to Octal Conversion

Method: Group binary digits into sets of 3 (from right to left). Convert each group to its octal equivalent.

Example: Convert (101101011)₂ to octal

Group:    101   101   011
Octal:     5     5     3

(101101011)₂ = (553)₈

If the number of bits is not a multiple of 3, add leading zeros:

Example: Convert (11010)₂ to octal

Add leading zero: 011  010
Octal:              3    2

(11010)₂ = (32)₈

Octal to Binary Conversion

Method: Convert each octal digit to its 3-bit binary equivalent.

Example: Convert (753)₈ to binary

Octal:    7      5      3
Binary:  111    101    011

(753)₈ = (111101011)₂

Octal	Binary
0	000
1	001
2	010
3	011
4	100
5	101
6	110
7	111

Binary to Hexadecimal Conversion

Method: Group binary digits into sets of 4 (from right to left). Convert each group to its hex equivalent.

Example: Convert (110111011110)₂ to hexadecimal

Group:    1101   1101   1110
Hex:       D      D      E

(110111011110)₂ = (DDE)₁₆

Hexadecimal to Binary Conversion

Method: Convert each hex digit to its 4-bit binary equivalent.

Example: Convert (3AF)₁₆ to binary

Hex:      3       A       F
Binary:  0011    1010    1111

(3AF)₁₆ = (001110101111)₂

Hex	Binary
0	0000
1	0001
2	0010
3	0011
4	0100
5	0101
6	0110
7	0111
8	1000
9	1001
A	1010
B	1011
C	1100
D	1101
E	1110
F	1111

Octal to Hexadecimal (and vice versa)

For these conversions, use binary as an intermediate step:

Octal to Hex: Octal -> Binary -> Hexadecimal

Example: Convert (572)₈ to hexadecimal

Step 1: Octal to Binary

5 = 101,  7 = 111,  2 = 010
(572)₈ = (101111010)₂

Step 2: Binary to Hexadecimal (group in 4s from right)

0001  0111  1010
  1     7     A

(572)₈ = (17A)₁₆

Binary Arithmetic

Binary Addition

The rules of binary addition:

A	B	Sum	Carry
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

Example: Add (1011)₂ and (1101)₂

    1 1 1 1   (carry)
      1 0 1 1
  +   1 1 0 1
  -----------
    1 1 0 0 0

(1011)₂ + (1101)₂ = (11000)₂

Verification: 11 + 13 = 24 and (11000)₂ = 16 + 8 = 24. Correct.

Binary Subtraction

Example: Subtract (0101)₂ from (1100)₂

      1 1 0 0
  -   0 1 0 1
  -----------
      0 1 1 1

(1100)₂ - (0101)₂ = (0111)₂

Verification: 12, 5 = 7 and (0111)₂ = 7. Correct.

Representation of Negative Numbers

1's Complement

Flip all bits (0 becomes 1, 1 becomes 0):

Number:       0 1 0 1 1
1's Complement: 1 0 1 0 0

2's Complement

Add 1 to the 1's complement:

Number:          0 1 0 1 1
1's Complement:  1 0 1 0 0
Add 1:         + 0 0 0 0 1
                -----------
2's Complement:  1 0 1 0 1

Exam tip: 2's complement is used to represent negative numbers in computers. The 2's complement of a number N is calculated as: 2's complement = 1's complement + 1

Practice Problems

Try these conversions yourself:

Convert (156)₁₀ to binary
Convert (10110110)₂ to decimal
Convert (456)₈ to decimal
Convert (1000)₁₀ to hexadecimal
Convert (ABC)₁₆ to binary
Convert (110100111)₂ to octal
Convert (FF)₁₆ to decimal
Add (1110)₂ and (1011)₂
Find the 2's complement of (01101)₂
Convert (0.75)₁₀ to binary

Answers:

(10011100)₂
(182)₁₀
(302)₁₀
(3E8)₁₆
(101010111100)₂
(647)₈
(255)₁₀
(11001)₂
(10011)₂
(0.11)₂

Quick Revision

Decimal to any base: Divide by base, read remainders bottom to top
Any base to decimal: Multiply each digit by positional power, then add
Binary to Octal: Group bits in 3s from right
Binary to Hex: Group bits in 4s from right
Octal to Binary: Each octal digit = 3 binary bits
Hex to Binary: Each hex digit = 4 binary bits
1's complement: Flip all bits
2's complement: 1's complement + 1, For fractional conversions: multiply by base for integer-to-fraction direction