Decimal to Binary and Binary to Decimal Conversation Solved Examples:

This topic is in continuation of number system. I will explain and provide some solved examples how to convert from binary to decimal and decimal to binary.

It is very easy and interesting topic. Anyone can understand easily

Key Questions

How to convert binary numbers into decimal numbers?

How to convert fractional binary numbers into equivalent decimal numbers?

How to convert from decimal to binary by sum of weights method?

How to convert from decimal to binary by 'Division by-2’ method?

Conversions of fractional decimal numbers

Table for sum of weights method
26	64
25	32
24	16
23	8
22	4
21	2
20	1
2-1	0.5
2-2	0.25
2-3	0.125
2-4	0.0625
2-5	0.03125
2-6	0.015625

Binary to Decimal Conversation:

Multiply each bit by its weight and add all of them.

Example#01: 1001110)2=?)10

1001110)2=(1*26)+(0*25)+(0*24)+(1*23)+(1*22.)+(1*21)+(0*20)

              =(1*64)+(0*32)+(0*16)+(1*8)+ (1*4) +(1*2)+(0*1)

                =64+0+0+8+4+2

                =78)10

Answer  1001110)2=78)10

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Example#02: 1101110)2=?)10

1101110)2=(1*26)+(1*25)+(0*24)+(1*23)+(1*22)+(1*21)+(0*20)

                =(1*64)+(1*32)+(0*16)+(1*8)+ (1*4)+(1*2)+(0*1)

                 =64+32+0+8+4+2+0

                 =110)10

Answer 1101110)2=110)10

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Example#03:1011100.01110)2

1011100.01110)2=(1*26)+(0*25)+(1*24)+(1*23)+(1*22)+(0*21)+(0*20).(0*2-1)+(1*2-2)+(1*2-3)+(1*2-4)+(0*2-5)

               =(1*64)+(0*32)+(1*16)+(1*8)+(1*4)+(0*2)+(0*1).(0*0.5)+(1*0.25)+(1*0.125)+(1*0.0625)+(0*0.03125)

=64+16+8+4.0.25+0.125+0.0625

Answer 1011100.01110)2=92.4375)10

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Example#04:1011001.10101)2=?)10

1011001.10101)2= (1*26)+(0*25)+(1*24)+(1*23)+(0*22)+(0*21)+(1*20).(1*2-1)+(0*2-2)+(1*2-3)+(0*2-4)+(1*2-5)

                           =(1*64)+(0*32)+(1*16)+(1*8)+(0*4)+(0*2)+(1*1).(1*0.5)+(0*0.25)+(1*0.125)+(1*0.0625)+(1*0.013625)

                  =64+16+8+1.0.5+0.125+0.013625

                         = 89.6562)10

Answer 1011001.10101)2 =89.6562)10

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Decimal to Binary Conversation:

There are two different methods of decimal to binary conversation.

• Sum of weights method
• Division by 2 method

The sum of weights method might be confusing. So to avoid confusion and make things clear and easy please refer table above.

Example#01: 78)10=?)2

Division by 2 Method (Decimal To Binary Conversion):

78/2=39    remainder 0

39/2=19    remainder 1

19/2=9      remainder 1

9/2=4        remainder 1

4/2=2        remainder 0

2/2=1        remainder 0

Rewrite the bold numbers, start from bottom 1001110)2

Sum of Weights Method (Decimal To Binary Conversion):

78 = 64+8+4+2

78 = 26+23+22+21

It can also be written as

     =26 +23+22+21  

Carefully looking at the above expression. Sum weights  that are present.. at the place 6, at the  place 3,at the place 2 and at the place 1. Assign binary 1 at these places.

Sum of weights that are absent.. at the place 0, at the place 4, at the place 5. There is no 20,24,25. Assign binary 0 at these places.

26	25	24	23	22	21	20
Present	Absent	Absent	Present	Present	Present	Absent
1	0	0	1	1	1	0

Answer: 78)2=1001110)2

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Example#02: 110)10=?)2

Division by 2 Method:

110/2=55    remainder 0

55/2=27      remainder 1

27/2=13      remainder 1

13/2=6        remainder 1

6/2=3          remainder 0

3/2=1          remainder 1

Rewrite the bold numbers, start from bottom 1101110)2

Sum of Weights Method:

110)10=64+32+8+4+2

         =26+25+23+22+21

                =1101110)2

26	25	24	23	22	21	20
Present	Present	Absent	Present	Present	Present	Absent
1	1	0	1	1	1	0

Answer: 110)10=1101110)2

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Example#03: 92.4646)10=?)2

The integer part can either be solved by sum of weights method or division by-2 method. While fractional part will be solved by some other technique.

Integer part will be solved by division by 2 method, while mantissa is solved by repeated multiplication by 2.

Division by 2 Method:

92/2=46      remainder 0

46/2=23      remainder 0

23/2=11      remainder 1

11/2=5        remainder 1

5/2=2          remainder 1

2/2=1          remainder 0

Integer =92)10=1011100)2

Mantissa =0.4646

Mantissa will be solved by repeated multiplication by 2

Multiply 0.4646 by 2. Again pick the resultant mantissa and multiply it by 2. Again pick the mantissa and multiply it by 2. Repeat until the mantissa becomes zero or desired number of places after binary point is achieved.

0.4646*2=0.9292 (MSB)

0.9292*2=1.8584

0.8584*2=1.7186

0.7186*2=1.4336

0.4336*2=0.8672 (LSB)

0.4646)10=01110)2

Answer 92.4646)10=1011100.01110)2

Solved by sum of weights method:

92)10=64+16+8+4

      =26+24+23+22

      =1011100)2

0.4646)10=0.25+0.125+0.0625

              =2-2+2-3+2-4

              =0.01110)2

26	25	24	23	22	21	20	2-1	2-2	2-3	2-4	2-5
Present	Absent	Present	Present	Absent	Absent	Absent	Present	Absent	Present	Absent	Present
1	0	1	1	1	0	0	0	1	1	1	0

Answer 92.4646)10=1011100.01110)2

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Example#04: 89.6785)10=?)2

Division by 2 Method:

89/2=44      remainder 1

44/2=22      remainder 0

22/2=11      remainder 0

11/2=5        remainder 1

5/2=2          remainder 1

2/2=1          remainder 0

89)10=1011001)2

Mantissa will be calculated by repeated multiplication by 2:

0.6785*2=1.357 (MSB)

0.357*2=0.714

0.714*2=1.428

0.428*2=0.856

0.856*2=1.712 (LSB)

0.6785)10=0.10101)2

Answer 89.6785)10=1011001.10101)2

Sum of weights method:

Refer table for sum of weights.

89)10=64+16+8+1

       =26+24+23+20

       =1011001)2

0.6785)10=0.5+0.125+0.03125

              =2-1+2-3+2-5

              =0.10101)2

26	25	24	23	22	21	20	2-1	2-2	2-3	2-4	2-5
Present	Absent	Present	Present	Absent	Absent	Absent	Present	Absent	Present	Absent	Present
1	1	0	1	1	1	0	1	0	1	0	1

Answer 89.6785)10=1011001.10101)2