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

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

Key Questions

How do I convert binary numbers into decimal numbers?

How do I convert fractional binary numbers into equivalent decimal numbers?

How do I convert a decimal to a binary using the sum of weights method?

How to convert from decimal to binary using the ‘Division by-2’ method?

Conversions of fractional decimal numbers

Table for the 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

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

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

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

Decimal to Binary Conversation:

There are two different methods of decimal-to-binary conversation.

  • The sum of weights method
  • Division by 2 methods

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:

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

The sum of Weights Method:

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 place 2 and at the place 1. Assign binary 1 at these places.

The sum of weights that are absent.. at the place 0, at 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

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

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 the desired number of places after the 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

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

Leave a comment

Your email address will not be published. Required fields are marked *