Decimal to Hexadecimal and Hexadecimal to Decimal Conversion Solved Examples

The method of conversion in between hexadecimal and decimal numbers are same as conversion between Octal and decimal numbers. I tried to make explanation as easy as possible. I provide 2 tables, one is for calculated powers of 16 and the other is for hexadecimal numbers equivalent in decimal numbers.

Key Questions:

• Conversion between decimal and hexadecimal numbers
• How to convert decimal fraction into hexadecimal fraction
• How to convert hexadecimal fraction into decimal fraction

164	163	162	161	160	16-1	16-2
65536	4096	256	16	1	0.0625	0.0039

Hexa-decimal	00	01	02	03	04	05	06	07	08	09	0A	0B	0C	0D	0E	0F
Decimal	00	01	02	03	04	05	06	07	08	09	10	11	12	13	14	15

Decimal to Hexadecimal Conversion:

As we see in decimal to binary conversion and decimal to octal conversion when  converting From decimal to binary number system we use repeated division by 2 and if decimal to octal number system we use repeated division by 8. So this time we are working with hexadecimal numbers so the repeated division by 16 method is used. And if we have mantissa or fractional part then we use repeated multiplication by 16 method. Let's start with examples.

Example#01:4859)10= ?)16

4859/16 =303       remainder 11)10=B)16

303/16= 18           remainder 15)10=F)16

18/16=1                remainder 2)10=2)16

Look at the highlighted numbers 12FB)16

Answer 4859)10= 12FB)16

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Example#02:23456.235)10= ?)16

Solve integer part using division by 16

23456/16=1466      remainder 0 =0)16

1466/16=91            remainder 10)10=A)16

91/16=5                  reminder 11)10=B)16

Look at the highlighted numbers  5BA0)16

Mantissa will be calculated by using repeated multiplication by 16

0.235*16=3.76 (most significant digit)

0.76*16=12.16)10=C.12)16

0.16*16=2.56. (least significant digit)

0.235)10=0.3C2)16

Answer 23456.235)10= 5BA0.3C2)16

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Example#03:1359.79)10= ?)16

For integer use repeated division by 16

1359/16 =84         remainder 15)10=F)16

84/16= 5                remainder 4

1359)10=54F)16

 For mantissa use repeated multiplication by 16

0.79*16=12.64 = C.64)16

0.64*16=10.24 = A.24)16

0.79)16=0.CA)16

Answer 1359.79)10= 54F.CA)16

Hexadecimal to Decimal Conversion:

As we know when converting from any number system to decimal number system we use sum of weights method. Like for binary numbers the sum of weights are powers of 2. For octal numbers the sum of weights are power of 8. Similarly for hexadecimal numbers the sum of weights are powers of 16.

Example#01: 12FB)16= ?)10

12FB)16=(1*163)+(2*162)+(F*161)+(B*160)

            =(1*4096)+(2*256)+(15*16)+(11*1)

            =4096+512+240+11

            =4859)10

Answer 12FB)16= 4859)10

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Example#02: 5BA0.3C2)16= ?)10

5BA0.3C2)16=(5*163)+(B*162)+(A*161)+(0*160)

                        .(3*16-1)+(C*16-2)+(2*16-3)

                      =(5*4096)+(11*256)+(10*16)+(0*1)

                       .(3*0.0625)+(12*0.00390625) 

                         neglect 16-3 which is too small.

                       =20480+5632+160.+0.1875+

                        0.046875

                       =23455.23)10

Answer 5BA0.3C2)16= 4859)10

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Example#03:  54F.CA)16= ?)10

54F.CA)16= (5*162)+(4*161)+(F*160).(C*16-1)+

                    (A*16-2)

               =(5*256)+(4*16)+(15*1).+(12*0.625)+

                  .(10*0.0039)

                =1280+64+15.75+0.039

                =1359.79)10

Answer 54F.CA)16= 1359.79)10

Octal Numbers to Decimal Numbers Solved Examples

Before beginning to this topic if you want to learn some basic information about number system you can read it from number system detailed explanation. Again I tried to make this topic easy to understand.

Key Questions:

• Convert from octal number system to decimal number system
• Convert from decimal number system to octal number system
• How to convert decimal fraction into octal fraction
• How to convert octal fraction into decimal fraction

Decimal to Octal Conversion:

The method of converting decimal number to octal number is same as decimal to binary numbers conversion. The difference is that when converting from decimal to binary we use repeated division by 2. But when converting from decimal to octal we use repeated division by 8. For mantissa calculation we use repeated multiplication by 2 when converting a decimal fraction into binary fraction. The method is explained in decimal to binary conversion. Similarly when working with octal numbers we use repeated multiplication by 8 for mantissa calculation.

Example#01:1899)10=?)8

Repeated division by 8

1899/8 = 237       remainder 3

237/8  = 29          remainder 5

29/8 = 3              remainder 5

Answer 1899)10= 3553)8

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Example#02:9876.87)10=?)8

First solve integer part by Repeated division by 8

9876/8 = 1234     remainder 4

1234/8  = 154      remainder 2

154/8 = 19           remainder 2

19/2 = 2               remainder 3

9876)10=23224)8

For mantissa calculation use repeated multiplication by 8 method

0.87*8=6.96 (Most significant digit)

0.96*8=7.68

0.68*8=5.44

0.44*8=3.52 (least significant digit)

Answer  9876.87)10= 23224.6752)8

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Example#03: 9248.356)10 = ?)8

Repeated division by 8

9248/8 = 1156    remainder 0

1156/8  = 144      remainder 4

144/8 = 18           remainder 0

18/2 = 2               remainder 2

9248)10=22040)8

For mantissa calculation use repeated multiplication by 8.

0.356*8=2.848 (most significant digit)

0.848*8=6.784

0.784*8=6.272 (least significant digit)

0.356)10=0.266)8

Answer: 9248.356)10 = 22040.266)8

Octal to Decimal Conversation:

When we are converting a number from octal number system to decimal number system we use sum of weights method. The method is same as binary to decimal conversion but the difference is base. For binary numbers system we multiply each digit by power of base 2. That is 2n. You can learn binary to decimal conversion here. Similarly when working on octal number system we use power of base 8. That is 8n. At this point I would like to add a table for calculated values of sum of weights. So it's easy to add a table for quick reference.

84	83	82	81	80	8-1	8-2	8-3	8-4
4096	512	64	8	1	0.125	0.015625	0.001953125	0.000244140625

Example#01:3553)8=?)10

5554)8=(3*83)+(5*82)+(5*81)+(3*80)

          =(5*512)+(5*64)+(5*8)+(3*1)

          =1536+320+40+3

          =1899)10

Answer:  3553)8=1899)10

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Example#02: 23224.6752)8=?)10

23224.6752)8=(2*84)+(3*83)+(2*82)+(2*81)+(4*80).(6*8-1)+(7*8-2)+(5*8-3)+(2*8-4)

                      =(2*4096)+(3*512)+(2*64)+(2*8)(4*1).(6*0.125)+(7*0.015625).....Neglect small fractions

                     =8192+1536+128+16+4.0.75+0.109375

                     =9876.86)10

Answer:  23224.6752)8=9876.86)10

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Example#03: 22040.266)8 =?)10

22040.266)8=(2*84)+(2*83)+(0*82)+(4*81)+(0*80)+(2*8-1)+(6*8-2)+(6*8-3)

                  =(2*4096)+(2*512)+(0*64)+(4*8)+(0*1).(2*0.125)+(6*0.015625)+(6*0.001953125)

                 =8191+1024+0+32+0.+0.25+0.09375+ 0.012

                =9248.36)10

Answer:  22040.266)8=9248.36)10

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