Convert Binary to Decimal and Decimal to Binary Numbers

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Convert Binary to Decimal Numbers

A tutorial on how to convert decimal to binary and binary to decimal numbers with full explanations, exercises and answers.

The digits 0, 1, 9 are to write any number in the decimal system. Likewise, we use 0 and 1 to write numbers in the binary number system.

In decimal number system which is also called the base-10 system , the decimal number 149 is written as follows:

149 = 1*102 + 4*101 + 9*100


The binary or base-2 number 1 0 0 1 0 1 0 1 can be converted to a decimal number as follows:
1*27 + 0*26 + 0*25 + 1*24 + 0*23 + 1*22 + 0*21 + 1*20
= 128 + 16 + 4 + 1
= 149


The binary number 1 0 0 1 0 1 0 1 is written as 149 in the decimal system.

Convert Decimal Numbers to Binary

To convert a decimal number into a binary number, we carry out successive divisions by 2 and use the reminders of the successive divisions.

Example: Convert the decimal number 123 to a binary number.

123 ÷ 2 = 61 , reminder = 1 , least significant bit, goes to the right.
61 ÷ 2 = 30 , reminder = 1
30 ÷ 2 = 15 , reminder = 0
15 ÷ 2 = 7 , reminder = 1
7 ÷ 2 = 3 , reminder = 1
3 ÷ 2 = 1 , reminder = 1
1 ÷ 2 = 0 , reminder = 1 , Most significant bit, goes to the left.

The decimal number 123 in written as 1 1 1 1 0 1 1 in binary.

We can easily check by converting 1 1 1 1 0 1 1 into decimal as follows:

1*26 + 1*25 + 1*24 + 1*23 + 0*22 + 1*21 + 1*20 = 64 + 32 + 16 + 8 + 2 + 1 = 123

A Decimal to Binary Converter that may be used to convert decimal numbers to binary form.

Exercises with Answers

A - Convert the binary numbers into decimal numbers

  1. 11
  2. 101
  3. 1111
  4. 110111011
  5. 1111100011110011

B - Convert the decimal numbers into binary numbers

  1. 7
  2. 13
  3. 128
  4. 1678
  5. 12359

Answers to Above Exercises

A - Convert the binary numbers into decimal numbers

  1. 11 = 1*21 + 1*20 = 2 + 1 = 3
  2. 101 = 1*22 + 0*21 + 1*20
    = 4 + 1
    = 5
  3. 1111 = 1*23 + 1*22 + 1*21 + 1*20
    = 8 + 4 + 2 + 1
    = 15
  4. 110111011 = 1*28 + 1*27 + 0*26 + 1*25 + 1*24 + 1*23 + 0*22 + 1*21 + 1*20
    = 256 + 128 + 32 + 16 + 8 + 2 + 1
    = 443
  5. 1111100011110011 = 1*215 + 1*214 + 1*213 + 1*212 + 1*211 + 0*210 + 0*29 + 0*28 + 1*27 + 1*26 + 1*25 + 1*24 + 0*23 + 0*22 + 1*21 + 1*20
    = 32768 + 16384 + 8192 + 4096 + 2048 + 128 + 64 + 32 + 16 + 2 + 1
    = 63731

B - Convert the decimal numbers into binary numbers

    In what followr r means the reminder of the division.
  1. 7 ÷ 2 = 3 , r = 1
    3 ÷ 2 = 1 , r = 1
    1 ÷ 2 = 0 , r = 1
    Hence 7 in binary is written as 111

  2. 13 ÷ 2 = 6 , r = 1
    6 ÷ 2 = 3 , r = 0
    3 ÷ 2 = 1 , r = 1
    1 ÷ 2 = 0 , r = 1
    Hence 13 in binary is written as 1101

  3. 128 ÷ 2 = 64 , r = 0
    64 ÷ 2 = 32 , r = 0
    32 ÷ 2 = 16 , r = 0
    16 ÷ 2 = 8 , r = 0
    8 ÷ 2 = 4 , r = 0
    4 ÷ 2 = 2 , r = 0
    2 ÷ 2 = 1 , r = 0
    1 ÷ 2 = 0 , r = 1
    Hence 128 in binary is written as 10000000

  4. 1678 ÷ 2 = 839 , r = 0
    839 ÷ 2 = 419 , r = 1
    419 ÷ 2 = 209 , r = 1
    209 ÷ 2 = 104 , r = 1
    104 ÷ 2 = 52 , r = 0
    52 ÷ 2 = 26 , r = 0
    26 ÷ 2 = 13 , r = 0
    13 ÷ 2 = 6 , r = 1
    6 ÷ 2 = 3 , r = 0
    3 ÷ 2 = 1 , r = 1
    1 ÷ 2 = 0 , r = 1
    Hence 1678 in binary is written as 11010001110

  5. 12359 ÷ 2 = 6179 , r = 1
    6179 ÷ 2 = 3089 , r = 1
    3089 ÷ 2 = 1544 , r = 1
    1544 ÷ 2 = 772 , r = 0
    772 ÷ 2 = 386 , r = 0
    386 ÷ 2 = 193 , r = 0
    193 ÷ 2 = 96 , r = 1
    96 ÷ 2 = 48 , r = 0
    48 ÷ 2 = 24 , r = 0
    24 ÷ 2 = 12 , r = 0
    12 ÷ 2 = 6 , r = 0
    6 ÷ 2 = 3 , r = 0
    3 ÷ 2 = 1 , r = 1
    1 ÷ 2 = 0 , r = 1
    Hence 12359 in binary form is written as follows 11000001000111