Convert 101 000 011 010 131 to a Signed Binary (Base 2)

How to convert 101 000 011 010 131(10), a signed base 10 integer number? How to write it as a signed binary code in base 2

What are the required steps to convert base 10 integer
number 101 000 011 010 131 to signed binary code (in base 2)?

  • A signed integer, written in base ten, or a decimal system number, is a number written using the digits 0 through 9 and the sign, which can be positive (+) or negative (-). If positive, the sign is usually not written. A number written in base two, or binary, is a number written using only the digits 0 and 1.

1. Divide the number repeatedly by 2:

Keep track of each remainder.

Stop when you get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 101 000 011 010 131 ÷ 2 = 50 500 005 505 065 + 1;
  • 50 500 005 505 065 ÷ 2 = 25 250 002 752 532 + 1;
  • 25 250 002 752 532 ÷ 2 = 12 625 001 376 266 + 0;
  • 12 625 001 376 266 ÷ 2 = 6 312 500 688 133 + 0;
  • 6 312 500 688 133 ÷ 2 = 3 156 250 344 066 + 1;
  • 3 156 250 344 066 ÷ 2 = 1 578 125 172 033 + 0;
  • 1 578 125 172 033 ÷ 2 = 789 062 586 016 + 1;
  • 789 062 586 016 ÷ 2 = 394 531 293 008 + 0;
  • 394 531 293 008 ÷ 2 = 197 265 646 504 + 0;
  • 197 265 646 504 ÷ 2 = 98 632 823 252 + 0;
  • 98 632 823 252 ÷ 2 = 49 316 411 626 + 0;
  • 49 316 411 626 ÷ 2 = 24 658 205 813 + 0;
  • 24 658 205 813 ÷ 2 = 12 329 102 906 + 1;
  • 12 329 102 906 ÷ 2 = 6 164 551 453 + 0;
  • 6 164 551 453 ÷ 2 = 3 082 275 726 + 1;
  • 3 082 275 726 ÷ 2 = 1 541 137 863 + 0;
  • 1 541 137 863 ÷ 2 = 770 568 931 + 1;
  • 770 568 931 ÷ 2 = 385 284 465 + 1;
  • 385 284 465 ÷ 2 = 192 642 232 + 1;
  • 192 642 232 ÷ 2 = 96 321 116 + 0;
  • 96 321 116 ÷ 2 = 48 160 558 + 0;
  • 48 160 558 ÷ 2 = 24 080 279 + 0;
  • 24 080 279 ÷ 2 = 12 040 139 + 1;
  • 12 040 139 ÷ 2 = 6 020 069 + 1;
  • 6 020 069 ÷ 2 = 3 010 034 + 1;
  • 3 010 034 ÷ 2 = 1 505 017 + 0;
  • 1 505 017 ÷ 2 = 752 508 + 1;
  • 752 508 ÷ 2 = 376 254 + 0;
  • 376 254 ÷ 2 = 188 127 + 0;
  • 188 127 ÷ 2 = 94 063 + 1;
  • 94 063 ÷ 2 = 47 031 + 1;
  • 47 031 ÷ 2 = 23 515 + 1;
  • 23 515 ÷ 2 = 11 757 + 1;
  • 11 757 ÷ 2 = 5 878 + 1;
  • 5 878 ÷ 2 = 2 939 + 0;
  • 2 939 ÷ 2 = 1 469 + 1;
  • 1 469 ÷ 2 = 734 + 1;
  • 734 ÷ 2 = 367 + 0;
  • 367 ÷ 2 = 183 + 1;
  • 183 ÷ 2 = 91 + 1;
  • 91 ÷ 2 = 45 + 1;
  • 45 ÷ 2 = 22 + 1;
  • 22 ÷ 2 = 11 + 0;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number:

Take all the remainders starting from the bottom of the list constructed above.

101 000 011 010 131(10) = 101 1011 1101 1011 1110 0101 1100 0111 0101 0000 0101 0011(2)


3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 47.

  • A signed binary's bit length must be equal to a power of 2, as of:
  • 21 = 2; 22 = 4; 23 = 8; 24 = 16; 25 = 32; 26 = 64; ...
  • The first bit (the leftmost) is reserved for the sign:
  • 0 = positive integer number, 1 = negative integer number

The least number that is:

1) a power of 2

2) and is larger than the actual length, 47,

3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)

=== is: 64.


4. Get the positive binary computer representation on 64 bits (8 Bytes):

If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 64:


101 000 011 010 131(10) Base 10 integer number converted and written as a signed binary code (in base 2):

101 000 011 010 131(10) = 0000 0000 0000 0000 0101 1011 1101 1011 1110 0101 1100 0111 0101 0000 0101 0011

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

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How to convert signed base 10 integers in decimal to binary code system

Follow the steps below to convert a signed base ten integer number to signed binary:

  • 1. In a signed binary, first bit (the leftmost) is reserved for sign: 0 = positive integer number, 1 = positive integer number. If the number to be converted is negative, start with its positive version.
  • 2. Divide repeatedly by 2 the positive integer number keeping track of each remainder. STOP when we get a quotient that is ZERO.
  • 3. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Binary numbers represented in computer language have a length of 4, 8, 16, 32, 64, ... bits (power of 2) - if needed, fill in extra '0' bits in front of the base 2 number (to the left), up to the right length; this way the first bit (the leftmost one) is always '0', as for a positive representation.
  • 5. To get the negative reprezentation of the number, simply switch the first bit (the leftmost one), from '0' to '1'.

Example: convert the negative number -63 from decimal system (base ten) to signed binary code system:

  • 1. Start with the positive version of the number: |-63| = 63;
  • 2. Divide repeatedly 63 by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder
    • 63 ÷ 2 = 31 + 1
    • 31 ÷ 2 = 15 + 1
    • 15 ÷ 2 = 7 + 1
    • 7 ÷ 2 = 3 + 1
    • 3 ÷ 2 = 1 + 1
    • 1 ÷ 2 = 0 + 1
  • 3. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above:
    63(10) = 11 1111(2)
  • 4. The actual length of base 2 representation number is 6, so the positive binary computer representation length of the signed binary will take in this case 8 bits (the least power of 2 higher than 6) - add extra '0's in front (to the left), up to the required length; this way the first bit (the leftmost one) is to be '0', as for a positive number:
    63(10) = 0011 1111(2)
  • 5. To get the negative integer number representation simply change the first bit (the leftmost), from '0' to '1':
    -63(10) = 1011 1111
  • Number -63(10), signed integer, converted from decimal system (base 10) to signed binary = 1011 1111