Convert -106 543 246 315 365 to a Signed Binary (Base 2)

How to convert -106 543 246 315 365(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 -106 543 246 315 365 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. Start with the positive version of the number:

|-106 543 246 315 365| = 106 543 246 315 365

2. 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;
  • 106 543 246 315 365 ÷ 2 = 53 271 623 157 682 + 1;
  • 53 271 623 157 682 ÷ 2 = 26 635 811 578 841 + 0;
  • 26 635 811 578 841 ÷ 2 = 13 317 905 789 420 + 1;
  • 13 317 905 789 420 ÷ 2 = 6 658 952 894 710 + 0;
  • 6 658 952 894 710 ÷ 2 = 3 329 476 447 355 + 0;
  • 3 329 476 447 355 ÷ 2 = 1 664 738 223 677 + 1;
  • 1 664 738 223 677 ÷ 2 = 832 369 111 838 + 1;
  • 832 369 111 838 ÷ 2 = 416 184 555 919 + 0;
  • 416 184 555 919 ÷ 2 = 208 092 277 959 + 1;
  • 208 092 277 959 ÷ 2 = 104 046 138 979 + 1;
  • 104 046 138 979 ÷ 2 = 52 023 069 489 + 1;
  • 52 023 069 489 ÷ 2 = 26 011 534 744 + 1;
  • 26 011 534 744 ÷ 2 = 13 005 767 372 + 0;
  • 13 005 767 372 ÷ 2 = 6 502 883 686 + 0;
  • 6 502 883 686 ÷ 2 = 3 251 441 843 + 0;
  • 3 251 441 843 ÷ 2 = 1 625 720 921 + 1;
  • 1 625 720 921 ÷ 2 = 812 860 460 + 1;
  • 812 860 460 ÷ 2 = 406 430 230 + 0;
  • 406 430 230 ÷ 2 = 203 215 115 + 0;
  • 203 215 115 ÷ 2 = 101 607 557 + 1;
  • 101 607 557 ÷ 2 = 50 803 778 + 1;
  • 50 803 778 ÷ 2 = 25 401 889 + 0;
  • 25 401 889 ÷ 2 = 12 700 944 + 1;
  • 12 700 944 ÷ 2 = 6 350 472 + 0;
  • 6 350 472 ÷ 2 = 3 175 236 + 0;
  • 3 175 236 ÷ 2 = 1 587 618 + 0;
  • 1 587 618 ÷ 2 = 793 809 + 0;
  • 793 809 ÷ 2 = 396 904 + 1;
  • 396 904 ÷ 2 = 198 452 + 0;
  • 198 452 ÷ 2 = 99 226 + 0;
  • 99 226 ÷ 2 = 49 613 + 0;
  • 49 613 ÷ 2 = 24 806 + 1;
  • 24 806 ÷ 2 = 12 403 + 0;
  • 12 403 ÷ 2 = 6 201 + 1;
  • 6 201 ÷ 2 = 3 100 + 1;
  • 3 100 ÷ 2 = 1 550 + 0;
  • 1 550 ÷ 2 = 775 + 0;
  • 775 ÷ 2 = 387 + 1;
  • 387 ÷ 2 = 193 + 1;
  • 193 ÷ 2 = 96 + 1;
  • 96 ÷ 2 = 48 + 0;
  • 48 ÷ 2 = 24 + 0;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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

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

106 543 246 315 365(10) = 110 0000 1110 0110 1000 1000 0101 1001 1000 1111 0110 0101(2)


4. 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.


5. 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:


106 543 246 315 365(10) = 0000 0000 0000 0000 0110 0000 1110 0110 1000 1000 0101 1001 1000 1111 0110 0101

6. Get the negative integer number representation:

To get the negative integer number representation on 64 bits (8 Bytes),

... change the first bit (the leftmost), from 0 to 1...


-106 543 246 315 365(10) Base 10 integer number converted and written as a signed binary code (in base 2):

-106 543 246 315 365(10) = 1000 0000 0000 0000 0110 0000 1110 0110 1000 1000 0101 1001 1000 1111 0110 0101

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