Convert 12 345 678 987 654 644 to a Signed Binary (Base 2)

How to convert 12 345 678 987 654 644(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 12 345 678 987 654 644 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;
  • 12 345 678 987 654 644 ÷ 2 = 6 172 839 493 827 322 + 0;
  • 6 172 839 493 827 322 ÷ 2 = 3 086 419 746 913 661 + 0;
  • 3 086 419 746 913 661 ÷ 2 = 1 543 209 873 456 830 + 1;
  • 1 543 209 873 456 830 ÷ 2 = 771 604 936 728 415 + 0;
  • 771 604 936 728 415 ÷ 2 = 385 802 468 364 207 + 1;
  • 385 802 468 364 207 ÷ 2 = 192 901 234 182 103 + 1;
  • 192 901 234 182 103 ÷ 2 = 96 450 617 091 051 + 1;
  • 96 450 617 091 051 ÷ 2 = 48 225 308 545 525 + 1;
  • 48 225 308 545 525 ÷ 2 = 24 112 654 272 762 + 1;
  • 24 112 654 272 762 ÷ 2 = 12 056 327 136 381 + 0;
  • 12 056 327 136 381 ÷ 2 = 6 028 163 568 190 + 1;
  • 6 028 163 568 190 ÷ 2 = 3 014 081 784 095 + 0;
  • 3 014 081 784 095 ÷ 2 = 1 507 040 892 047 + 1;
  • 1 507 040 892 047 ÷ 2 = 753 520 446 023 + 1;
  • 753 520 446 023 ÷ 2 = 376 760 223 011 + 1;
  • 376 760 223 011 ÷ 2 = 188 380 111 505 + 1;
  • 188 380 111 505 ÷ 2 = 94 190 055 752 + 1;
  • 94 190 055 752 ÷ 2 = 47 095 027 876 + 0;
  • 47 095 027 876 ÷ 2 = 23 547 513 938 + 0;
  • 23 547 513 938 ÷ 2 = 11 773 756 969 + 0;
  • 11 773 756 969 ÷ 2 = 5 886 878 484 + 1;
  • 5 886 878 484 ÷ 2 = 2 943 439 242 + 0;
  • 2 943 439 242 ÷ 2 = 1 471 719 621 + 0;
  • 1 471 719 621 ÷ 2 = 735 859 810 + 1;
  • 735 859 810 ÷ 2 = 367 929 905 + 0;
  • 367 929 905 ÷ 2 = 183 964 952 + 1;
  • 183 964 952 ÷ 2 = 91 982 476 + 0;
  • 91 982 476 ÷ 2 = 45 991 238 + 0;
  • 45 991 238 ÷ 2 = 22 995 619 + 0;
  • 22 995 619 ÷ 2 = 11 497 809 + 1;
  • 11 497 809 ÷ 2 = 5 748 904 + 1;
  • 5 748 904 ÷ 2 = 2 874 452 + 0;
  • 2 874 452 ÷ 2 = 1 437 226 + 0;
  • 1 437 226 ÷ 2 = 718 613 + 0;
  • 718 613 ÷ 2 = 359 306 + 1;
  • 359 306 ÷ 2 = 179 653 + 0;
  • 179 653 ÷ 2 = 89 826 + 1;
  • 89 826 ÷ 2 = 44 913 + 0;
  • 44 913 ÷ 2 = 22 456 + 1;
  • 22 456 ÷ 2 = 11 228 + 0;
  • 11 228 ÷ 2 = 5 614 + 0;
  • 5 614 ÷ 2 = 2 807 + 0;
  • 2 807 ÷ 2 = 1 403 + 1;
  • 1 403 ÷ 2 = 701 + 1;
  • 701 ÷ 2 = 350 + 1;
  • 350 ÷ 2 = 175 + 0;
  • 175 ÷ 2 = 87 + 1;
  • 87 ÷ 2 = 43 + 1;
  • 43 ÷ 2 = 21 + 1;
  • 21 ÷ 2 = 10 + 1;
  • 10 ÷ 2 = 5 + 0;
  • 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.

12 345 678 987 654 644(10) = 10 1011 1101 1100 0101 0100 0110 0010 1001 0001 1111 0101 1111 0100(2)


3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 54.

  • 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, 54,

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:


12 345 678 987 654 644(10) Base 10 integer number converted and written as a signed binary code (in base 2):

12 345 678 987 654 644(10) = 0000 0000 0010 1011 1101 1100 0101 0100 0110 0010 1001 0001 1111 0101 1111 0100

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