Convert -8 925 825 746 957 753 993 to a Signed Binary (Base 2)

How to convert -8 925 825 746 957 753 993(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 -8 925 825 746 957 753 993 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:

|-8 925 825 746 957 753 993| = 8 925 825 746 957 753 993

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;
  • 8 925 825 746 957 753 993 ÷ 2 = 4 462 912 873 478 876 996 + 1;
  • 4 462 912 873 478 876 996 ÷ 2 = 2 231 456 436 739 438 498 + 0;
  • 2 231 456 436 739 438 498 ÷ 2 = 1 115 728 218 369 719 249 + 0;
  • 1 115 728 218 369 719 249 ÷ 2 = 557 864 109 184 859 624 + 1;
  • 557 864 109 184 859 624 ÷ 2 = 278 932 054 592 429 812 + 0;
  • 278 932 054 592 429 812 ÷ 2 = 139 466 027 296 214 906 + 0;
  • 139 466 027 296 214 906 ÷ 2 = 69 733 013 648 107 453 + 0;
  • 69 733 013 648 107 453 ÷ 2 = 34 866 506 824 053 726 + 1;
  • 34 866 506 824 053 726 ÷ 2 = 17 433 253 412 026 863 + 0;
  • 17 433 253 412 026 863 ÷ 2 = 8 716 626 706 013 431 + 1;
  • 8 716 626 706 013 431 ÷ 2 = 4 358 313 353 006 715 + 1;
  • 4 358 313 353 006 715 ÷ 2 = 2 179 156 676 503 357 + 1;
  • 2 179 156 676 503 357 ÷ 2 = 1 089 578 338 251 678 + 1;
  • 1 089 578 338 251 678 ÷ 2 = 544 789 169 125 839 + 0;
  • 544 789 169 125 839 ÷ 2 = 272 394 584 562 919 + 1;
  • 272 394 584 562 919 ÷ 2 = 136 197 292 281 459 + 1;
  • 136 197 292 281 459 ÷ 2 = 68 098 646 140 729 + 1;
  • 68 098 646 140 729 ÷ 2 = 34 049 323 070 364 + 1;
  • 34 049 323 070 364 ÷ 2 = 17 024 661 535 182 + 0;
  • 17 024 661 535 182 ÷ 2 = 8 512 330 767 591 + 0;
  • 8 512 330 767 591 ÷ 2 = 4 256 165 383 795 + 1;
  • 4 256 165 383 795 ÷ 2 = 2 128 082 691 897 + 1;
  • 2 128 082 691 897 ÷ 2 = 1 064 041 345 948 + 1;
  • 1 064 041 345 948 ÷ 2 = 532 020 672 974 + 0;
  • 532 020 672 974 ÷ 2 = 266 010 336 487 + 0;
  • 266 010 336 487 ÷ 2 = 133 005 168 243 + 1;
  • 133 005 168 243 ÷ 2 = 66 502 584 121 + 1;
  • 66 502 584 121 ÷ 2 = 33 251 292 060 + 1;
  • 33 251 292 060 ÷ 2 = 16 625 646 030 + 0;
  • 16 625 646 030 ÷ 2 = 8 312 823 015 + 0;
  • 8 312 823 015 ÷ 2 = 4 156 411 507 + 1;
  • 4 156 411 507 ÷ 2 = 2 078 205 753 + 1;
  • 2 078 205 753 ÷ 2 = 1 039 102 876 + 1;
  • 1 039 102 876 ÷ 2 = 519 551 438 + 0;
  • 519 551 438 ÷ 2 = 259 775 719 + 0;
  • 259 775 719 ÷ 2 = 129 887 859 + 1;
  • 129 887 859 ÷ 2 = 64 943 929 + 1;
  • 64 943 929 ÷ 2 = 32 471 964 + 1;
  • 32 471 964 ÷ 2 = 16 235 982 + 0;
  • 16 235 982 ÷ 2 = 8 117 991 + 0;
  • 8 117 991 ÷ 2 = 4 058 995 + 1;
  • 4 058 995 ÷ 2 = 2 029 497 + 1;
  • 2 029 497 ÷ 2 = 1 014 748 + 1;
  • 1 014 748 ÷ 2 = 507 374 + 0;
  • 507 374 ÷ 2 = 253 687 + 0;
  • 253 687 ÷ 2 = 126 843 + 1;
  • 126 843 ÷ 2 = 63 421 + 1;
  • 63 421 ÷ 2 = 31 710 + 1;
  • 31 710 ÷ 2 = 15 855 + 0;
  • 15 855 ÷ 2 = 7 927 + 1;
  • 7 927 ÷ 2 = 3 963 + 1;
  • 3 963 ÷ 2 = 1 981 + 1;
  • 1 981 ÷ 2 = 990 + 1;
  • 990 ÷ 2 = 495 + 0;
  • 495 ÷ 2 = 247 + 1;
  • 247 ÷ 2 = 123 + 1;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 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:

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

8 925 825 746 957 753 993(10) = 111 1011 1101 1110 1110 0111 0011 1001 1100 1110 0111 0011 1101 1110 1000 1001(2)


4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 63.

  • 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, 63,

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:


8 925 825 746 957 753 993(10) = 0111 1011 1101 1110 1110 0111 0011 1001 1100 1110 0111 0011 1101 1110 1000 1001

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...


-8 925 825 746 957 753 993(10) Base 10 integer number converted and written as a signed binary code (in base 2):

-8 925 825 746 957 753 993(10) = 1111 1011 1101 1110 1110 0111 0011 1001 1100 1110 0111 0011 1101 1110 1000 1001

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