Convert -4 323 455 729 517 199 967 to a Signed Binary (Base 2)

How to convert -4 323 455 729 517 199 967(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 -4 323 455 729 517 199 967 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:

|-4 323 455 729 517 199 967| = 4 323 455 729 517 199 967

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;
  • 4 323 455 729 517 199 967 ÷ 2 = 2 161 727 864 758 599 983 + 1;
  • 2 161 727 864 758 599 983 ÷ 2 = 1 080 863 932 379 299 991 + 1;
  • 1 080 863 932 379 299 991 ÷ 2 = 540 431 966 189 649 995 + 1;
  • 540 431 966 189 649 995 ÷ 2 = 270 215 983 094 824 997 + 1;
  • 270 215 983 094 824 997 ÷ 2 = 135 107 991 547 412 498 + 1;
  • 135 107 991 547 412 498 ÷ 2 = 67 553 995 773 706 249 + 0;
  • 67 553 995 773 706 249 ÷ 2 = 33 776 997 886 853 124 + 1;
  • 33 776 997 886 853 124 ÷ 2 = 16 888 498 943 426 562 + 0;
  • 16 888 498 943 426 562 ÷ 2 = 8 444 249 471 713 281 + 0;
  • 8 444 249 471 713 281 ÷ 2 = 4 222 124 735 856 640 + 1;
  • 4 222 124 735 856 640 ÷ 2 = 2 111 062 367 928 320 + 0;
  • 2 111 062 367 928 320 ÷ 2 = 1 055 531 183 964 160 + 0;
  • 1 055 531 183 964 160 ÷ 2 = 527 765 591 982 080 + 0;
  • 527 765 591 982 080 ÷ 2 = 263 882 795 991 040 + 0;
  • 263 882 795 991 040 ÷ 2 = 131 941 397 995 520 + 0;
  • 131 941 397 995 520 ÷ 2 = 65 970 698 997 760 + 0;
  • 65 970 698 997 760 ÷ 2 = 32 985 349 498 880 + 0;
  • 32 985 349 498 880 ÷ 2 = 16 492 674 749 440 + 0;
  • 16 492 674 749 440 ÷ 2 = 8 246 337 374 720 + 0;
  • 8 246 337 374 720 ÷ 2 = 4 123 168 687 360 + 0;
  • 4 123 168 687 360 ÷ 2 = 2 061 584 343 680 + 0;
  • 2 061 584 343 680 ÷ 2 = 1 030 792 171 840 + 0;
  • 1 030 792 171 840 ÷ 2 = 515 396 085 920 + 0;
  • 515 396 085 920 ÷ 2 = 257 698 042 960 + 0;
  • 257 698 042 960 ÷ 2 = 128 849 021 480 + 0;
  • 128 849 021 480 ÷ 2 = 64 424 510 740 + 0;
  • 64 424 510 740 ÷ 2 = 32 212 255 370 + 0;
  • 32 212 255 370 ÷ 2 = 16 106 127 685 + 0;
  • 16 106 127 685 ÷ 2 = 8 053 063 842 + 1;
  • 8 053 063 842 ÷ 2 = 4 026 531 921 + 0;
  • 4 026 531 921 ÷ 2 = 2 013 265 960 + 1;
  • 2 013 265 960 ÷ 2 = 1 006 632 980 + 0;
  • 1 006 632 980 ÷ 2 = 503 316 490 + 0;
  • 503 316 490 ÷ 2 = 251 658 245 + 0;
  • 251 658 245 ÷ 2 = 125 829 122 + 1;
  • 125 829 122 ÷ 2 = 62 914 561 + 0;
  • 62 914 561 ÷ 2 = 31 457 280 + 1;
  • 31 457 280 ÷ 2 = 15 728 640 + 0;
  • 15 728 640 ÷ 2 = 7 864 320 + 0;
  • 7 864 320 ÷ 2 = 3 932 160 + 0;
  • 3 932 160 ÷ 2 = 1 966 080 + 0;
  • 1 966 080 ÷ 2 = 983 040 + 0;
  • 983 040 ÷ 2 = 491 520 + 0;
  • 491 520 ÷ 2 = 245 760 + 0;
  • 245 760 ÷ 2 = 122 880 + 0;
  • 122 880 ÷ 2 = 61 440 + 0;
  • 61 440 ÷ 2 = 30 720 + 0;
  • 30 720 ÷ 2 = 15 360 + 0;
  • 15 360 ÷ 2 = 7 680 + 0;
  • 7 680 ÷ 2 = 3 840 + 0;
  • 3 840 ÷ 2 = 1 920 + 0;
  • 1 920 ÷ 2 = 960 + 0;
  • 960 ÷ 2 = 480 + 0;
  • 480 ÷ 2 = 240 + 0;
  • 240 ÷ 2 = 120 + 0;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 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.

4 323 455 729 517 199 967(10) = 11 1100 0000 0000 0000 0000 0001 0100 0101 0000 0000 0000 0000 0010 0101 1111(2)


4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 62.

  • 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, 62,

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:


4 323 455 729 517 199 967(10) = 0011 1100 0000 0000 0000 0000 0001 0100 0101 0000 0000 0000 0000 0010 0101 1111

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...


-4 323 455 729 517 199 967(10) Base 10 integer number converted and written as a signed binary code (in base 2):

-4 323 455 729 517 199 967(10) = 1011 1100 0000 0000 0000 0000 0001 0100 0101 0000 0000 0000 0000 0010 0101 1111

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