Convert -56 971 709 003 653 to a Signed Binary (Base 2)

How to convert -56 971 709 003 653(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 -56 971 709 003 653 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:

|-56 971 709 003 653| = 56 971 709 003 653

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;
  • 56 971 709 003 653 ÷ 2 = 28 485 854 501 826 + 1;
  • 28 485 854 501 826 ÷ 2 = 14 242 927 250 913 + 0;
  • 14 242 927 250 913 ÷ 2 = 7 121 463 625 456 + 1;
  • 7 121 463 625 456 ÷ 2 = 3 560 731 812 728 + 0;
  • 3 560 731 812 728 ÷ 2 = 1 780 365 906 364 + 0;
  • 1 780 365 906 364 ÷ 2 = 890 182 953 182 + 0;
  • 890 182 953 182 ÷ 2 = 445 091 476 591 + 0;
  • 445 091 476 591 ÷ 2 = 222 545 738 295 + 1;
  • 222 545 738 295 ÷ 2 = 111 272 869 147 + 1;
  • 111 272 869 147 ÷ 2 = 55 636 434 573 + 1;
  • 55 636 434 573 ÷ 2 = 27 818 217 286 + 1;
  • 27 818 217 286 ÷ 2 = 13 909 108 643 + 0;
  • 13 909 108 643 ÷ 2 = 6 954 554 321 + 1;
  • 6 954 554 321 ÷ 2 = 3 477 277 160 + 1;
  • 3 477 277 160 ÷ 2 = 1 738 638 580 + 0;
  • 1 738 638 580 ÷ 2 = 869 319 290 + 0;
  • 869 319 290 ÷ 2 = 434 659 645 + 0;
  • 434 659 645 ÷ 2 = 217 329 822 + 1;
  • 217 329 822 ÷ 2 = 108 664 911 + 0;
  • 108 664 911 ÷ 2 = 54 332 455 + 1;
  • 54 332 455 ÷ 2 = 27 166 227 + 1;
  • 27 166 227 ÷ 2 = 13 583 113 + 1;
  • 13 583 113 ÷ 2 = 6 791 556 + 1;
  • 6 791 556 ÷ 2 = 3 395 778 + 0;
  • 3 395 778 ÷ 2 = 1 697 889 + 0;
  • 1 697 889 ÷ 2 = 848 944 + 1;
  • 848 944 ÷ 2 = 424 472 + 0;
  • 424 472 ÷ 2 = 212 236 + 0;
  • 212 236 ÷ 2 = 106 118 + 0;
  • 106 118 ÷ 2 = 53 059 + 0;
  • 53 059 ÷ 2 = 26 529 + 1;
  • 26 529 ÷ 2 = 13 264 + 1;
  • 13 264 ÷ 2 = 6 632 + 0;
  • 6 632 ÷ 2 = 3 316 + 0;
  • 3 316 ÷ 2 = 1 658 + 0;
  • 1 658 ÷ 2 = 829 + 0;
  • 829 ÷ 2 = 414 + 1;
  • 414 ÷ 2 = 207 + 0;
  • 207 ÷ 2 = 103 + 1;
  • 103 ÷ 2 = 51 + 1;
  • 51 ÷ 2 = 25 + 1;
  • 25 ÷ 2 = 12 + 1;
  • 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.

56 971 709 003 653(10) = 11 0011 1101 0000 1100 0010 0111 1010 0011 0111 1000 0101(2)


4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 46.

  • 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, 46,

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:


56 971 709 003 653(10) = 0000 0000 0000 0000 0011 0011 1101 0000 1100 0010 0111 1010 0011 0111 1000 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...


-56 971 709 003 653(10) Base 10 integer number converted and written as a signed binary code (in base 2):

-56 971 709 003 653(10) = 1000 0000 0000 0000 0011 0011 1101 0000 1100 0010 0111 1010 0011 0111 1000 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