Convert -665 631 709 538 435 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number -665 631 709 538 435(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
-665 631 709 538 435 to a signed binary in one's (1's) complement representation?

  • 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:

|-665 631 709 538 435| = 665 631 709 538 435

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;
  • 665 631 709 538 435 ÷ 2 = 332 815 854 769 217 + 1;
  • 332 815 854 769 217 ÷ 2 = 166 407 927 384 608 + 1;
  • 166 407 927 384 608 ÷ 2 = 83 203 963 692 304 + 0;
  • 83 203 963 692 304 ÷ 2 = 41 601 981 846 152 + 0;
  • 41 601 981 846 152 ÷ 2 = 20 800 990 923 076 + 0;
  • 20 800 990 923 076 ÷ 2 = 10 400 495 461 538 + 0;
  • 10 400 495 461 538 ÷ 2 = 5 200 247 730 769 + 0;
  • 5 200 247 730 769 ÷ 2 = 2 600 123 865 384 + 1;
  • 2 600 123 865 384 ÷ 2 = 1 300 061 932 692 + 0;
  • 1 300 061 932 692 ÷ 2 = 650 030 966 346 + 0;
  • 650 030 966 346 ÷ 2 = 325 015 483 173 + 0;
  • 325 015 483 173 ÷ 2 = 162 507 741 586 + 1;
  • 162 507 741 586 ÷ 2 = 81 253 870 793 + 0;
  • 81 253 870 793 ÷ 2 = 40 626 935 396 + 1;
  • 40 626 935 396 ÷ 2 = 20 313 467 698 + 0;
  • 20 313 467 698 ÷ 2 = 10 156 733 849 + 0;
  • 10 156 733 849 ÷ 2 = 5 078 366 924 + 1;
  • 5 078 366 924 ÷ 2 = 2 539 183 462 + 0;
  • 2 539 183 462 ÷ 2 = 1 269 591 731 + 0;
  • 1 269 591 731 ÷ 2 = 634 795 865 + 1;
  • 634 795 865 ÷ 2 = 317 397 932 + 1;
  • 317 397 932 ÷ 2 = 158 698 966 + 0;
  • 158 698 966 ÷ 2 = 79 349 483 + 0;
  • 79 349 483 ÷ 2 = 39 674 741 + 1;
  • 39 674 741 ÷ 2 = 19 837 370 + 1;
  • 19 837 370 ÷ 2 = 9 918 685 + 0;
  • 9 918 685 ÷ 2 = 4 959 342 + 1;
  • 4 959 342 ÷ 2 = 2 479 671 + 0;
  • 2 479 671 ÷ 2 = 1 239 835 + 1;
  • 1 239 835 ÷ 2 = 619 917 + 1;
  • 619 917 ÷ 2 = 309 958 + 1;
  • 309 958 ÷ 2 = 154 979 + 0;
  • 154 979 ÷ 2 = 77 489 + 1;
  • 77 489 ÷ 2 = 38 744 + 1;
  • 38 744 ÷ 2 = 19 372 + 0;
  • 19 372 ÷ 2 = 9 686 + 0;
  • 9 686 ÷ 2 = 4 843 + 0;
  • 4 843 ÷ 2 = 2 421 + 1;
  • 2 421 ÷ 2 = 1 210 + 1;
  • 1 210 ÷ 2 = 605 + 0;
  • 605 ÷ 2 = 302 + 1;
  • 302 ÷ 2 = 151 + 0;
  • 151 ÷ 2 = 75 + 1;
  • 75 ÷ 2 = 37 + 1;
  • 37 ÷ 2 = 18 + 1;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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.

665 631 709 538 435(10) = 10 0101 1101 0110 0011 0111 0101 1001 1001 0010 1000 1000 0011(2)

4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 50.

  • 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) indicates 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, 50,

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.


665 631 709 538 435(10) = 0000 0000 0000 0010 0101 1101 0110 0011 0111 0101 1001 1001 0010 1000 1000 0011

6. Get the negative integer number representation:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in one's complement representation,

  • ... Reverse all the bits from 0 to 1 and from 1 to 0 (flip the digits).


-665 631 709 538 435(10) = !(0000 0000 0000 0010 0101 1101 0110 0011 0111 0101 1001 1001 0010 1000 1000 0011)


Decimal Number -665 631 709 538 435(10) converted to signed binary in one's complement representation:

-665 631 709 538 435(10) = 1111 1111 1111 1101 1010 0010 1001 1100 1000 1010 0110 0110 1101 0111 0111 1100

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

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How to convert signed integers from the decimal system to signed binary in one's complement representation

Follow the steps below to convert a signed base 10 integer number to signed binary in one's complement representation:

  • 1. If the number to be converted is negative, start with the positive version of the number.
  • 2. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, keeping track of each remainder, until we get a quotient that is equal to 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 must have 4, 8, 16, 32, 64, ... bit length (a power of 2) - if needed, fill in '0' bits in front (to the left) of the base 2 number calculated above, up to the right length; this way the first bit (leftmost) will always be '0', correctly representing a positive number.
  • 5. To get the negative integer number representation in signed binary one's complement, replace all '0' bits with '1's and all '1' bits with '0's.

Example: convert the negative number -49 from the decimal system (base ten) to signed binary one's complement:

  • 1. Start with the positive version of the number: |-49| = 49
  • 2. Divide repeatedly 49 by 2, keeping track of each remainder:
    • division = quotient + remainder
    • 49 ÷ 2 = 24 + 1
    • 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, by taking all the remainders starting from the bottom of the list constructed above:
    49(10) = 11 0001(2)
  • 4. The actual bit length of base 2 representation is 6, so the positive binary computer representation of a signed binary will take in this case 8 bits (the least power of 2 that is larger than 6) - add '0's in front of the base 2 number, up to the required length:
    49(10) = 0011 0001(2)
  • 5. To get the negative integer number representation in signed binary one's complement, replace all '0' bits with '1's and all '1' bits with '0's:
    -49(10) = 1100 1110
  • Number -49(10), signed integer, converted from the decimal system (base 10) to signed binary in one's complement representation = 1100 1110