Convert 255 771 709 540 859 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 255 771 709 540 859(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
255 771 709 540 859 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. 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;
  • 255 771 709 540 859 ÷ 2 = 127 885 854 770 429 + 1;
  • 127 885 854 770 429 ÷ 2 = 63 942 927 385 214 + 1;
  • 63 942 927 385 214 ÷ 2 = 31 971 463 692 607 + 0;
  • 31 971 463 692 607 ÷ 2 = 15 985 731 846 303 + 1;
  • 15 985 731 846 303 ÷ 2 = 7 992 865 923 151 + 1;
  • 7 992 865 923 151 ÷ 2 = 3 996 432 961 575 + 1;
  • 3 996 432 961 575 ÷ 2 = 1 998 216 480 787 + 1;
  • 1 998 216 480 787 ÷ 2 = 999 108 240 393 + 1;
  • 999 108 240 393 ÷ 2 = 499 554 120 196 + 1;
  • 499 554 120 196 ÷ 2 = 249 777 060 098 + 0;
  • 249 777 060 098 ÷ 2 = 124 888 530 049 + 0;
  • 124 888 530 049 ÷ 2 = 62 444 265 024 + 1;
  • 62 444 265 024 ÷ 2 = 31 222 132 512 + 0;
  • 31 222 132 512 ÷ 2 = 15 611 066 256 + 0;
  • 15 611 066 256 ÷ 2 = 7 805 533 128 + 0;
  • 7 805 533 128 ÷ 2 = 3 902 766 564 + 0;
  • 3 902 766 564 ÷ 2 = 1 951 383 282 + 0;
  • 1 951 383 282 ÷ 2 = 975 691 641 + 0;
  • 975 691 641 ÷ 2 = 487 845 820 + 1;
  • 487 845 820 ÷ 2 = 243 922 910 + 0;
  • 243 922 910 ÷ 2 = 121 961 455 + 0;
  • 121 961 455 ÷ 2 = 60 980 727 + 1;
  • 60 980 727 ÷ 2 = 30 490 363 + 1;
  • 30 490 363 ÷ 2 = 15 245 181 + 1;
  • 15 245 181 ÷ 2 = 7 622 590 + 1;
  • 7 622 590 ÷ 2 = 3 811 295 + 0;
  • 3 811 295 ÷ 2 = 1 905 647 + 1;
  • 1 905 647 ÷ 2 = 952 823 + 1;
  • 952 823 ÷ 2 = 476 411 + 1;
  • 476 411 ÷ 2 = 238 205 + 1;
  • 238 205 ÷ 2 = 119 102 + 1;
  • 119 102 ÷ 2 = 59 551 + 0;
  • 59 551 ÷ 2 = 29 775 + 1;
  • 29 775 ÷ 2 = 14 887 + 1;
  • 14 887 ÷ 2 = 7 443 + 1;
  • 7 443 ÷ 2 = 3 721 + 1;
  • 3 721 ÷ 2 = 1 860 + 1;
  • 1 860 ÷ 2 = 930 + 0;
  • 930 ÷ 2 = 465 + 0;
  • 465 ÷ 2 = 232 + 1;
  • 232 ÷ 2 = 116 + 0;
  • 116 ÷ 2 = 58 + 0;
  • 58 ÷ 2 = 29 + 0;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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.

255 771 709 540 859(10) = 1110 1000 1001 1111 0111 1101 1110 0100 0000 1001 1111 1011(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 48.

  • 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, 48,

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.


Decimal Number 255 771 709 540 859(10) converted to signed binary in one's complement representation:

255 771 709 540 859(10) = 0000 0000 0000 0000 1110 1000 1001 1111 0111 1101 1110 0100 0000 1001 1111 1011

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