Convert -101 731 713 805 853 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number -101 731 713 805 853(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
-101 731 713 805 853 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:

|-101 731 713 805 853| = 101 731 713 805 853

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;
  • 101 731 713 805 853 ÷ 2 = 50 865 856 902 926 + 1;
  • 50 865 856 902 926 ÷ 2 = 25 432 928 451 463 + 0;
  • 25 432 928 451 463 ÷ 2 = 12 716 464 225 731 + 1;
  • 12 716 464 225 731 ÷ 2 = 6 358 232 112 865 + 1;
  • 6 358 232 112 865 ÷ 2 = 3 179 116 056 432 + 1;
  • 3 179 116 056 432 ÷ 2 = 1 589 558 028 216 + 0;
  • 1 589 558 028 216 ÷ 2 = 794 779 014 108 + 0;
  • 794 779 014 108 ÷ 2 = 397 389 507 054 + 0;
  • 397 389 507 054 ÷ 2 = 198 694 753 527 + 0;
  • 198 694 753 527 ÷ 2 = 99 347 376 763 + 1;
  • 99 347 376 763 ÷ 2 = 49 673 688 381 + 1;
  • 49 673 688 381 ÷ 2 = 24 836 844 190 + 1;
  • 24 836 844 190 ÷ 2 = 12 418 422 095 + 0;
  • 12 418 422 095 ÷ 2 = 6 209 211 047 + 1;
  • 6 209 211 047 ÷ 2 = 3 104 605 523 + 1;
  • 3 104 605 523 ÷ 2 = 1 552 302 761 + 1;
  • 1 552 302 761 ÷ 2 = 776 151 380 + 1;
  • 776 151 380 ÷ 2 = 388 075 690 + 0;
  • 388 075 690 ÷ 2 = 194 037 845 + 0;
  • 194 037 845 ÷ 2 = 97 018 922 + 1;
  • 97 018 922 ÷ 2 = 48 509 461 + 0;
  • 48 509 461 ÷ 2 = 24 254 730 + 1;
  • 24 254 730 ÷ 2 = 12 127 365 + 0;
  • 12 127 365 ÷ 2 = 6 063 682 + 1;
  • 6 063 682 ÷ 2 = 3 031 841 + 0;
  • 3 031 841 ÷ 2 = 1 515 920 + 1;
  • 1 515 920 ÷ 2 = 757 960 + 0;
  • 757 960 ÷ 2 = 378 980 + 0;
  • 378 980 ÷ 2 = 189 490 + 0;
  • 189 490 ÷ 2 = 94 745 + 0;
  • 94 745 ÷ 2 = 47 372 + 1;
  • 47 372 ÷ 2 = 23 686 + 0;
  • 23 686 ÷ 2 = 11 843 + 0;
  • 11 843 ÷ 2 = 5 921 + 1;
  • 5 921 ÷ 2 = 2 960 + 1;
  • 2 960 ÷ 2 = 1 480 + 0;
  • 1 480 ÷ 2 = 740 + 0;
  • 740 ÷ 2 = 370 + 0;
  • 370 ÷ 2 = 185 + 0;
  • 185 ÷ 2 = 92 + 1;
  • 92 ÷ 2 = 46 + 0;
  • 46 ÷ 2 = 23 + 0;
  • 23 ÷ 2 = 11 + 1;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 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.

101 731 713 805 853(10) = 101 1100 1000 0110 0100 0010 1010 1001 1110 1110 0001 1101(2)

4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 47.

  • 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, 47,

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.


101 731 713 805 853(10) = 0000 0000 0000 0000 0101 1100 1000 0110 0100 0010 1010 1001 1110 1110 0001 1101

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).


-101 731 713 805 853(10) = !(0000 0000 0000 0000 0101 1100 1000 0110 0100 0010 1010 1001 1110 1110 0001 1101)


Decimal Number -101 731 713 805 853(10) converted to signed binary in one's complement representation:

-101 731 713 805 853(10) = 1111 1111 1111 1111 1010 0011 0111 1001 1011 1101 0101 0110 0001 0001 1110 0010

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