Convert -101 731 713 805 104 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number -101 731 713 805 104(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
-101 731 713 805 104 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:

|-101 731 713 805 104| = 101 731 713 805 104

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;
  • 101 731 713 805 104 ÷ 2 = 50 865 856 902 552 + 0;
  • 50 865 856 902 552 ÷ 2 = 25 432 928 451 276 + 0;
  • 25 432 928 451 276 ÷ 2 = 12 716 464 225 638 + 0;
  • 12 716 464 225 638 ÷ 2 = 6 358 232 112 819 + 0;
  • 6 358 232 112 819 ÷ 2 = 3 179 116 056 409 + 1;
  • 3 179 116 056 409 ÷ 2 = 1 589 558 028 204 + 1;
  • 1 589 558 028 204 ÷ 2 = 794 779 014 102 + 0;
  • 794 779 014 102 ÷ 2 = 397 389 507 051 + 0;
  • 397 389 507 051 ÷ 2 = 198 694 753 525 + 1;
  • 198 694 753 525 ÷ 2 = 99 347 376 762 + 1;
  • 99 347 376 762 ÷ 2 = 49 673 688 381 + 0;
  • 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.

101 731 713 805 104(10) = 101 1100 1000 0110 0100 0010 1010 1001 1110 1011 0011 0000(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.


101 731 713 805 104(10) = 0000 0000 0000 0000 0101 1100 1000 0110 0100 0010 1010 1001 1110 1011 0011 0000

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).


-101 731 713 805 104(10) = !(0000 0000 0000 0000 0101 1100 1000 0110 0100 0010 1010 1001 1110 1011 0011 0000)


Decimal Number -101 731 713 805 104(10) converted to signed binary in one's complement representation:

-101 731 713 805 104(10) = 1111 1111 1111 1111 1010 0011 0111 1001 1011 1101 0101 0110 0001 0100 1100 1111

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