Convert 102 727 544 217 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 102 727 544 217(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
102 727 544 217 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;
  • 102 727 544 217 ÷ 2 = 51 363 772 108 + 1;
  • 51 363 772 108 ÷ 2 = 25 681 886 054 + 0;
  • 25 681 886 054 ÷ 2 = 12 840 943 027 + 0;
  • 12 840 943 027 ÷ 2 = 6 420 471 513 + 1;
  • 6 420 471 513 ÷ 2 = 3 210 235 756 + 1;
  • 3 210 235 756 ÷ 2 = 1 605 117 878 + 0;
  • 1 605 117 878 ÷ 2 = 802 558 939 + 0;
  • 802 558 939 ÷ 2 = 401 279 469 + 1;
  • 401 279 469 ÷ 2 = 200 639 734 + 1;
  • 200 639 734 ÷ 2 = 100 319 867 + 0;
  • 100 319 867 ÷ 2 = 50 159 933 + 1;
  • 50 159 933 ÷ 2 = 25 079 966 + 1;
  • 25 079 966 ÷ 2 = 12 539 983 + 0;
  • 12 539 983 ÷ 2 = 6 269 991 + 1;
  • 6 269 991 ÷ 2 = 3 134 995 + 1;
  • 3 134 995 ÷ 2 = 1 567 497 + 1;
  • 1 567 497 ÷ 2 = 783 748 + 1;
  • 783 748 ÷ 2 = 391 874 + 0;
  • 391 874 ÷ 2 = 195 937 + 0;
  • 195 937 ÷ 2 = 97 968 + 1;
  • 97 968 ÷ 2 = 48 984 + 0;
  • 48 984 ÷ 2 = 24 492 + 0;
  • 24 492 ÷ 2 = 12 246 + 0;
  • 12 246 ÷ 2 = 6 123 + 0;
  • 6 123 ÷ 2 = 3 061 + 1;
  • 3 061 ÷ 2 = 1 530 + 1;
  • 1 530 ÷ 2 = 765 + 0;
  • 765 ÷ 2 = 382 + 1;
  • 382 ÷ 2 = 191 + 0;
  • 191 ÷ 2 = 95 + 1;
  • 95 ÷ 2 = 47 + 1;
  • 47 ÷ 2 = 23 + 1;
  • 23 ÷ 2 = 11 + 1;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 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.

102 727 544 217(10) = 1 0111 1110 1011 0000 1001 1110 1101 1001 1001(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 37.

  • 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, 37,

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 102 727 544 217(10) converted to signed binary in one's complement representation:

102 727 544 217(10) = 0000 0000 0000 0000 0000 0000 0001 0111 1110 1011 0000 1001 1110 1101 1001 1001

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