Convert 100 000 001 000 555 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 100 000 001 000 555(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
100 000 001 000 555 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;
  • 100 000 001 000 555 ÷ 2 = 50 000 000 500 277 + 1;
  • 50 000 000 500 277 ÷ 2 = 25 000 000 250 138 + 1;
  • 25 000 000 250 138 ÷ 2 = 12 500 000 125 069 + 0;
  • 12 500 000 125 069 ÷ 2 = 6 250 000 062 534 + 1;
  • 6 250 000 062 534 ÷ 2 = 3 125 000 031 267 + 0;
  • 3 125 000 031 267 ÷ 2 = 1 562 500 015 633 + 1;
  • 1 562 500 015 633 ÷ 2 = 781 250 007 816 + 1;
  • 781 250 007 816 ÷ 2 = 390 625 003 908 + 0;
  • 390 625 003 908 ÷ 2 = 195 312 501 954 + 0;
  • 195 312 501 954 ÷ 2 = 97 656 250 977 + 0;
  • 97 656 250 977 ÷ 2 = 48 828 125 488 + 1;
  • 48 828 125 488 ÷ 2 = 24 414 062 744 + 0;
  • 24 414 062 744 ÷ 2 = 12 207 031 372 + 0;
  • 12 207 031 372 ÷ 2 = 6 103 515 686 + 0;
  • 6 103 515 686 ÷ 2 = 3 051 757 843 + 0;
  • 3 051 757 843 ÷ 2 = 1 525 878 921 + 1;
  • 1 525 878 921 ÷ 2 = 762 939 460 + 1;
  • 762 939 460 ÷ 2 = 381 469 730 + 0;
  • 381 469 730 ÷ 2 = 190 734 865 + 0;
  • 190 734 865 ÷ 2 = 95 367 432 + 1;
  • 95 367 432 ÷ 2 = 47 683 716 + 0;
  • 47 683 716 ÷ 2 = 23 841 858 + 0;
  • 23 841 858 ÷ 2 = 11 920 929 + 0;
  • 11 920 929 ÷ 2 = 5 960 464 + 1;
  • 5 960 464 ÷ 2 = 2 980 232 + 0;
  • 2 980 232 ÷ 2 = 1 490 116 + 0;
  • 1 490 116 ÷ 2 = 745 058 + 0;
  • 745 058 ÷ 2 = 372 529 + 0;
  • 372 529 ÷ 2 = 186 264 + 1;
  • 186 264 ÷ 2 = 93 132 + 0;
  • 93 132 ÷ 2 = 46 566 + 0;
  • 46 566 ÷ 2 = 23 283 + 0;
  • 23 283 ÷ 2 = 11 641 + 1;
  • 11 641 ÷ 2 = 5 820 + 1;
  • 5 820 ÷ 2 = 2 910 + 0;
  • 2 910 ÷ 2 = 1 455 + 0;
  • 1 455 ÷ 2 = 727 + 1;
  • 727 ÷ 2 = 363 + 1;
  • 363 ÷ 2 = 181 + 1;
  • 181 ÷ 2 = 90 + 1;
  • 90 ÷ 2 = 45 + 0;
  • 45 ÷ 2 = 22 + 1;
  • 22 ÷ 2 = 11 + 0;
  • 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.

100 000 001 000 555(10) = 101 1010 1111 0011 0001 0000 1000 1001 1000 0100 0110 1011(2)

3. 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.


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 100 000 001 000 555(10) converted to signed binary in one's complement representation:

100 000 001 000 555(10) = 0000 0000 0000 0000 0101 1010 1111 0011 0001 0000 1000 1001 1000 0100 0110 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