One's Complement: Integer Number 11 111 111 110 000 025 Converted and Written as a Signed Binary in One's Complement Representation

Integer number 11 111 111 110 000 025(10) written as a signed binary in one's complement representation

1. Divide the number repeatedly by 2:

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 11 111 111 110 000 025 ÷ 2 = 5 555 555 555 000 012 + 1;
  • 5 555 555 555 000 012 ÷ 2 = 2 777 777 777 500 006 + 0;
  • 2 777 777 777 500 006 ÷ 2 = 1 388 888 888 750 003 + 0;
  • 1 388 888 888 750 003 ÷ 2 = 694 444 444 375 001 + 1;
  • 694 444 444 375 001 ÷ 2 = 347 222 222 187 500 + 1;
  • 347 222 222 187 500 ÷ 2 = 173 611 111 093 750 + 0;
  • 173 611 111 093 750 ÷ 2 = 86 805 555 546 875 + 0;
  • 86 805 555 546 875 ÷ 2 = 43 402 777 773 437 + 1;
  • 43 402 777 773 437 ÷ 2 = 21 701 388 886 718 + 1;
  • 21 701 388 886 718 ÷ 2 = 10 850 694 443 359 + 0;
  • 10 850 694 443 359 ÷ 2 = 5 425 347 221 679 + 1;
  • 5 425 347 221 679 ÷ 2 = 2 712 673 610 839 + 1;
  • 2 712 673 610 839 ÷ 2 = 1 356 336 805 419 + 1;
  • 1 356 336 805 419 ÷ 2 = 678 168 402 709 + 1;
  • 678 168 402 709 ÷ 2 = 339 084 201 354 + 1;
  • 339 084 201 354 ÷ 2 = 169 542 100 677 + 0;
  • 169 542 100 677 ÷ 2 = 84 771 050 338 + 1;
  • 84 771 050 338 ÷ 2 = 42 385 525 169 + 0;
  • 42 385 525 169 ÷ 2 = 21 192 762 584 + 1;
  • 21 192 762 584 ÷ 2 = 10 596 381 292 + 0;
  • 10 596 381 292 ÷ 2 = 5 298 190 646 + 0;
  • 5 298 190 646 ÷ 2 = 2 649 095 323 + 0;
  • 2 649 095 323 ÷ 2 = 1 324 547 661 + 1;
  • 1 324 547 661 ÷ 2 = 662 273 830 + 1;
  • 662 273 830 ÷ 2 = 331 136 915 + 0;
  • 331 136 915 ÷ 2 = 165 568 457 + 1;
  • 165 568 457 ÷ 2 = 82 784 228 + 1;
  • 82 784 228 ÷ 2 = 41 392 114 + 0;
  • 41 392 114 ÷ 2 = 20 696 057 + 0;
  • 20 696 057 ÷ 2 = 10 348 028 + 1;
  • 10 348 028 ÷ 2 = 5 174 014 + 0;
  • 5 174 014 ÷ 2 = 2 587 007 + 0;
  • 2 587 007 ÷ 2 = 1 293 503 + 1;
  • 1 293 503 ÷ 2 = 646 751 + 1;
  • 646 751 ÷ 2 = 323 375 + 1;
  • 323 375 ÷ 2 = 161 687 + 1;
  • 161 687 ÷ 2 = 80 843 + 1;
  • 80 843 ÷ 2 = 40 421 + 1;
  • 40 421 ÷ 2 = 20 210 + 1;
  • 20 210 ÷ 2 = 10 105 + 0;
  • 10 105 ÷ 2 = 5 052 + 1;
  • 5 052 ÷ 2 = 2 526 + 0;
  • 2 526 ÷ 2 = 1 263 + 0;
  • 1 263 ÷ 2 = 631 + 1;
  • 631 ÷ 2 = 315 + 1;
  • 315 ÷ 2 = 157 + 1;
  • 157 ÷ 2 = 78 + 1;
  • 78 ÷ 2 = 39 + 0;
  • 39 ÷ 2 = 19 + 1;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 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.


11 111 111 110 000 025(10) = 10 0111 0111 1001 0111 1111 0010 0110 1100 0101 0111 1101 1001 1001(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 54.

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, 54,

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.


Number 11 111 111 110 000 025(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in one's complement representation:

11 111 111 110 000 025(10) = 0000 0000 0010 0111 0111 1001 0111 1111 0010 0110 1100 0101 0111 1101 1001 1001

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

More operations on signed integers written as signed binary numbers in one's complement representation:

» Signed integer number 11 111 111 110 000 024 converted from decimal system (from base 10) and written as a signed binary in one's complement representation (written in base 2) = ?

» Signed integer number 11 111 111 110 000 026 converted from decimal system (from base 10) and written as a signed binary in one's complement representation (written in base 2) = ?

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