One's Complement: Integer ↗ Binary: 10 001 000 101 109 865 Convert the Integer Number to a Signed Binary in One's Complement Representation. Write the Base Ten Decimal System Number as a Binary Code (Written in Base Two)

Signed integer number 10 001 000 101 109 865(10) converted and written as a signed binary in one's complement representation (base 2) = ?

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;
  • 10 001 000 101 109 865 ÷ 2 = 5 000 500 050 554 932 + 1;
  • 5 000 500 050 554 932 ÷ 2 = 2 500 250 025 277 466 + 0;
  • 2 500 250 025 277 466 ÷ 2 = 1 250 125 012 638 733 + 0;
  • 1 250 125 012 638 733 ÷ 2 = 625 062 506 319 366 + 1;
  • 625 062 506 319 366 ÷ 2 = 312 531 253 159 683 + 0;
  • 312 531 253 159 683 ÷ 2 = 156 265 626 579 841 + 1;
  • 156 265 626 579 841 ÷ 2 = 78 132 813 289 920 + 1;
  • 78 132 813 289 920 ÷ 2 = 39 066 406 644 960 + 0;
  • 39 066 406 644 960 ÷ 2 = 19 533 203 322 480 + 0;
  • 19 533 203 322 480 ÷ 2 = 9 766 601 661 240 + 0;
  • 9 766 601 661 240 ÷ 2 = 4 883 300 830 620 + 0;
  • 4 883 300 830 620 ÷ 2 = 2 441 650 415 310 + 0;
  • 2 441 650 415 310 ÷ 2 = 1 220 825 207 655 + 0;
  • 1 220 825 207 655 ÷ 2 = 610 412 603 827 + 1;
  • 610 412 603 827 ÷ 2 = 305 206 301 913 + 1;
  • 305 206 301 913 ÷ 2 = 152 603 150 956 + 1;
  • 152 603 150 956 ÷ 2 = 76 301 575 478 + 0;
  • 76 301 575 478 ÷ 2 = 38 150 787 739 + 0;
  • 38 150 787 739 ÷ 2 = 19 075 393 869 + 1;
  • 19 075 393 869 ÷ 2 = 9 537 696 934 + 1;
  • 9 537 696 934 ÷ 2 = 4 768 848 467 + 0;
  • 4 768 848 467 ÷ 2 = 2 384 424 233 + 1;
  • 2 384 424 233 ÷ 2 = 1 192 212 116 + 1;
  • 1 192 212 116 ÷ 2 = 596 106 058 + 0;
  • 596 106 058 ÷ 2 = 298 053 029 + 0;
  • 298 053 029 ÷ 2 = 149 026 514 + 1;
  • 149 026 514 ÷ 2 = 74 513 257 + 0;
  • 74 513 257 ÷ 2 = 37 256 628 + 1;
  • 37 256 628 ÷ 2 = 18 628 314 + 0;
  • 18 628 314 ÷ 2 = 9 314 157 + 0;
  • 9 314 157 ÷ 2 = 4 657 078 + 1;
  • 4 657 078 ÷ 2 = 2 328 539 + 0;
  • 2 328 539 ÷ 2 = 1 164 269 + 1;
  • 1 164 269 ÷ 2 = 582 134 + 1;
  • 582 134 ÷ 2 = 291 067 + 0;
  • 291 067 ÷ 2 = 145 533 + 1;
  • 145 533 ÷ 2 = 72 766 + 1;
  • 72 766 ÷ 2 = 36 383 + 0;
  • 36 383 ÷ 2 = 18 191 + 1;
  • 18 191 ÷ 2 = 9 095 + 1;
  • 9 095 ÷ 2 = 4 547 + 1;
  • 4 547 ÷ 2 = 2 273 + 1;
  • 2 273 ÷ 2 = 1 136 + 1;
  • 1 136 ÷ 2 = 568 + 0;
  • 568 ÷ 2 = 284 + 0;
  • 284 ÷ 2 = 142 + 0;
  • 142 ÷ 2 = 71 + 0;
  • 71 ÷ 2 = 35 + 1;
  • 35 ÷ 2 = 17 + 1;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 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.


10 001 000 101 109 865(10) = 10 0011 1000 0111 1101 1011 0100 1010 0110 1100 1110 0000 0110 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 10 001 000 101 109 865(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:

10 001 000 101 109 865(10) = 0000 0000 0010 0011 1000 0111 1101 1011 0100 1010 0110 1100 1110 0000 0110 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 10 001 000 101 109 864 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 10 001 000 101 109 866 converted from decimal system (from base 10) and written as a signed binary in one's complement representation (written in base 2) = ?

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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