Integer to One's Complement Binary: Number 1 111 111 111 111 208 Converted and Written as a Signed Binary in One's Complement Representation

Integer number 1 111 111 111 111 208(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;
  • 1 111 111 111 111 208 ÷ 2 = 555 555 555 555 604 + 0;
  • 555 555 555 555 604 ÷ 2 = 277 777 777 777 802 + 0;
  • 277 777 777 777 802 ÷ 2 = 138 888 888 888 901 + 0;
  • 138 888 888 888 901 ÷ 2 = 69 444 444 444 450 + 1;
  • 69 444 444 444 450 ÷ 2 = 34 722 222 222 225 + 0;
  • 34 722 222 222 225 ÷ 2 = 17 361 111 111 112 + 1;
  • 17 361 111 111 112 ÷ 2 = 8 680 555 555 556 + 0;
  • 8 680 555 555 556 ÷ 2 = 4 340 277 777 778 + 0;
  • 4 340 277 777 778 ÷ 2 = 2 170 138 888 889 + 0;
  • 2 170 138 888 889 ÷ 2 = 1 085 069 444 444 + 1;
  • 1 085 069 444 444 ÷ 2 = 542 534 722 222 + 0;
  • 542 534 722 222 ÷ 2 = 271 267 361 111 + 0;
  • 271 267 361 111 ÷ 2 = 135 633 680 555 + 1;
  • 135 633 680 555 ÷ 2 = 67 816 840 277 + 1;
  • 67 816 840 277 ÷ 2 = 33 908 420 138 + 1;
  • 33 908 420 138 ÷ 2 = 16 954 210 069 + 0;
  • 16 954 210 069 ÷ 2 = 8 477 105 034 + 1;
  • 8 477 105 034 ÷ 2 = 4 238 552 517 + 0;
  • 4 238 552 517 ÷ 2 = 2 119 276 258 + 1;
  • 2 119 276 258 ÷ 2 = 1 059 638 129 + 0;
  • 1 059 638 129 ÷ 2 = 529 819 064 + 1;
  • 529 819 064 ÷ 2 = 264 909 532 + 0;
  • 264 909 532 ÷ 2 = 132 454 766 + 0;
  • 132 454 766 ÷ 2 = 66 227 383 + 0;
  • 66 227 383 ÷ 2 = 33 113 691 + 1;
  • 33 113 691 ÷ 2 = 16 556 845 + 1;
  • 16 556 845 ÷ 2 = 8 278 422 + 1;
  • 8 278 422 ÷ 2 = 4 139 211 + 0;
  • 4 139 211 ÷ 2 = 2 069 605 + 1;
  • 2 069 605 ÷ 2 = 1 034 802 + 1;
  • 1 034 802 ÷ 2 = 517 401 + 0;
  • 517 401 ÷ 2 = 258 700 + 1;
  • 258 700 ÷ 2 = 129 350 + 0;
  • 129 350 ÷ 2 = 64 675 + 0;
  • 64 675 ÷ 2 = 32 337 + 1;
  • 32 337 ÷ 2 = 16 168 + 1;
  • 16 168 ÷ 2 = 8 084 + 0;
  • 8 084 ÷ 2 = 4 042 + 0;
  • 4 042 ÷ 2 = 2 021 + 0;
  • 2 021 ÷ 2 = 1 010 + 1;
  • 1 010 ÷ 2 = 505 + 0;
  • 505 ÷ 2 = 252 + 1;
  • 252 ÷ 2 = 126 + 0;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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.


1 111 111 111 111 208(10) = 11 1111 0010 1000 1100 1011 0111 0001 0101 0111 0010 0010 1000(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 50.

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, 50,

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 1 111 111 111 111 208(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:

1 111 111 111 111 208(10) = 0000 0000 0000 0011 1111 0010 1000 1100 1011 0111 0001 0101 0111 0010 0010 1000

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