One's Complement: Integer ↗ Binary: 100 101 101 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 100 101 101(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;
  • 100 101 101 ÷ 2 = 50 050 550 + 1;
  • 50 050 550 ÷ 2 = 25 025 275 + 0;
  • 25 025 275 ÷ 2 = 12 512 637 + 1;
  • 12 512 637 ÷ 2 = 6 256 318 + 1;
  • 6 256 318 ÷ 2 = 3 128 159 + 0;
  • 3 128 159 ÷ 2 = 1 564 079 + 1;
  • 1 564 079 ÷ 2 = 782 039 + 1;
  • 782 039 ÷ 2 = 391 019 + 1;
  • 391 019 ÷ 2 = 195 509 + 1;
  • 195 509 ÷ 2 = 97 754 + 1;
  • 97 754 ÷ 2 = 48 877 + 0;
  • 48 877 ÷ 2 = 24 438 + 1;
  • 24 438 ÷ 2 = 12 219 + 0;
  • 12 219 ÷ 2 = 6 109 + 1;
  • 6 109 ÷ 2 = 3 054 + 1;
  • 3 054 ÷ 2 = 1 527 + 0;
  • 1 527 ÷ 2 = 763 + 1;
  • 763 ÷ 2 = 381 + 1;
  • 381 ÷ 2 = 190 + 1;
  • 190 ÷ 2 = 95 + 0;
  • 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.


100 101 101(10) = 101 1111 0111 0110 1011 1110 1101(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 27.


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, 27,

3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)


=== is: 32.


4. Get the positive binary computer representation on 32 bits (4 Bytes):

If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 32.


Number 100 101 101(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:

100 101 101(10) = 0000 0101 1111 0111 0110 1011 1110 1101

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