Convert 10 000 010 110 228 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 10 000 010 110 228(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
10 000 010 110 228 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;
  • 10 000 010 110 228 ÷ 2 = 5 000 005 055 114 + 0;
  • 5 000 005 055 114 ÷ 2 = 2 500 002 527 557 + 0;
  • 2 500 002 527 557 ÷ 2 = 1 250 001 263 778 + 1;
  • 1 250 001 263 778 ÷ 2 = 625 000 631 889 + 0;
  • 625 000 631 889 ÷ 2 = 312 500 315 944 + 1;
  • 312 500 315 944 ÷ 2 = 156 250 157 972 + 0;
  • 156 250 157 972 ÷ 2 = 78 125 078 986 + 0;
  • 78 125 078 986 ÷ 2 = 39 062 539 493 + 0;
  • 39 062 539 493 ÷ 2 = 19 531 269 746 + 1;
  • 19 531 269 746 ÷ 2 = 9 765 634 873 + 0;
  • 9 765 634 873 ÷ 2 = 4 882 817 436 + 1;
  • 4 882 817 436 ÷ 2 = 2 441 408 718 + 0;
  • 2 441 408 718 ÷ 2 = 1 220 704 359 + 0;
  • 1 220 704 359 ÷ 2 = 610 352 179 + 1;
  • 610 352 179 ÷ 2 = 305 176 089 + 1;
  • 305 176 089 ÷ 2 = 152 588 044 + 1;
  • 152 588 044 ÷ 2 = 76 294 022 + 0;
  • 76 294 022 ÷ 2 = 38 147 011 + 0;
  • 38 147 011 ÷ 2 = 19 073 505 + 1;
  • 19 073 505 ÷ 2 = 9 536 752 + 1;
  • 9 536 752 ÷ 2 = 4 768 376 + 0;
  • 4 768 376 ÷ 2 = 2 384 188 + 0;
  • 2 384 188 ÷ 2 = 1 192 094 + 0;
  • 1 192 094 ÷ 2 = 596 047 + 0;
  • 596 047 ÷ 2 = 298 023 + 1;
  • 298 023 ÷ 2 = 149 011 + 1;
  • 149 011 ÷ 2 = 74 505 + 1;
  • 74 505 ÷ 2 = 37 252 + 1;
  • 37 252 ÷ 2 = 18 626 + 0;
  • 18 626 ÷ 2 = 9 313 + 0;
  • 9 313 ÷ 2 = 4 656 + 1;
  • 4 656 ÷ 2 = 2 328 + 0;
  • 2 328 ÷ 2 = 1 164 + 0;
  • 1 164 ÷ 2 = 582 + 0;
  • 582 ÷ 2 = 291 + 0;
  • 291 ÷ 2 = 145 + 1;
  • 145 ÷ 2 = 72 + 1;
  • 72 ÷ 2 = 36 + 0;
  • 36 ÷ 2 = 18 + 0;
  • 18 ÷ 2 = 9 + 0;
  • 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.

10 000 010 110 228(10) = 1001 0001 1000 0100 1111 0000 1100 1110 0101 0001 0100(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 44.

  • 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, 44,

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 10 000 010 110 228(10) converted to signed binary in one's complement representation:

10 000 010 110 228(10) = 0000 0000 0000 0000 0000 1001 0001 1000 0100 1111 0000 1100 1110 0101 0001 0100

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