Convert 1 010 111 110 903 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 1 010 111 110 903(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
1 010 111 110 903 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;
  • 1 010 111 110 903 ÷ 2 = 505 055 555 451 + 1;
  • 505 055 555 451 ÷ 2 = 252 527 777 725 + 1;
  • 252 527 777 725 ÷ 2 = 126 263 888 862 + 1;
  • 126 263 888 862 ÷ 2 = 63 131 944 431 + 0;
  • 63 131 944 431 ÷ 2 = 31 565 972 215 + 1;
  • 31 565 972 215 ÷ 2 = 15 782 986 107 + 1;
  • 15 782 986 107 ÷ 2 = 7 891 493 053 + 1;
  • 7 891 493 053 ÷ 2 = 3 945 746 526 + 1;
  • 3 945 746 526 ÷ 2 = 1 972 873 263 + 0;
  • 1 972 873 263 ÷ 2 = 986 436 631 + 1;
  • 986 436 631 ÷ 2 = 493 218 315 + 1;
  • 493 218 315 ÷ 2 = 246 609 157 + 1;
  • 246 609 157 ÷ 2 = 123 304 578 + 1;
  • 123 304 578 ÷ 2 = 61 652 289 + 0;
  • 61 652 289 ÷ 2 = 30 826 144 + 1;
  • 30 826 144 ÷ 2 = 15 413 072 + 0;
  • 15 413 072 ÷ 2 = 7 706 536 + 0;
  • 7 706 536 ÷ 2 = 3 853 268 + 0;
  • 3 853 268 ÷ 2 = 1 926 634 + 0;
  • 1 926 634 ÷ 2 = 963 317 + 0;
  • 963 317 ÷ 2 = 481 658 + 1;
  • 481 658 ÷ 2 = 240 829 + 0;
  • 240 829 ÷ 2 = 120 414 + 1;
  • 120 414 ÷ 2 = 60 207 + 0;
  • 60 207 ÷ 2 = 30 103 + 1;
  • 30 103 ÷ 2 = 15 051 + 1;
  • 15 051 ÷ 2 = 7 525 + 1;
  • 7 525 ÷ 2 = 3 762 + 1;
  • 3 762 ÷ 2 = 1 881 + 0;
  • 1 881 ÷ 2 = 940 + 1;
  • 940 ÷ 2 = 470 + 0;
  • 470 ÷ 2 = 235 + 0;
  • 235 ÷ 2 = 117 + 1;
  • 117 ÷ 2 = 58 + 1;
  • 58 ÷ 2 = 29 + 0;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 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.

1 010 111 110 903(10) = 1110 1011 0010 1111 0101 0000 0101 1110 1111 0111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 40.

  • 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, 40,

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 1 010 111 110 903(10) converted to signed binary in one's complement representation:

1 010 111 110 903(10) = 0000 0000 0000 0000 0000 0000 1110 1011 0010 1111 0101 0000 0101 1110 1111 0111

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