Convert 1 000 110 110 106 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 1 000 110 110 106(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
1 000 110 110 106 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;
  • 1 000 110 110 106 ÷ 2 = 500 055 055 053 + 0;
  • 500 055 055 053 ÷ 2 = 250 027 527 526 + 1;
  • 250 027 527 526 ÷ 2 = 125 013 763 763 + 0;
  • 125 013 763 763 ÷ 2 = 62 506 881 881 + 1;
  • 62 506 881 881 ÷ 2 = 31 253 440 940 + 1;
  • 31 253 440 940 ÷ 2 = 15 626 720 470 + 0;
  • 15 626 720 470 ÷ 2 = 7 813 360 235 + 0;
  • 7 813 360 235 ÷ 2 = 3 906 680 117 + 1;
  • 3 906 680 117 ÷ 2 = 1 953 340 058 + 1;
  • 1 953 340 058 ÷ 2 = 976 670 029 + 0;
  • 976 670 029 ÷ 2 = 488 335 014 + 1;
  • 488 335 014 ÷ 2 = 244 167 507 + 0;
  • 244 167 507 ÷ 2 = 122 083 753 + 1;
  • 122 083 753 ÷ 2 = 61 041 876 + 1;
  • 61 041 876 ÷ 2 = 30 520 938 + 0;
  • 30 520 938 ÷ 2 = 15 260 469 + 0;
  • 15 260 469 ÷ 2 = 7 630 234 + 1;
  • 7 630 234 ÷ 2 = 3 815 117 + 0;
  • 3 815 117 ÷ 2 = 1 907 558 + 1;
  • 1 907 558 ÷ 2 = 953 779 + 0;
  • 953 779 ÷ 2 = 476 889 + 1;
  • 476 889 ÷ 2 = 238 444 + 1;
  • 238 444 ÷ 2 = 119 222 + 0;
  • 119 222 ÷ 2 = 59 611 + 0;
  • 59 611 ÷ 2 = 29 805 + 1;
  • 29 805 ÷ 2 = 14 902 + 1;
  • 14 902 ÷ 2 = 7 451 + 0;
  • 7 451 ÷ 2 = 3 725 + 1;
  • 3 725 ÷ 2 = 1 862 + 1;
  • 1 862 ÷ 2 = 931 + 0;
  • 931 ÷ 2 = 465 + 1;
  • 465 ÷ 2 = 232 + 1;
  • 232 ÷ 2 = 116 + 0;
  • 116 ÷ 2 = 58 + 0;
  • 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.

1 000 110 110 106(10) = 1110 1000 1101 1011 0011 0101 0011 0101 1001 1010(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 1 000 110 110 106(10) converted to signed binary in one's complement representation:

1 000 110 110 106(10) = 0000 0000 0000 0000 0000 0000 1110 1000 1101 1011 0011 0101 0011 0101 1001 1010

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