Convert 111 111 111 111 222 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 111 111 111 111 222(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
111 111 111 111 222 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;
  • 111 111 111 111 222 ÷ 2 = 55 555 555 555 611 + 0;
  • 55 555 555 555 611 ÷ 2 = 27 777 777 777 805 + 1;
  • 27 777 777 777 805 ÷ 2 = 13 888 888 888 902 + 1;
  • 13 888 888 888 902 ÷ 2 = 6 944 444 444 451 + 0;
  • 6 944 444 444 451 ÷ 2 = 3 472 222 222 225 + 1;
  • 3 472 222 222 225 ÷ 2 = 1 736 111 111 112 + 1;
  • 1 736 111 111 112 ÷ 2 = 868 055 555 556 + 0;
  • 868 055 555 556 ÷ 2 = 434 027 777 778 + 0;
  • 434 027 777 778 ÷ 2 = 217 013 888 889 + 0;
  • 217 013 888 889 ÷ 2 = 108 506 944 444 + 1;
  • 108 506 944 444 ÷ 2 = 54 253 472 222 + 0;
  • 54 253 472 222 ÷ 2 = 27 126 736 111 + 0;
  • 27 126 736 111 ÷ 2 = 13 563 368 055 + 1;
  • 13 563 368 055 ÷ 2 = 6 781 684 027 + 1;
  • 6 781 684 027 ÷ 2 = 3 390 842 013 + 1;
  • 3 390 842 013 ÷ 2 = 1 695 421 006 + 1;
  • 1 695 421 006 ÷ 2 = 847 710 503 + 0;
  • 847 710 503 ÷ 2 = 423 855 251 + 1;
  • 423 855 251 ÷ 2 = 211 927 625 + 1;
  • 211 927 625 ÷ 2 = 105 963 812 + 1;
  • 105 963 812 ÷ 2 = 52 981 906 + 0;
  • 52 981 906 ÷ 2 = 26 490 953 + 0;
  • 26 490 953 ÷ 2 = 13 245 476 + 1;
  • 13 245 476 ÷ 2 = 6 622 738 + 0;
  • 6 622 738 ÷ 2 = 3 311 369 + 0;
  • 3 311 369 ÷ 2 = 1 655 684 + 1;
  • 1 655 684 ÷ 2 = 827 842 + 0;
  • 827 842 ÷ 2 = 413 921 + 0;
  • 413 921 ÷ 2 = 206 960 + 1;
  • 206 960 ÷ 2 = 103 480 + 0;
  • 103 480 ÷ 2 = 51 740 + 0;
  • 51 740 ÷ 2 = 25 870 + 0;
  • 25 870 ÷ 2 = 12 935 + 0;
  • 12 935 ÷ 2 = 6 467 + 1;
  • 6 467 ÷ 2 = 3 233 + 1;
  • 3 233 ÷ 2 = 1 616 + 1;
  • 1 616 ÷ 2 = 808 + 0;
  • 808 ÷ 2 = 404 + 0;
  • 404 ÷ 2 = 202 + 0;
  • 202 ÷ 2 = 101 + 0;
  • 101 ÷ 2 = 50 + 1;
  • 50 ÷ 2 = 25 + 0;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 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.

111 111 111 111 222(10) = 110 0101 0000 1110 0001 0010 0100 1110 1111 0010 0011 0110(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 47.

  • 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, 47,

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 111 111 111 111 222(10) converted to signed binary in one's complement representation:

111 111 111 111 222(10) = 0000 0000 0000 0000 0110 0101 0000 1110 0001 0010 0100 1110 1111 0010 0011 0110

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