Convert 111 111 101 110 451 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 111 111 101 110 451(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
111 111 101 110 451 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;
  • 111 111 101 110 451 ÷ 2 = 55 555 550 555 225 + 1;
  • 55 555 550 555 225 ÷ 2 = 27 777 775 277 612 + 1;
  • 27 777 775 277 612 ÷ 2 = 13 888 887 638 806 + 0;
  • 13 888 887 638 806 ÷ 2 = 6 944 443 819 403 + 0;
  • 6 944 443 819 403 ÷ 2 = 3 472 221 909 701 + 1;
  • 3 472 221 909 701 ÷ 2 = 1 736 110 954 850 + 1;
  • 1 736 110 954 850 ÷ 2 = 868 055 477 425 + 0;
  • 868 055 477 425 ÷ 2 = 434 027 738 712 + 1;
  • 434 027 738 712 ÷ 2 = 217 013 869 356 + 0;
  • 217 013 869 356 ÷ 2 = 108 506 934 678 + 0;
  • 108 506 934 678 ÷ 2 = 54 253 467 339 + 0;
  • 54 253 467 339 ÷ 2 = 27 126 733 669 + 1;
  • 27 126 733 669 ÷ 2 = 13 563 366 834 + 1;
  • 13 563 366 834 ÷ 2 = 6 781 683 417 + 0;
  • 6 781 683 417 ÷ 2 = 3 390 841 708 + 1;
  • 3 390 841 708 ÷ 2 = 1 695 420 854 + 0;
  • 1 695 420 854 ÷ 2 = 847 710 427 + 0;
  • 847 710 427 ÷ 2 = 423 855 213 + 1;
  • 423 855 213 ÷ 2 = 211 927 606 + 1;
  • 211 927 606 ÷ 2 = 105 963 803 + 0;
  • 105 963 803 ÷ 2 = 52 981 901 + 1;
  • 52 981 901 ÷ 2 = 26 490 950 + 1;
  • 26 490 950 ÷ 2 = 13 245 475 + 0;
  • 13 245 475 ÷ 2 = 6 622 737 + 1;
  • 6 622 737 ÷ 2 = 3 311 368 + 1;
  • 3 311 368 ÷ 2 = 1 655 684 + 0;
  • 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.

111 111 101 110 451(10) = 110 0101 0000 1110 0001 0001 1011 0110 0101 1000 1011 0011(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 111 111 101 110 451(10) converted to signed binary in one's complement representation:

111 111 101 110 451(10) = 0000 0000 0000 0000 0110 0101 0000 1110 0001 0001 1011 0110 0101 1000 1011 0011

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