Convert 25 746 071 050 631 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 25 746 071 050 631(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
25 746 071 050 631 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;
  • 25 746 071 050 631 ÷ 2 = 12 873 035 525 315 + 1;
  • 12 873 035 525 315 ÷ 2 = 6 436 517 762 657 + 1;
  • 6 436 517 762 657 ÷ 2 = 3 218 258 881 328 + 1;
  • 3 218 258 881 328 ÷ 2 = 1 609 129 440 664 + 0;
  • 1 609 129 440 664 ÷ 2 = 804 564 720 332 + 0;
  • 804 564 720 332 ÷ 2 = 402 282 360 166 + 0;
  • 402 282 360 166 ÷ 2 = 201 141 180 083 + 0;
  • 201 141 180 083 ÷ 2 = 100 570 590 041 + 1;
  • 100 570 590 041 ÷ 2 = 50 285 295 020 + 1;
  • 50 285 295 020 ÷ 2 = 25 142 647 510 + 0;
  • 25 142 647 510 ÷ 2 = 12 571 323 755 + 0;
  • 12 571 323 755 ÷ 2 = 6 285 661 877 + 1;
  • 6 285 661 877 ÷ 2 = 3 142 830 938 + 1;
  • 3 142 830 938 ÷ 2 = 1 571 415 469 + 0;
  • 1 571 415 469 ÷ 2 = 785 707 734 + 1;
  • 785 707 734 ÷ 2 = 392 853 867 + 0;
  • 392 853 867 ÷ 2 = 196 426 933 + 1;
  • 196 426 933 ÷ 2 = 98 213 466 + 1;
  • 98 213 466 ÷ 2 = 49 106 733 + 0;
  • 49 106 733 ÷ 2 = 24 553 366 + 1;
  • 24 553 366 ÷ 2 = 12 276 683 + 0;
  • 12 276 683 ÷ 2 = 6 138 341 + 1;
  • 6 138 341 ÷ 2 = 3 069 170 + 1;
  • 3 069 170 ÷ 2 = 1 534 585 + 0;
  • 1 534 585 ÷ 2 = 767 292 + 1;
  • 767 292 ÷ 2 = 383 646 + 0;
  • 383 646 ÷ 2 = 191 823 + 0;
  • 191 823 ÷ 2 = 95 911 + 1;
  • 95 911 ÷ 2 = 47 955 + 1;
  • 47 955 ÷ 2 = 23 977 + 1;
  • 23 977 ÷ 2 = 11 988 + 1;
  • 11 988 ÷ 2 = 5 994 + 0;
  • 5 994 ÷ 2 = 2 997 + 0;
  • 2 997 ÷ 2 = 1 498 + 1;
  • 1 498 ÷ 2 = 749 + 0;
  • 749 ÷ 2 = 374 + 1;
  • 374 ÷ 2 = 187 + 0;
  • 187 ÷ 2 = 93 + 1;
  • 93 ÷ 2 = 46 + 1;
  • 46 ÷ 2 = 23 + 0;
  • 23 ÷ 2 = 11 + 1;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 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.

25 746 071 050 631(10) = 1 0111 0110 1010 0111 1001 0110 1011 0101 1001 1000 0111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 45.

  • 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, 45,

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 25 746 071 050 631(10) converted to signed binary in one's complement representation:

25 746 071 050 631(10) = 0000 0000 0000 0000 0001 0111 0110 1010 0111 1001 0110 1011 0101 1001 1000 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