Convert 2 018 091 203 020 695 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 2 018 091 203 020 695(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
2 018 091 203 020 695 to a signed binary in two's (2'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.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 2 018 091 203 020 695 ÷ 2 = 1 009 045 601 510 347 + 1;
  • 1 009 045 601 510 347 ÷ 2 = 504 522 800 755 173 + 1;
  • 504 522 800 755 173 ÷ 2 = 252 261 400 377 586 + 1;
  • 252 261 400 377 586 ÷ 2 = 126 130 700 188 793 + 0;
  • 126 130 700 188 793 ÷ 2 = 63 065 350 094 396 + 1;
  • 63 065 350 094 396 ÷ 2 = 31 532 675 047 198 + 0;
  • 31 532 675 047 198 ÷ 2 = 15 766 337 523 599 + 0;
  • 15 766 337 523 599 ÷ 2 = 7 883 168 761 799 + 1;
  • 7 883 168 761 799 ÷ 2 = 3 941 584 380 899 + 1;
  • 3 941 584 380 899 ÷ 2 = 1 970 792 190 449 + 1;
  • 1 970 792 190 449 ÷ 2 = 985 396 095 224 + 1;
  • 985 396 095 224 ÷ 2 = 492 698 047 612 + 0;
  • 492 698 047 612 ÷ 2 = 246 349 023 806 + 0;
  • 246 349 023 806 ÷ 2 = 123 174 511 903 + 0;
  • 123 174 511 903 ÷ 2 = 61 587 255 951 + 1;
  • 61 587 255 951 ÷ 2 = 30 793 627 975 + 1;
  • 30 793 627 975 ÷ 2 = 15 396 813 987 + 1;
  • 15 396 813 987 ÷ 2 = 7 698 406 993 + 1;
  • 7 698 406 993 ÷ 2 = 3 849 203 496 + 1;
  • 3 849 203 496 ÷ 2 = 1 924 601 748 + 0;
  • 1 924 601 748 ÷ 2 = 962 300 874 + 0;
  • 962 300 874 ÷ 2 = 481 150 437 + 0;
  • 481 150 437 ÷ 2 = 240 575 218 + 1;
  • 240 575 218 ÷ 2 = 120 287 609 + 0;
  • 120 287 609 ÷ 2 = 60 143 804 + 1;
  • 60 143 804 ÷ 2 = 30 071 902 + 0;
  • 30 071 902 ÷ 2 = 15 035 951 + 0;
  • 15 035 951 ÷ 2 = 7 517 975 + 1;
  • 7 517 975 ÷ 2 = 3 758 987 + 1;
  • 3 758 987 ÷ 2 = 1 879 493 + 1;
  • 1 879 493 ÷ 2 = 939 746 + 1;
  • 939 746 ÷ 2 = 469 873 + 0;
  • 469 873 ÷ 2 = 234 936 + 1;
  • 234 936 ÷ 2 = 117 468 + 0;
  • 117 468 ÷ 2 = 58 734 + 0;
  • 58 734 ÷ 2 = 29 367 + 0;
  • 29 367 ÷ 2 = 14 683 + 1;
  • 14 683 ÷ 2 = 7 341 + 1;
  • 7 341 ÷ 2 = 3 670 + 1;
  • 3 670 ÷ 2 = 1 835 + 0;
  • 1 835 ÷ 2 = 917 + 1;
  • 917 ÷ 2 = 458 + 1;
  • 458 ÷ 2 = 229 + 0;
  • 229 ÷ 2 = 114 + 1;
  • 114 ÷ 2 = 57 + 0;
  • 57 ÷ 2 = 28 + 1;
  • 28 ÷ 2 = 14 + 0;
  • 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.

2 018 091 203 020 695(10) = 111 0010 1011 0111 0001 0111 1001 0100 0111 1100 0111 1001 0111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 51.

  • 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, 51,

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 2 018 091 203 020 695(10) converted to signed binary in two's complement representation:

2 018 091 203 020 695(10) = 0000 0000 0000 0111 0010 1011 0111 0001 0111 1001 0100 0111 1100 0111 1001 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 decimal system to signed binary in two's complement representation

Follow the steps below to convert a signed base 10 integer number to signed binary in two'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, keeping track of each remainder, until we get a quotient that is 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, add extra bits on 0 in front (to the left) of the base 2 number above, up to the required length, so that the first bit (the leftmost) will 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 1s and all 1 bits with 0s (reversing the digits).
  • 6. To get the negative integer number, in signed binary two's complement representation, add 1 to the number above.

Example: convert the negative number -60 from the decimal system (base ten) to signed binary in two's complement:

  • 1. Start with the positive version of the number: |-60| = 60
  • 2. Divide repeatedly 60 by 2, keeping track of each remainder:
    • division = quotient + remainder
    • 60 ÷ 2 = 30 + 0
    • 30 ÷ 2 = 15 + 0
    • 15 ÷ 2 = 7 + 1
    • 7 ÷ 2 = 3 + 1
    • 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:
    60(10) = 11 1100(2)
  • 4. Bit length of base 2 representation number is 6, so the positive binary computer representation of a signed binary will take in this particular case 8 bits (the least power of 2 larger than 6) - add extra 0 digits in front of the base 2 number, up to the required length:
    60(10) = 0011 1100(2)
  • 5. To get the negative integer number representation in signed binary one's complement, replace all the 0 bits with 1s and all 1 bits with 0s (reversing the digits):
    !(0011 1100) = 1100 0011
  • 6. To get the negative integer number, signed binary in two's complement representation, add 1 to the number above:
    -60(10) = 1100 0011 + 1 = 1100 0100
  • Number -60(10), signed integer, converted from decimal system (base 10) to signed binary two's complement representation = 1100 0100