Two's Complement: Integer ↗ Binary: 1 011 111 111 109 981 Convert the Integer Number to a Signed Binary in Two's Complement Representation. Write the Base Ten Decimal System Number as a Binary Code (Written in Base Two)

Signed integer number 1 011 111 111 109 981(10) converted and written as a signed binary in two's complement representation (base 2) = ?

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;
  • 1 011 111 111 109 981 ÷ 2 = 505 555 555 554 990 + 1;
  • 505 555 555 554 990 ÷ 2 = 252 777 777 777 495 + 0;
  • 252 777 777 777 495 ÷ 2 = 126 388 888 888 747 + 1;
  • 126 388 888 888 747 ÷ 2 = 63 194 444 444 373 + 1;
  • 63 194 444 444 373 ÷ 2 = 31 597 222 222 186 + 1;
  • 31 597 222 222 186 ÷ 2 = 15 798 611 111 093 + 0;
  • 15 798 611 111 093 ÷ 2 = 7 899 305 555 546 + 1;
  • 7 899 305 555 546 ÷ 2 = 3 949 652 777 773 + 0;
  • 3 949 652 777 773 ÷ 2 = 1 974 826 388 886 + 1;
  • 1 974 826 388 886 ÷ 2 = 987 413 194 443 + 0;
  • 987 413 194 443 ÷ 2 = 493 706 597 221 + 1;
  • 493 706 597 221 ÷ 2 = 246 853 298 610 + 1;
  • 246 853 298 610 ÷ 2 = 123 426 649 305 + 0;
  • 123 426 649 305 ÷ 2 = 61 713 324 652 + 1;
  • 61 713 324 652 ÷ 2 = 30 856 662 326 + 0;
  • 30 856 662 326 ÷ 2 = 15 428 331 163 + 0;
  • 15 428 331 163 ÷ 2 = 7 714 165 581 + 1;
  • 7 714 165 581 ÷ 2 = 3 857 082 790 + 1;
  • 3 857 082 790 ÷ 2 = 1 928 541 395 + 0;
  • 1 928 541 395 ÷ 2 = 964 270 697 + 1;
  • 964 270 697 ÷ 2 = 482 135 348 + 1;
  • 482 135 348 ÷ 2 = 241 067 674 + 0;
  • 241 067 674 ÷ 2 = 120 533 837 + 0;
  • 120 533 837 ÷ 2 = 60 266 918 + 1;
  • 60 266 918 ÷ 2 = 30 133 459 + 0;
  • 30 133 459 ÷ 2 = 15 066 729 + 1;
  • 15 066 729 ÷ 2 = 7 533 364 + 1;
  • 7 533 364 ÷ 2 = 3 766 682 + 0;
  • 3 766 682 ÷ 2 = 1 883 341 + 0;
  • 1 883 341 ÷ 2 = 941 670 + 1;
  • 941 670 ÷ 2 = 470 835 + 0;
  • 470 835 ÷ 2 = 235 417 + 1;
  • 235 417 ÷ 2 = 117 708 + 1;
  • 117 708 ÷ 2 = 58 854 + 0;
  • 58 854 ÷ 2 = 29 427 + 0;
  • 29 427 ÷ 2 = 14 713 + 1;
  • 14 713 ÷ 2 = 7 356 + 1;
  • 7 356 ÷ 2 = 3 678 + 0;
  • 3 678 ÷ 2 = 1 839 + 0;
  • 1 839 ÷ 2 = 919 + 1;
  • 919 ÷ 2 = 459 + 1;
  • 459 ÷ 2 = 229 + 1;
  • 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.


1 011 111 111 109 981(10) = 11 1001 0111 1001 1001 1010 0110 1001 1011 0010 1101 0101 1101(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 50.

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, 50,

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.


Number 1 011 111 111 109 981(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in two's complement representation:

1 011 111 111 109 981(10) = 0000 0000 0000 0011 1001 0111 1001 1001 1010 0110 1001 1011 0010 1101 0101 1101

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

More operations on integers written as signed binary numbers in two's complement representation:

» Signed integer number 1 011 111 111 109 980 converted from decimal system (from base 10) and written as a signed binary in two's complement representation (written in base 2) = ?

» Signed integer number 1 011 111 111 109 982 converted from decimal system (from base 10) and written as a signed binary in two's complement representation (written in base 2) = ?

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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