Convert 1 011 111 001 001 952 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 011 111 001 001 952(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 011 111 001 001 952 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;
  • 1 011 111 001 001 952 ÷ 2 = 505 555 500 500 976 + 0;
  • 505 555 500 500 976 ÷ 2 = 252 777 750 250 488 + 0;
  • 252 777 750 250 488 ÷ 2 = 126 388 875 125 244 + 0;
  • 126 388 875 125 244 ÷ 2 = 63 194 437 562 622 + 0;
  • 63 194 437 562 622 ÷ 2 = 31 597 218 781 311 + 0;
  • 31 597 218 781 311 ÷ 2 = 15 798 609 390 655 + 1;
  • 15 798 609 390 655 ÷ 2 = 7 899 304 695 327 + 1;
  • 7 899 304 695 327 ÷ 2 = 3 949 652 347 663 + 1;
  • 3 949 652 347 663 ÷ 2 = 1 974 826 173 831 + 1;
  • 1 974 826 173 831 ÷ 2 = 987 413 086 915 + 1;
  • 987 413 086 915 ÷ 2 = 493 706 543 457 + 1;
  • 493 706 543 457 ÷ 2 = 246 853 271 728 + 1;
  • 246 853 271 728 ÷ 2 = 123 426 635 864 + 0;
  • 123 426 635 864 ÷ 2 = 61 713 317 932 + 0;
  • 61 713 317 932 ÷ 2 = 30 856 658 966 + 0;
  • 30 856 658 966 ÷ 2 = 15 428 329 483 + 0;
  • 15 428 329 483 ÷ 2 = 7 714 164 741 + 1;
  • 7 714 164 741 ÷ 2 = 3 857 082 370 + 1;
  • 3 857 082 370 ÷ 2 = 1 928 541 185 + 0;
  • 1 928 541 185 ÷ 2 = 964 270 592 + 1;
  • 964 270 592 ÷ 2 = 482 135 296 + 0;
  • 482 135 296 ÷ 2 = 241 067 648 + 0;
  • 241 067 648 ÷ 2 = 120 533 824 + 0;
  • 120 533 824 ÷ 2 = 60 266 912 + 0;
  • 60 266 912 ÷ 2 = 30 133 456 + 0;
  • 30 133 456 ÷ 2 = 15 066 728 + 0;
  • 15 066 728 ÷ 2 = 7 533 364 + 0;
  • 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.

1 011 111 001 001 952(10) = 11 1001 0111 1001 1001 1010 0000 0000 1011 0000 1111 1110 0000(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.


Decimal Number 1 011 111 001 001 952(10) converted to signed binary in two's complement representation:

1 011 111 001 001 952(10) = 0000 0000 0000 0011 1001 0111 1001 1001 1010 0000 0000 1011 0000 1111 1110 0000

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