Convert 11 001 011 110 111 317 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 11 001 011 110 111 317(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
11 001 011 110 111 317 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;
  • 11 001 011 110 111 317 ÷ 2 = 5 500 505 555 055 658 + 1;
  • 5 500 505 555 055 658 ÷ 2 = 2 750 252 777 527 829 + 0;
  • 2 750 252 777 527 829 ÷ 2 = 1 375 126 388 763 914 + 1;
  • 1 375 126 388 763 914 ÷ 2 = 687 563 194 381 957 + 0;
  • 687 563 194 381 957 ÷ 2 = 343 781 597 190 978 + 1;
  • 343 781 597 190 978 ÷ 2 = 171 890 798 595 489 + 0;
  • 171 890 798 595 489 ÷ 2 = 85 945 399 297 744 + 1;
  • 85 945 399 297 744 ÷ 2 = 42 972 699 648 872 + 0;
  • 42 972 699 648 872 ÷ 2 = 21 486 349 824 436 + 0;
  • 21 486 349 824 436 ÷ 2 = 10 743 174 912 218 + 0;
  • 10 743 174 912 218 ÷ 2 = 5 371 587 456 109 + 0;
  • 5 371 587 456 109 ÷ 2 = 2 685 793 728 054 + 1;
  • 2 685 793 728 054 ÷ 2 = 1 342 896 864 027 + 0;
  • 1 342 896 864 027 ÷ 2 = 671 448 432 013 + 1;
  • 671 448 432 013 ÷ 2 = 335 724 216 006 + 1;
  • 335 724 216 006 ÷ 2 = 167 862 108 003 + 0;
  • 167 862 108 003 ÷ 2 = 83 931 054 001 + 1;
  • 83 931 054 001 ÷ 2 = 41 965 527 000 + 1;
  • 41 965 527 000 ÷ 2 = 20 982 763 500 + 0;
  • 20 982 763 500 ÷ 2 = 10 491 381 750 + 0;
  • 10 491 381 750 ÷ 2 = 5 245 690 875 + 0;
  • 5 245 690 875 ÷ 2 = 2 622 845 437 + 1;
  • 2 622 845 437 ÷ 2 = 1 311 422 718 + 1;
  • 1 311 422 718 ÷ 2 = 655 711 359 + 0;
  • 655 711 359 ÷ 2 = 327 855 679 + 1;
  • 327 855 679 ÷ 2 = 163 927 839 + 1;
  • 163 927 839 ÷ 2 = 81 963 919 + 1;
  • 81 963 919 ÷ 2 = 40 981 959 + 1;
  • 40 981 959 ÷ 2 = 20 490 979 + 1;
  • 20 490 979 ÷ 2 = 10 245 489 + 1;
  • 10 245 489 ÷ 2 = 5 122 744 + 1;
  • 5 122 744 ÷ 2 = 2 561 372 + 0;
  • 2 561 372 ÷ 2 = 1 280 686 + 0;
  • 1 280 686 ÷ 2 = 640 343 + 0;
  • 640 343 ÷ 2 = 320 171 + 1;
  • 320 171 ÷ 2 = 160 085 + 1;
  • 160 085 ÷ 2 = 80 042 + 1;
  • 80 042 ÷ 2 = 40 021 + 0;
  • 40 021 ÷ 2 = 20 010 + 1;
  • 20 010 ÷ 2 = 10 005 + 0;
  • 10 005 ÷ 2 = 5 002 + 1;
  • 5 002 ÷ 2 = 2 501 + 0;
  • 2 501 ÷ 2 = 1 250 + 1;
  • 1 250 ÷ 2 = 625 + 0;
  • 625 ÷ 2 = 312 + 1;
  • 312 ÷ 2 = 156 + 0;
  • 156 ÷ 2 = 78 + 0;
  • 78 ÷ 2 = 39 + 0;
  • 39 ÷ 2 = 19 + 1;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 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.

11 001 011 110 111 317(10) = 10 0111 0001 0101 0101 1100 0111 1111 0110 0011 0110 1000 0101 0101(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 54.

  • 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, 54,

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 11 001 011 110 111 317(10) converted to signed binary in two's complement representation:

11 001 011 110 111 317(10) = 0000 0000 0010 0111 0001 0101 0101 1100 0111 1111 0110 0011 0110 1000 0101 0101

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