Convert 11 111 111 111 013 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 11 111 111 111 013(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
11 111 111 111 013 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;
  • 11 111 111 111 013 ÷ 2 = 5 555 555 555 506 + 1;
  • 5 555 555 555 506 ÷ 2 = 2 777 777 777 753 + 0;
  • 2 777 777 777 753 ÷ 2 = 1 388 888 888 876 + 1;
  • 1 388 888 888 876 ÷ 2 = 694 444 444 438 + 0;
  • 694 444 444 438 ÷ 2 = 347 222 222 219 + 0;
  • 347 222 222 219 ÷ 2 = 173 611 111 109 + 1;
  • 173 611 111 109 ÷ 2 = 86 805 555 554 + 1;
  • 86 805 555 554 ÷ 2 = 43 402 777 777 + 0;
  • 43 402 777 777 ÷ 2 = 21 701 388 888 + 1;
  • 21 701 388 888 ÷ 2 = 10 850 694 444 + 0;
  • 10 850 694 444 ÷ 2 = 5 425 347 222 + 0;
  • 5 425 347 222 ÷ 2 = 2 712 673 611 + 0;
  • 2 712 673 611 ÷ 2 = 1 356 336 805 + 1;
  • 1 356 336 805 ÷ 2 = 678 168 402 + 1;
  • 678 168 402 ÷ 2 = 339 084 201 + 0;
  • 339 084 201 ÷ 2 = 169 542 100 + 1;
  • 169 542 100 ÷ 2 = 84 771 050 + 0;
  • 84 771 050 ÷ 2 = 42 385 525 + 0;
  • 42 385 525 ÷ 2 = 21 192 762 + 1;
  • 21 192 762 ÷ 2 = 10 596 381 + 0;
  • 10 596 381 ÷ 2 = 5 298 190 + 1;
  • 5 298 190 ÷ 2 = 2 649 095 + 0;
  • 2 649 095 ÷ 2 = 1 324 547 + 1;
  • 1 324 547 ÷ 2 = 662 273 + 1;
  • 662 273 ÷ 2 = 331 136 + 1;
  • 331 136 ÷ 2 = 165 568 + 0;
  • 165 568 ÷ 2 = 82 784 + 0;
  • 82 784 ÷ 2 = 41 392 + 0;
  • 41 392 ÷ 2 = 20 696 + 0;
  • 20 696 ÷ 2 = 10 348 + 0;
  • 10 348 ÷ 2 = 5 174 + 0;
  • 5 174 ÷ 2 = 2 587 + 0;
  • 2 587 ÷ 2 = 1 293 + 1;
  • 1 293 ÷ 2 = 646 + 1;
  • 646 ÷ 2 = 323 + 0;
  • 323 ÷ 2 = 161 + 1;
  • 161 ÷ 2 = 80 + 1;
  • 80 ÷ 2 = 40 + 0;
  • 40 ÷ 2 = 20 + 0;
  • 20 ÷ 2 = 10 + 0;
  • 10 ÷ 2 = 5 + 0;
  • 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.

11 111 111 111 013(10) = 1010 0001 1011 0000 0001 1101 0100 1011 0001 0110 0101(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 44.

  • 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, 44,

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 11 111 111 111 013(10) converted to signed binary in two's complement representation:

11 111 111 111 013(10) = 0000 0000 0000 0000 0000 1010 0001 1011 0000 0001 1101 0100 1011 0001 0110 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