Convert 111 111 111 215 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 111 111 111 215(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
111 111 111 215 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;
  • 111 111 111 215 ÷ 2 = 55 555 555 607 + 1;
  • 55 555 555 607 ÷ 2 = 27 777 777 803 + 1;
  • 27 777 777 803 ÷ 2 = 13 888 888 901 + 1;
  • 13 888 888 901 ÷ 2 = 6 944 444 450 + 1;
  • 6 944 444 450 ÷ 2 = 3 472 222 225 + 0;
  • 3 472 222 225 ÷ 2 = 1 736 111 112 + 1;
  • 1 736 111 112 ÷ 2 = 868 055 556 + 0;
  • 868 055 556 ÷ 2 = 434 027 778 + 0;
  • 434 027 778 ÷ 2 = 217 013 889 + 0;
  • 217 013 889 ÷ 2 = 108 506 944 + 1;
  • 108 506 944 ÷ 2 = 54 253 472 + 0;
  • 54 253 472 ÷ 2 = 27 126 736 + 0;
  • 27 126 736 ÷ 2 = 13 563 368 + 0;
  • 13 563 368 ÷ 2 = 6 781 684 + 0;
  • 6 781 684 ÷ 2 = 3 390 842 + 0;
  • 3 390 842 ÷ 2 = 1 695 421 + 0;
  • 1 695 421 ÷ 2 = 847 710 + 1;
  • 847 710 ÷ 2 = 423 855 + 0;
  • 423 855 ÷ 2 = 211 927 + 1;
  • 211 927 ÷ 2 = 105 963 + 1;
  • 105 963 ÷ 2 = 52 981 + 1;
  • 52 981 ÷ 2 = 26 490 + 1;
  • 26 490 ÷ 2 = 13 245 + 0;
  • 13 245 ÷ 2 = 6 622 + 1;
  • 6 622 ÷ 2 = 3 311 + 0;
  • 3 311 ÷ 2 = 1 655 + 1;
  • 1 655 ÷ 2 = 827 + 1;
  • 827 ÷ 2 = 413 + 1;
  • 413 ÷ 2 = 206 + 1;
  • 206 ÷ 2 = 103 + 0;
  • 103 ÷ 2 = 51 + 1;
  • 51 ÷ 2 = 25 + 1;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 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.

111 111 111 215(10) = 1 1001 1101 1110 1011 1101 0000 0010 0010 1111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 37.

  • 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, 37,

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 111 111 111 215(10) converted to signed binary in two's complement representation:

111 111 111 215(10) = 0000 0000 0000 0000 0000 0000 0001 1001 1101 1110 1011 1101 0000 0010 0010 1111

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