Convert 1 111 111 110 009 939 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 111 111 110 009 939(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 111 111 110 009 939 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;
  • 1 111 111 110 009 939 ÷ 2 = 555 555 555 004 969 + 1;
  • 555 555 555 004 969 ÷ 2 = 277 777 777 502 484 + 1;
  • 277 777 777 502 484 ÷ 2 = 138 888 888 751 242 + 0;
  • 138 888 888 751 242 ÷ 2 = 69 444 444 375 621 + 0;
  • 69 444 444 375 621 ÷ 2 = 34 722 222 187 810 + 1;
  • 34 722 222 187 810 ÷ 2 = 17 361 111 093 905 + 0;
  • 17 361 111 093 905 ÷ 2 = 8 680 555 546 952 + 1;
  • 8 680 555 546 952 ÷ 2 = 4 340 277 773 476 + 0;
  • 4 340 277 773 476 ÷ 2 = 2 170 138 886 738 + 0;
  • 2 170 138 886 738 ÷ 2 = 1 085 069 443 369 + 0;
  • 1 085 069 443 369 ÷ 2 = 542 534 721 684 + 1;
  • 542 534 721 684 ÷ 2 = 271 267 360 842 + 0;
  • 271 267 360 842 ÷ 2 = 135 633 680 421 + 0;
  • 135 633 680 421 ÷ 2 = 67 816 840 210 + 1;
  • 67 816 840 210 ÷ 2 = 33 908 420 105 + 0;
  • 33 908 420 105 ÷ 2 = 16 954 210 052 + 1;
  • 16 954 210 052 ÷ 2 = 8 477 105 026 + 0;
  • 8 477 105 026 ÷ 2 = 4 238 552 513 + 0;
  • 4 238 552 513 ÷ 2 = 2 119 276 256 + 1;
  • 2 119 276 256 ÷ 2 = 1 059 638 128 + 0;
  • 1 059 638 128 ÷ 2 = 529 819 064 + 0;
  • 529 819 064 ÷ 2 = 264 909 532 + 0;
  • 264 909 532 ÷ 2 = 132 454 766 + 0;
  • 132 454 766 ÷ 2 = 66 227 383 + 0;
  • 66 227 383 ÷ 2 = 33 113 691 + 1;
  • 33 113 691 ÷ 2 = 16 556 845 + 1;
  • 16 556 845 ÷ 2 = 8 278 422 + 1;
  • 8 278 422 ÷ 2 = 4 139 211 + 0;
  • 4 139 211 ÷ 2 = 2 069 605 + 1;
  • 2 069 605 ÷ 2 = 1 034 802 + 1;
  • 1 034 802 ÷ 2 = 517 401 + 0;
  • 517 401 ÷ 2 = 258 700 + 1;
  • 258 700 ÷ 2 = 129 350 + 0;
  • 129 350 ÷ 2 = 64 675 + 0;
  • 64 675 ÷ 2 = 32 337 + 1;
  • 32 337 ÷ 2 = 16 168 + 1;
  • 16 168 ÷ 2 = 8 084 + 0;
  • 8 084 ÷ 2 = 4 042 + 0;
  • 4 042 ÷ 2 = 2 021 + 0;
  • 2 021 ÷ 2 = 1 010 + 1;
  • 1 010 ÷ 2 = 505 + 0;
  • 505 ÷ 2 = 252 + 1;
  • 252 ÷ 2 = 126 + 0;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 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.

1 111 111 110 009 939(10) = 11 1111 0010 1000 1100 1011 0111 0000 0100 1010 0100 0101 0011(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 1 111 111 110 009 939(10) converted to signed binary in two's complement representation:

1 111 111 110 009 939(10) = 0000 0000 0000 0011 1111 0010 1000 1100 1011 0111 0000 0100 1010 0100 0101 0011

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