Convert 111 111 099 999 727 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 111 111 099 999 727(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
111 111 099 999 727 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;
  • 111 111 099 999 727 ÷ 2 = 55 555 549 999 863 + 1;
  • 55 555 549 999 863 ÷ 2 = 27 777 774 999 931 + 1;
  • 27 777 774 999 931 ÷ 2 = 13 888 887 499 965 + 1;
  • 13 888 887 499 965 ÷ 2 = 6 944 443 749 982 + 1;
  • 6 944 443 749 982 ÷ 2 = 3 472 221 874 991 + 0;
  • 3 472 221 874 991 ÷ 2 = 1 736 110 937 495 + 1;
  • 1 736 110 937 495 ÷ 2 = 868 055 468 747 + 1;
  • 868 055 468 747 ÷ 2 = 434 027 734 373 + 1;
  • 434 027 734 373 ÷ 2 = 217 013 867 186 + 1;
  • 217 013 867 186 ÷ 2 = 108 506 933 593 + 0;
  • 108 506 933 593 ÷ 2 = 54 253 466 796 + 1;
  • 54 253 466 796 ÷ 2 = 27 126 733 398 + 0;
  • 27 126 733 398 ÷ 2 = 13 563 366 699 + 0;
  • 13 563 366 699 ÷ 2 = 6 781 683 349 + 1;
  • 6 781 683 349 ÷ 2 = 3 390 841 674 + 1;
  • 3 390 841 674 ÷ 2 = 1 695 420 837 + 0;
  • 1 695 420 837 ÷ 2 = 847 710 418 + 1;
  • 847 710 418 ÷ 2 = 423 855 209 + 0;
  • 423 855 209 ÷ 2 = 211 927 604 + 1;
  • 211 927 604 ÷ 2 = 105 963 802 + 0;
  • 105 963 802 ÷ 2 = 52 981 901 + 0;
  • 52 981 901 ÷ 2 = 26 490 950 + 1;
  • 26 490 950 ÷ 2 = 13 245 475 + 0;
  • 13 245 475 ÷ 2 = 6 622 737 + 1;
  • 6 622 737 ÷ 2 = 3 311 368 + 1;
  • 3 311 368 ÷ 2 = 1 655 684 + 0;
  • 1 655 684 ÷ 2 = 827 842 + 0;
  • 827 842 ÷ 2 = 413 921 + 0;
  • 413 921 ÷ 2 = 206 960 + 1;
  • 206 960 ÷ 2 = 103 480 + 0;
  • 103 480 ÷ 2 = 51 740 + 0;
  • 51 740 ÷ 2 = 25 870 + 0;
  • 25 870 ÷ 2 = 12 935 + 0;
  • 12 935 ÷ 2 = 6 467 + 1;
  • 6 467 ÷ 2 = 3 233 + 1;
  • 3 233 ÷ 2 = 1 616 + 1;
  • 1 616 ÷ 2 = 808 + 0;
  • 808 ÷ 2 = 404 + 0;
  • 404 ÷ 2 = 202 + 0;
  • 202 ÷ 2 = 101 + 0;
  • 101 ÷ 2 = 50 + 1;
  • 50 ÷ 2 = 25 + 0;
  • 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.

111 111 099 999 727(10) = 110 0101 0000 1110 0001 0001 1010 0101 0110 0101 1110 1111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 47.

  • 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, 47,

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 111 111 099 999 727(10) converted to signed binary in two's complement representation:

111 111 099 999 727(10) = 0000 0000 0000 0000 0110 0101 0000 1110 0001 0001 1010 0101 0110 0101 1110 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