Two's Complement: Integer ↗ Binary: 111 011 000 999 965 Convert the Integer Number to a Signed Binary in Two's Complement Representation. Write the Base Ten Decimal System Number as a Binary Code (Written in Base Two)

Signed integer number 111 011 000 999 965(10) converted and written as a signed binary in two's complement representation (base 2) = ?

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;
  • 111 011 000 999 965 ÷ 2 = 55 505 500 499 982 + 1;
  • 55 505 500 499 982 ÷ 2 = 27 752 750 249 991 + 0;
  • 27 752 750 249 991 ÷ 2 = 13 876 375 124 995 + 1;
  • 13 876 375 124 995 ÷ 2 = 6 938 187 562 497 + 1;
  • 6 938 187 562 497 ÷ 2 = 3 469 093 781 248 + 1;
  • 3 469 093 781 248 ÷ 2 = 1 734 546 890 624 + 0;
  • 1 734 546 890 624 ÷ 2 = 867 273 445 312 + 0;
  • 867 273 445 312 ÷ 2 = 433 636 722 656 + 0;
  • 433 636 722 656 ÷ 2 = 216 818 361 328 + 0;
  • 216 818 361 328 ÷ 2 = 108 409 180 664 + 0;
  • 108 409 180 664 ÷ 2 = 54 204 590 332 + 0;
  • 54 204 590 332 ÷ 2 = 27 102 295 166 + 0;
  • 27 102 295 166 ÷ 2 = 13 551 147 583 + 0;
  • 13 551 147 583 ÷ 2 = 6 775 573 791 + 1;
  • 6 775 573 791 ÷ 2 = 3 387 786 895 + 1;
  • 3 387 786 895 ÷ 2 = 1 693 893 447 + 1;
  • 1 693 893 447 ÷ 2 = 846 946 723 + 1;
  • 846 946 723 ÷ 2 = 423 473 361 + 1;
  • 423 473 361 ÷ 2 = 211 736 680 + 1;
  • 211 736 680 ÷ 2 = 105 868 340 + 0;
  • 105 868 340 ÷ 2 = 52 934 170 + 0;
  • 52 934 170 ÷ 2 = 26 467 085 + 0;
  • 26 467 085 ÷ 2 = 13 233 542 + 1;
  • 13 233 542 ÷ 2 = 6 616 771 + 0;
  • 6 616 771 ÷ 2 = 3 308 385 + 1;
  • 3 308 385 ÷ 2 = 1 654 192 + 1;
  • 1 654 192 ÷ 2 = 827 096 + 0;
  • 827 096 ÷ 2 = 413 548 + 0;
  • 413 548 ÷ 2 = 206 774 + 0;
  • 206 774 ÷ 2 = 103 387 + 0;
  • 103 387 ÷ 2 = 51 693 + 1;
  • 51 693 ÷ 2 = 25 846 + 1;
  • 25 846 ÷ 2 = 12 923 + 0;
  • 12 923 ÷ 2 = 6 461 + 1;
  • 6 461 ÷ 2 = 3 230 + 1;
  • 3 230 ÷ 2 = 1 615 + 0;
  • 1 615 ÷ 2 = 807 + 1;
  • 807 ÷ 2 = 403 + 1;
  • 403 ÷ 2 = 201 + 1;
  • 201 ÷ 2 = 100 + 1;
  • 100 ÷ 2 = 50 + 0;
  • 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.


111 011 000 999 965(10) = 110 0100 1111 0110 1100 0011 0100 0111 1110 0000 0001 1101(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.


Number 111 011 000 999 965(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in two's complement representation:

111 011 000 999 965(10) = 0000 0000 0000 0000 0110 0100 1111 0110 1100 0011 0100 0111 1110 0000 0001 1101

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

More operations on integers written as signed binary numbers in two's complement representation:

» Signed integer number 111 011 000 999 964 converted from decimal system (from base 10) and written as a signed binary in two's complement representation (written in base 2) = ?

» Signed integer number 111 011 000 999 966 converted from decimal system (from base 10) and written as a signed binary in two's complement representation (written in base 2) = ?

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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