Two's Complement: Integer ↗ Binary: 9 999 999 999 999 999 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 9 999 999 999 999 999(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;
  • 9 999 999 999 999 999 ÷ 2 = 4 999 999 999 999 999 + 1;
  • 4 999 999 999 999 999 ÷ 2 = 2 499 999 999 999 999 + 1;
  • 2 499 999 999 999 999 ÷ 2 = 1 249 999 999 999 999 + 1;
  • 1 249 999 999 999 999 ÷ 2 = 624 999 999 999 999 + 1;
  • 624 999 999 999 999 ÷ 2 = 312 499 999 999 999 + 1;
  • 312 499 999 999 999 ÷ 2 = 156 249 999 999 999 + 1;
  • 156 249 999 999 999 ÷ 2 = 78 124 999 999 999 + 1;
  • 78 124 999 999 999 ÷ 2 = 39 062 499 999 999 + 1;
  • 39 062 499 999 999 ÷ 2 = 19 531 249 999 999 + 1;
  • 19 531 249 999 999 ÷ 2 = 9 765 624 999 999 + 1;
  • 9 765 624 999 999 ÷ 2 = 4 882 812 499 999 + 1;
  • 4 882 812 499 999 ÷ 2 = 2 441 406 249 999 + 1;
  • 2 441 406 249 999 ÷ 2 = 1 220 703 124 999 + 1;
  • 1 220 703 124 999 ÷ 2 = 610 351 562 499 + 1;
  • 610 351 562 499 ÷ 2 = 305 175 781 249 + 1;
  • 305 175 781 249 ÷ 2 = 152 587 890 624 + 1;
  • 152 587 890 624 ÷ 2 = 76 293 945 312 + 0;
  • 76 293 945 312 ÷ 2 = 38 146 972 656 + 0;
  • 38 146 972 656 ÷ 2 = 19 073 486 328 + 0;
  • 19 073 486 328 ÷ 2 = 9 536 743 164 + 0;
  • 9 536 743 164 ÷ 2 = 4 768 371 582 + 0;
  • 4 768 371 582 ÷ 2 = 2 384 185 791 + 0;
  • 2 384 185 791 ÷ 2 = 1 192 092 895 + 1;
  • 1 192 092 895 ÷ 2 = 596 046 447 + 1;
  • 596 046 447 ÷ 2 = 298 023 223 + 1;
  • 298 023 223 ÷ 2 = 149 011 611 + 1;
  • 149 011 611 ÷ 2 = 74 505 805 + 1;
  • 74 505 805 ÷ 2 = 37 252 902 + 1;
  • 37 252 902 ÷ 2 = 18 626 451 + 0;
  • 18 626 451 ÷ 2 = 9 313 225 + 1;
  • 9 313 225 ÷ 2 = 4 656 612 + 1;
  • 4 656 612 ÷ 2 = 2 328 306 + 0;
  • 2 328 306 ÷ 2 = 1 164 153 + 0;
  • 1 164 153 ÷ 2 = 582 076 + 1;
  • 582 076 ÷ 2 = 291 038 + 0;
  • 291 038 ÷ 2 = 145 519 + 0;
  • 145 519 ÷ 2 = 72 759 + 1;
  • 72 759 ÷ 2 = 36 379 + 1;
  • 36 379 ÷ 2 = 18 189 + 1;
  • 18 189 ÷ 2 = 9 094 + 1;
  • 9 094 ÷ 2 = 4 547 + 0;
  • 4 547 ÷ 2 = 2 273 + 1;
  • 2 273 ÷ 2 = 1 136 + 1;
  • 1 136 ÷ 2 = 568 + 0;
  • 568 ÷ 2 = 284 + 0;
  • 284 ÷ 2 = 142 + 0;
  • 142 ÷ 2 = 71 + 0;
  • 71 ÷ 2 = 35 + 1;
  • 35 ÷ 2 = 17 + 1;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 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.


9 999 999 999 999 999(10) = 10 0011 1000 0110 1111 0010 0110 1111 1100 0000 1111 1111 1111 1111(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 54.

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, 54,

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 9 999 999 999 999 999(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:

9 999 999 999 999 999(10) = 0000 0000 0010 0011 1000 0110 1111 0010 0110 1111 1100 0000 1111 1111 1111 1111

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 9 999 999 999 999 998 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 10 000 000 000 000 000 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