Integer as Two's Complement Binary: Number 9 999 999 800 000 055 Converted and Written as a Signed Binary in Two's Complement Representation

Integer number 9 999 999 800 000 055(10) written as a signed binary in two's complement representation

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;
  • 9 999 999 800 000 055 ÷ 2 = 4 999 999 900 000 027 + 1;
  • 4 999 999 900 000 027 ÷ 2 = 2 499 999 950 000 013 + 1;
  • 2 499 999 950 000 013 ÷ 2 = 1 249 999 975 000 006 + 1;
  • 1 249 999 975 000 006 ÷ 2 = 624 999 987 500 003 + 0;
  • 624 999 987 500 003 ÷ 2 = 312 499 993 750 001 + 1;
  • 312 499 993 750 001 ÷ 2 = 156 249 996 875 000 + 1;
  • 156 249 996 875 000 ÷ 2 = 78 124 998 437 500 + 0;
  • 78 124 998 437 500 ÷ 2 = 39 062 499 218 750 + 0;
  • 39 062 499 218 750 ÷ 2 = 19 531 249 609 375 + 0;
  • 19 531 249 609 375 ÷ 2 = 9 765 624 804 687 + 1;
  • 9 765 624 804 687 ÷ 2 = 4 882 812 402 343 + 1;
  • 4 882 812 402 343 ÷ 2 = 2 441 406 201 171 + 1;
  • 2 441 406 201 171 ÷ 2 = 1 220 703 100 585 + 1;
  • 1 220 703 100 585 ÷ 2 = 610 351 550 292 + 1;
  • 610 351 550 292 ÷ 2 = 305 175 775 146 + 0;
  • 305 175 775 146 ÷ 2 = 152 587 887 573 + 0;
  • 152 587 887 573 ÷ 2 = 76 293 943 786 + 1;
  • 76 293 943 786 ÷ 2 = 38 146 971 893 + 0;
  • 38 146 971 893 ÷ 2 = 19 073 485 946 + 1;
  • 19 073 485 946 ÷ 2 = 9 536 742 973 + 0;
  • 9 536 742 973 ÷ 2 = 4 768 371 486 + 1;
  • 4 768 371 486 ÷ 2 = 2 384 185 743 + 0;
  • 2 384 185 743 ÷ 2 = 1 192 092 871 + 1;
  • 1 192 092 871 ÷ 2 = 596 046 435 + 1;
  • 596 046 435 ÷ 2 = 298 023 217 + 1;
  • 298 023 217 ÷ 2 = 149 011 608 + 1;
  • 149 011 608 ÷ 2 = 74 505 804 + 0;
  • 74 505 804 ÷ 2 = 37 252 902 + 0;
  • 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.

9 999 999 800 000 055(10) = 10 0011 1000 0110 1111 0010 0110 0011 1101 0101 0011 1110 0011 0111(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 9 999 999 800 000 055(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:

9 999 999 800 000 055(10) = 0000 0000 0010 0011 1000 0110 1111 0010 0110 0011 1101 0101 0011 1110 0011 0111

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