Convert 9 199 999 999 999 998 876 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 9 199 999 999 999 998 876(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
9 199 999 999 999 998 876 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;
  • 9 199 999 999 999 998 876 ÷ 2 = 4 599 999 999 999 999 438 + 0;
  • 4 599 999 999 999 999 438 ÷ 2 = 2 299 999 999 999 999 719 + 0;
  • 2 299 999 999 999 999 719 ÷ 2 = 1 149 999 999 999 999 859 + 1;
  • 1 149 999 999 999 999 859 ÷ 2 = 574 999 999 999 999 929 + 1;
  • 574 999 999 999 999 929 ÷ 2 = 287 499 999 999 999 964 + 1;
  • 287 499 999 999 999 964 ÷ 2 = 143 749 999 999 999 982 + 0;
  • 143 749 999 999 999 982 ÷ 2 = 71 874 999 999 999 991 + 0;
  • 71 874 999 999 999 991 ÷ 2 = 35 937 499 999 999 995 + 1;
  • 35 937 499 999 999 995 ÷ 2 = 17 968 749 999 999 997 + 1;
  • 17 968 749 999 999 997 ÷ 2 = 8 984 374 999 999 998 + 1;
  • 8 984 374 999 999 998 ÷ 2 = 4 492 187 499 999 999 + 0;
  • 4 492 187 499 999 999 ÷ 2 = 2 246 093 749 999 999 + 1;
  • 2 246 093 749 999 999 ÷ 2 = 1 123 046 874 999 999 + 1;
  • 1 123 046 874 999 999 ÷ 2 = 561 523 437 499 999 + 1;
  • 561 523 437 499 999 ÷ 2 = 280 761 718 749 999 + 1;
  • 280 761 718 749 999 ÷ 2 = 140 380 859 374 999 + 1;
  • 140 380 859 374 999 ÷ 2 = 70 190 429 687 499 + 1;
  • 70 190 429 687 499 ÷ 2 = 35 095 214 843 749 + 1;
  • 35 095 214 843 749 ÷ 2 = 17 547 607 421 874 + 1;
  • 17 547 607 421 874 ÷ 2 = 8 773 803 710 937 + 0;
  • 8 773 803 710 937 ÷ 2 = 4 386 901 855 468 + 1;
  • 4 386 901 855 468 ÷ 2 = 2 193 450 927 734 + 0;
  • 2 193 450 927 734 ÷ 2 = 1 096 725 463 867 + 0;
  • 1 096 725 463 867 ÷ 2 = 548 362 731 933 + 1;
  • 548 362 731 933 ÷ 2 = 274 181 365 966 + 1;
  • 274 181 365 966 ÷ 2 = 137 090 682 983 + 0;
  • 137 090 682 983 ÷ 2 = 68 545 341 491 + 1;
  • 68 545 341 491 ÷ 2 = 34 272 670 745 + 1;
  • 34 272 670 745 ÷ 2 = 17 136 335 372 + 1;
  • 17 136 335 372 ÷ 2 = 8 568 167 686 + 0;
  • 8 568 167 686 ÷ 2 = 4 284 083 843 + 0;
  • 4 284 083 843 ÷ 2 = 2 142 041 921 + 1;
  • 2 142 041 921 ÷ 2 = 1 071 020 960 + 1;
  • 1 071 020 960 ÷ 2 = 535 510 480 + 0;
  • 535 510 480 ÷ 2 = 267 755 240 + 0;
  • 267 755 240 ÷ 2 = 133 877 620 + 0;
  • 133 877 620 ÷ 2 = 66 938 810 + 0;
  • 66 938 810 ÷ 2 = 33 469 405 + 0;
  • 33 469 405 ÷ 2 = 16 734 702 + 1;
  • 16 734 702 ÷ 2 = 8 367 351 + 0;
  • 8 367 351 ÷ 2 = 4 183 675 + 1;
  • 4 183 675 ÷ 2 = 2 091 837 + 1;
  • 2 091 837 ÷ 2 = 1 045 918 + 1;
  • 1 045 918 ÷ 2 = 522 959 + 0;
  • 522 959 ÷ 2 = 261 479 + 1;
  • 261 479 ÷ 2 = 130 739 + 1;
  • 130 739 ÷ 2 = 65 369 + 1;
  • 65 369 ÷ 2 = 32 684 + 1;
  • 32 684 ÷ 2 = 16 342 + 0;
  • 16 342 ÷ 2 = 8 171 + 0;
  • 8 171 ÷ 2 = 4 085 + 1;
  • 4 085 ÷ 2 = 2 042 + 1;
  • 2 042 ÷ 2 = 1 021 + 0;
  • 1 021 ÷ 2 = 510 + 1;
  • 510 ÷ 2 = 255 + 0;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 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.

9 199 999 999 999 998 876(10) = 111 1111 1010 1100 1111 0111 0100 0001 1001 1101 1001 0111 1111 1011 1001 1100(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 63.

  • 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, 63,

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 9 199 999 999 999 998 876(10) converted to signed binary in two's complement representation:

9 199 999 999 999 998 876(10) = 0111 1111 1010 1100 1111 0111 0100 0001 1001 1101 1001 0111 1111 1011 1001 1100

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