Convert 2 815 505 882 731 717 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 2 815 505 882 731 717(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
2 815 505 882 731 717 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;
  • 2 815 505 882 731 717 ÷ 2 = 1 407 752 941 365 858 + 1;
  • 1 407 752 941 365 858 ÷ 2 = 703 876 470 682 929 + 0;
  • 703 876 470 682 929 ÷ 2 = 351 938 235 341 464 + 1;
  • 351 938 235 341 464 ÷ 2 = 175 969 117 670 732 + 0;
  • 175 969 117 670 732 ÷ 2 = 87 984 558 835 366 + 0;
  • 87 984 558 835 366 ÷ 2 = 43 992 279 417 683 + 0;
  • 43 992 279 417 683 ÷ 2 = 21 996 139 708 841 + 1;
  • 21 996 139 708 841 ÷ 2 = 10 998 069 854 420 + 1;
  • 10 998 069 854 420 ÷ 2 = 5 499 034 927 210 + 0;
  • 5 499 034 927 210 ÷ 2 = 2 749 517 463 605 + 0;
  • 2 749 517 463 605 ÷ 2 = 1 374 758 731 802 + 1;
  • 1 374 758 731 802 ÷ 2 = 687 379 365 901 + 0;
  • 687 379 365 901 ÷ 2 = 343 689 682 950 + 1;
  • 343 689 682 950 ÷ 2 = 171 844 841 475 + 0;
  • 171 844 841 475 ÷ 2 = 85 922 420 737 + 1;
  • 85 922 420 737 ÷ 2 = 42 961 210 368 + 1;
  • 42 961 210 368 ÷ 2 = 21 480 605 184 + 0;
  • 21 480 605 184 ÷ 2 = 10 740 302 592 + 0;
  • 10 740 302 592 ÷ 2 = 5 370 151 296 + 0;
  • 5 370 151 296 ÷ 2 = 2 685 075 648 + 0;
  • 2 685 075 648 ÷ 2 = 1 342 537 824 + 0;
  • 1 342 537 824 ÷ 2 = 671 268 912 + 0;
  • 671 268 912 ÷ 2 = 335 634 456 + 0;
  • 335 634 456 ÷ 2 = 167 817 228 + 0;
  • 167 817 228 ÷ 2 = 83 908 614 + 0;
  • 83 908 614 ÷ 2 = 41 954 307 + 0;
  • 41 954 307 ÷ 2 = 20 977 153 + 1;
  • 20 977 153 ÷ 2 = 10 488 576 + 1;
  • 10 488 576 ÷ 2 = 5 244 288 + 0;
  • 5 244 288 ÷ 2 = 2 622 144 + 0;
  • 2 622 144 ÷ 2 = 1 311 072 + 0;
  • 1 311 072 ÷ 2 = 655 536 + 0;
  • 655 536 ÷ 2 = 327 768 + 0;
  • 327 768 ÷ 2 = 163 884 + 0;
  • 163 884 ÷ 2 = 81 942 + 0;
  • 81 942 ÷ 2 = 40 971 + 0;
  • 40 971 ÷ 2 = 20 485 + 1;
  • 20 485 ÷ 2 = 10 242 + 1;
  • 10 242 ÷ 2 = 5 121 + 0;
  • 5 121 ÷ 2 = 2 560 + 1;
  • 2 560 ÷ 2 = 1 280 + 0;
  • 1 280 ÷ 2 = 640 + 0;
  • 640 ÷ 2 = 320 + 0;
  • 320 ÷ 2 = 160 + 0;
  • 160 ÷ 2 = 80 + 0;
  • 80 ÷ 2 = 40 + 0;
  • 40 ÷ 2 = 20 + 0;
  • 20 ÷ 2 = 10 + 0;
  • 10 ÷ 2 = 5 + 0;
  • 5 ÷ 2 = 2 + 1;
  • 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.

2 815 505 882 731 717(10) = 1010 0000 0000 1011 0000 0000 1100 0000 0000 1101 0100 1100 0101(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 52.

  • 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, 52,

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 2 815 505 882 731 717(10) converted to signed binary in two's complement representation:

2 815 505 882 731 717(10) = 0000 0000 0000 1010 0000 0000 1011 0000 0000 1100 0000 0000 1101 0100 1100 0101

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