Convert 1 111 000 831 713 984 391 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 111 000 831 713 984 391(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 111 000 831 713 984 391 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;
  • 1 111 000 831 713 984 391 ÷ 2 = 555 500 415 856 992 195 + 1;
  • 555 500 415 856 992 195 ÷ 2 = 277 750 207 928 496 097 + 1;
  • 277 750 207 928 496 097 ÷ 2 = 138 875 103 964 248 048 + 1;
  • 138 875 103 964 248 048 ÷ 2 = 69 437 551 982 124 024 + 0;
  • 69 437 551 982 124 024 ÷ 2 = 34 718 775 991 062 012 + 0;
  • 34 718 775 991 062 012 ÷ 2 = 17 359 387 995 531 006 + 0;
  • 17 359 387 995 531 006 ÷ 2 = 8 679 693 997 765 503 + 0;
  • 8 679 693 997 765 503 ÷ 2 = 4 339 846 998 882 751 + 1;
  • 4 339 846 998 882 751 ÷ 2 = 2 169 923 499 441 375 + 1;
  • 2 169 923 499 441 375 ÷ 2 = 1 084 961 749 720 687 + 1;
  • 1 084 961 749 720 687 ÷ 2 = 542 480 874 860 343 + 1;
  • 542 480 874 860 343 ÷ 2 = 271 240 437 430 171 + 1;
  • 271 240 437 430 171 ÷ 2 = 135 620 218 715 085 + 1;
  • 135 620 218 715 085 ÷ 2 = 67 810 109 357 542 + 1;
  • 67 810 109 357 542 ÷ 2 = 33 905 054 678 771 + 0;
  • 33 905 054 678 771 ÷ 2 = 16 952 527 339 385 + 1;
  • 16 952 527 339 385 ÷ 2 = 8 476 263 669 692 + 1;
  • 8 476 263 669 692 ÷ 2 = 4 238 131 834 846 + 0;
  • 4 238 131 834 846 ÷ 2 = 2 119 065 917 423 + 0;
  • 2 119 065 917 423 ÷ 2 = 1 059 532 958 711 + 1;
  • 1 059 532 958 711 ÷ 2 = 529 766 479 355 + 1;
  • 529 766 479 355 ÷ 2 = 264 883 239 677 + 1;
  • 264 883 239 677 ÷ 2 = 132 441 619 838 + 1;
  • 132 441 619 838 ÷ 2 = 66 220 809 919 + 0;
  • 66 220 809 919 ÷ 2 = 33 110 404 959 + 1;
  • 33 110 404 959 ÷ 2 = 16 555 202 479 + 1;
  • 16 555 202 479 ÷ 2 = 8 277 601 239 + 1;
  • 8 277 601 239 ÷ 2 = 4 138 800 619 + 1;
  • 4 138 800 619 ÷ 2 = 2 069 400 309 + 1;
  • 2 069 400 309 ÷ 2 = 1 034 700 154 + 1;
  • 1 034 700 154 ÷ 2 = 517 350 077 + 0;
  • 517 350 077 ÷ 2 = 258 675 038 + 1;
  • 258 675 038 ÷ 2 = 129 337 519 + 0;
  • 129 337 519 ÷ 2 = 64 668 759 + 1;
  • 64 668 759 ÷ 2 = 32 334 379 + 1;
  • 32 334 379 ÷ 2 = 16 167 189 + 1;
  • 16 167 189 ÷ 2 = 8 083 594 + 1;
  • 8 083 594 ÷ 2 = 4 041 797 + 0;
  • 4 041 797 ÷ 2 = 2 020 898 + 1;
  • 2 020 898 ÷ 2 = 1 010 449 + 0;
  • 1 010 449 ÷ 2 = 505 224 + 1;
  • 505 224 ÷ 2 = 252 612 + 0;
  • 252 612 ÷ 2 = 126 306 + 0;
  • 126 306 ÷ 2 = 63 153 + 0;
  • 63 153 ÷ 2 = 31 576 + 1;
  • 31 576 ÷ 2 = 15 788 + 0;
  • 15 788 ÷ 2 = 7 894 + 0;
  • 7 894 ÷ 2 = 3 947 + 0;
  • 3 947 ÷ 2 = 1 973 + 1;
  • 1 973 ÷ 2 = 986 + 1;
  • 986 ÷ 2 = 493 + 0;
  • 493 ÷ 2 = 246 + 1;
  • 246 ÷ 2 = 123 + 0;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 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.

1 111 000 831 713 984 391(10) = 1111 0110 1011 0001 0001 0101 1110 1011 1111 0111 1001 1011 1111 1000 0111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 60.

  • 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, 60,

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 1 111 000 831 713 984 391(10) converted to signed binary in two's complement representation:

1 111 000 831 713 984 391(10) = 0000 1111 0110 1011 0001 0001 0101 1110 1011 1111 0111 1001 1011 1111 1000 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