Convert 1 010 111 111 099 843 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 010 111 111 099 843(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 010 111 111 099 843 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;
  • 1 010 111 111 099 843 ÷ 2 = 505 055 555 549 921 + 1;
  • 505 055 555 549 921 ÷ 2 = 252 527 777 774 960 + 1;
  • 252 527 777 774 960 ÷ 2 = 126 263 888 887 480 + 0;
  • 126 263 888 887 480 ÷ 2 = 63 131 944 443 740 + 0;
  • 63 131 944 443 740 ÷ 2 = 31 565 972 221 870 + 0;
  • 31 565 972 221 870 ÷ 2 = 15 782 986 110 935 + 0;
  • 15 782 986 110 935 ÷ 2 = 7 891 493 055 467 + 1;
  • 7 891 493 055 467 ÷ 2 = 3 945 746 527 733 + 1;
  • 3 945 746 527 733 ÷ 2 = 1 972 873 263 866 + 1;
  • 1 972 873 263 866 ÷ 2 = 986 436 631 933 + 0;
  • 986 436 631 933 ÷ 2 = 493 218 315 966 + 1;
  • 493 218 315 966 ÷ 2 = 246 609 157 983 + 0;
  • 246 609 157 983 ÷ 2 = 123 304 578 991 + 1;
  • 123 304 578 991 ÷ 2 = 61 652 289 495 + 1;
  • 61 652 289 495 ÷ 2 = 30 826 144 747 + 1;
  • 30 826 144 747 ÷ 2 = 15 413 072 373 + 1;
  • 15 413 072 373 ÷ 2 = 7 706 536 186 + 1;
  • 7 706 536 186 ÷ 2 = 3 853 268 093 + 0;
  • 3 853 268 093 ÷ 2 = 1 926 634 046 + 1;
  • 1 926 634 046 ÷ 2 = 963 317 023 + 0;
  • 963 317 023 ÷ 2 = 481 658 511 + 1;
  • 481 658 511 ÷ 2 = 240 829 255 + 1;
  • 240 829 255 ÷ 2 = 120 414 627 + 1;
  • 120 414 627 ÷ 2 = 60 207 313 + 1;
  • 60 207 313 ÷ 2 = 30 103 656 + 1;
  • 30 103 656 ÷ 2 = 15 051 828 + 0;
  • 15 051 828 ÷ 2 = 7 525 914 + 0;
  • 7 525 914 ÷ 2 = 3 762 957 + 0;
  • 3 762 957 ÷ 2 = 1 881 478 + 1;
  • 1 881 478 ÷ 2 = 940 739 + 0;
  • 940 739 ÷ 2 = 470 369 + 1;
  • 470 369 ÷ 2 = 235 184 + 1;
  • 235 184 ÷ 2 = 117 592 + 0;
  • 117 592 ÷ 2 = 58 796 + 0;
  • 58 796 ÷ 2 = 29 398 + 0;
  • 29 398 ÷ 2 = 14 699 + 0;
  • 14 699 ÷ 2 = 7 349 + 1;
  • 7 349 ÷ 2 = 3 674 + 1;
  • 3 674 ÷ 2 = 1 837 + 0;
  • 1 837 ÷ 2 = 918 + 1;
  • 918 ÷ 2 = 459 + 0;
  • 459 ÷ 2 = 229 + 1;
  • 229 ÷ 2 = 114 + 1;
  • 114 ÷ 2 = 57 + 0;
  • 57 ÷ 2 = 28 + 1;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 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.

1 010 111 111 099 843(10) = 11 1001 0110 1011 0000 1101 0001 1111 0101 1111 0101 1100 0011(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 50.

  • 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, 50,

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 1 010 111 111 099 843(10) converted to signed binary in two's complement representation:

1 010 111 111 099 843(10) = 0000 0000 0000 0011 1001 0110 1011 0000 1101 0001 1111 0101 1111 0101 1100 0011

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