Convert 8 888 888 888 888 985 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 8 888 888 888 888 985(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
8 888 888 888 888 985 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;
  • 8 888 888 888 888 985 ÷ 2 = 4 444 444 444 444 492 + 1;
  • 4 444 444 444 444 492 ÷ 2 = 2 222 222 222 222 246 + 0;
  • 2 222 222 222 222 246 ÷ 2 = 1 111 111 111 111 123 + 0;
  • 1 111 111 111 111 123 ÷ 2 = 555 555 555 555 561 + 1;
  • 555 555 555 555 561 ÷ 2 = 277 777 777 777 780 + 1;
  • 277 777 777 777 780 ÷ 2 = 138 888 888 888 890 + 0;
  • 138 888 888 888 890 ÷ 2 = 69 444 444 444 445 + 0;
  • 69 444 444 444 445 ÷ 2 = 34 722 222 222 222 + 1;
  • 34 722 222 222 222 ÷ 2 = 17 361 111 111 111 + 0;
  • 17 361 111 111 111 ÷ 2 = 8 680 555 555 555 + 1;
  • 8 680 555 555 555 ÷ 2 = 4 340 277 777 777 + 1;
  • 4 340 277 777 777 ÷ 2 = 2 170 138 888 888 + 1;
  • 2 170 138 888 888 ÷ 2 = 1 085 069 444 444 + 0;
  • 1 085 069 444 444 ÷ 2 = 542 534 722 222 + 0;
  • 542 534 722 222 ÷ 2 = 271 267 361 111 + 0;
  • 271 267 361 111 ÷ 2 = 135 633 680 555 + 1;
  • 135 633 680 555 ÷ 2 = 67 816 840 277 + 1;
  • 67 816 840 277 ÷ 2 = 33 908 420 138 + 1;
  • 33 908 420 138 ÷ 2 = 16 954 210 069 + 0;
  • 16 954 210 069 ÷ 2 = 8 477 105 034 + 1;
  • 8 477 105 034 ÷ 2 = 4 238 552 517 + 0;
  • 4 238 552 517 ÷ 2 = 2 119 276 258 + 1;
  • 2 119 276 258 ÷ 2 = 1 059 638 129 + 0;
  • 1 059 638 129 ÷ 2 = 529 819 064 + 1;
  • 529 819 064 ÷ 2 = 264 909 532 + 0;
  • 264 909 532 ÷ 2 = 132 454 766 + 0;
  • 132 454 766 ÷ 2 = 66 227 383 + 0;
  • 66 227 383 ÷ 2 = 33 113 691 + 1;
  • 33 113 691 ÷ 2 = 16 556 845 + 1;
  • 16 556 845 ÷ 2 = 8 278 422 + 1;
  • 8 278 422 ÷ 2 = 4 139 211 + 0;
  • 4 139 211 ÷ 2 = 2 069 605 + 1;
  • 2 069 605 ÷ 2 = 1 034 802 + 1;
  • 1 034 802 ÷ 2 = 517 401 + 0;
  • 517 401 ÷ 2 = 258 700 + 1;
  • 258 700 ÷ 2 = 129 350 + 0;
  • 129 350 ÷ 2 = 64 675 + 0;
  • 64 675 ÷ 2 = 32 337 + 1;
  • 32 337 ÷ 2 = 16 168 + 1;
  • 16 168 ÷ 2 = 8 084 + 0;
  • 8 084 ÷ 2 = 4 042 + 0;
  • 4 042 ÷ 2 = 2 021 + 0;
  • 2 021 ÷ 2 = 1 010 + 1;
  • 1 010 ÷ 2 = 505 + 0;
  • 505 ÷ 2 = 252 + 1;
  • 252 ÷ 2 = 126 + 0;
  • 126 ÷ 2 = 63 + 0;
  • 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.

8 888 888 888 888 985(10) = 1 1111 1001 0100 0110 0101 1011 1000 1010 1011 1000 1110 1001 1001(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 53.

  • 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, 53,

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 8 888 888 888 888 985(10) converted to signed binary in two's complement representation:

8 888 888 888 888 985(10) = 0000 0000 0001 1111 1001 0100 0110 0101 1011 1000 1010 1011 1000 1110 1001 1001

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