Convert 84 688 897 866 462 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 84 688 897 866 462(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
84 688 897 866 462 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;
  • 84 688 897 866 462 ÷ 2 = 42 344 448 933 231 + 0;
  • 42 344 448 933 231 ÷ 2 = 21 172 224 466 615 + 1;
  • 21 172 224 466 615 ÷ 2 = 10 586 112 233 307 + 1;
  • 10 586 112 233 307 ÷ 2 = 5 293 056 116 653 + 1;
  • 5 293 056 116 653 ÷ 2 = 2 646 528 058 326 + 1;
  • 2 646 528 058 326 ÷ 2 = 1 323 264 029 163 + 0;
  • 1 323 264 029 163 ÷ 2 = 661 632 014 581 + 1;
  • 661 632 014 581 ÷ 2 = 330 816 007 290 + 1;
  • 330 816 007 290 ÷ 2 = 165 408 003 645 + 0;
  • 165 408 003 645 ÷ 2 = 82 704 001 822 + 1;
  • 82 704 001 822 ÷ 2 = 41 352 000 911 + 0;
  • 41 352 000 911 ÷ 2 = 20 676 000 455 + 1;
  • 20 676 000 455 ÷ 2 = 10 338 000 227 + 1;
  • 10 338 000 227 ÷ 2 = 5 169 000 113 + 1;
  • 5 169 000 113 ÷ 2 = 2 584 500 056 + 1;
  • 2 584 500 056 ÷ 2 = 1 292 250 028 + 0;
  • 1 292 250 028 ÷ 2 = 646 125 014 + 0;
  • 646 125 014 ÷ 2 = 323 062 507 + 0;
  • 323 062 507 ÷ 2 = 161 531 253 + 1;
  • 161 531 253 ÷ 2 = 80 765 626 + 1;
  • 80 765 626 ÷ 2 = 40 382 813 + 0;
  • 40 382 813 ÷ 2 = 20 191 406 + 1;
  • 20 191 406 ÷ 2 = 10 095 703 + 0;
  • 10 095 703 ÷ 2 = 5 047 851 + 1;
  • 5 047 851 ÷ 2 = 2 523 925 + 1;
  • 2 523 925 ÷ 2 = 1 261 962 + 1;
  • 1 261 962 ÷ 2 = 630 981 + 0;
  • 630 981 ÷ 2 = 315 490 + 1;
  • 315 490 ÷ 2 = 157 745 + 0;
  • 157 745 ÷ 2 = 78 872 + 1;
  • 78 872 ÷ 2 = 39 436 + 0;
  • 39 436 ÷ 2 = 19 718 + 0;
  • 19 718 ÷ 2 = 9 859 + 0;
  • 9 859 ÷ 2 = 4 929 + 1;
  • 4 929 ÷ 2 = 2 464 + 1;
  • 2 464 ÷ 2 = 1 232 + 0;
  • 1 232 ÷ 2 = 616 + 0;
  • 616 ÷ 2 = 308 + 0;
  • 308 ÷ 2 = 154 + 0;
  • 154 ÷ 2 = 77 + 0;
  • 77 ÷ 2 = 38 + 1;
  • 38 ÷ 2 = 19 + 0;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 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.

84 688 897 866 462(10) = 100 1101 0000 0110 0010 1011 1010 1100 0111 1010 1101 1110(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 47.

  • 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, 47,

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 84 688 897 866 462(10) converted to signed binary in two's complement representation:

84 688 897 866 462(10) = 0000 0000 0000 0000 0100 1101 0000 0110 0010 1011 1010 1100 0111 1010 1101 1110

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