Convert 1 000 111 100 011 168 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 000 111 100 011 168(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 000 111 100 011 168 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;
  • 1 000 111 100 011 168 ÷ 2 = 500 055 550 005 584 + 0;
  • 500 055 550 005 584 ÷ 2 = 250 027 775 002 792 + 0;
  • 250 027 775 002 792 ÷ 2 = 125 013 887 501 396 + 0;
  • 125 013 887 501 396 ÷ 2 = 62 506 943 750 698 + 0;
  • 62 506 943 750 698 ÷ 2 = 31 253 471 875 349 + 0;
  • 31 253 471 875 349 ÷ 2 = 15 626 735 937 674 + 1;
  • 15 626 735 937 674 ÷ 2 = 7 813 367 968 837 + 0;
  • 7 813 367 968 837 ÷ 2 = 3 906 683 984 418 + 1;
  • 3 906 683 984 418 ÷ 2 = 1 953 341 992 209 + 0;
  • 1 953 341 992 209 ÷ 2 = 976 670 996 104 + 1;
  • 976 670 996 104 ÷ 2 = 488 335 498 052 + 0;
  • 488 335 498 052 ÷ 2 = 244 167 749 026 + 0;
  • 244 167 749 026 ÷ 2 = 122 083 874 513 + 0;
  • 122 083 874 513 ÷ 2 = 61 041 937 256 + 1;
  • 61 041 937 256 ÷ 2 = 30 520 968 628 + 0;
  • 30 520 968 628 ÷ 2 = 15 260 484 314 + 0;
  • 15 260 484 314 ÷ 2 = 7 630 242 157 + 0;
  • 7 630 242 157 ÷ 2 = 3 815 121 078 + 1;
  • 3 815 121 078 ÷ 2 = 1 907 560 539 + 0;
  • 1 907 560 539 ÷ 2 = 953 780 269 + 1;
  • 953 780 269 ÷ 2 = 476 890 134 + 1;
  • 476 890 134 ÷ 2 = 238 445 067 + 0;
  • 238 445 067 ÷ 2 = 119 222 533 + 1;
  • 119 222 533 ÷ 2 = 59 611 266 + 1;
  • 59 611 266 ÷ 2 = 29 805 633 + 0;
  • 29 805 633 ÷ 2 = 14 902 816 + 1;
  • 14 902 816 ÷ 2 = 7 451 408 + 0;
  • 7 451 408 ÷ 2 = 3 725 704 + 0;
  • 3 725 704 ÷ 2 = 1 862 852 + 0;
  • 1 862 852 ÷ 2 = 931 426 + 0;
  • 931 426 ÷ 2 = 465 713 + 0;
  • 465 713 ÷ 2 = 232 856 + 1;
  • 232 856 ÷ 2 = 116 428 + 0;
  • 116 428 ÷ 2 = 58 214 + 0;
  • 58 214 ÷ 2 = 29 107 + 0;
  • 29 107 ÷ 2 = 14 553 + 1;
  • 14 553 ÷ 2 = 7 276 + 1;
  • 7 276 ÷ 2 = 3 638 + 0;
  • 3 638 ÷ 2 = 1 819 + 0;
  • 1 819 ÷ 2 = 909 + 1;
  • 909 ÷ 2 = 454 + 1;
  • 454 ÷ 2 = 227 + 0;
  • 227 ÷ 2 = 113 + 1;
  • 113 ÷ 2 = 56 + 1;
  • 56 ÷ 2 = 28 + 0;
  • 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.

1 000 111 100 011 168(10) = 11 1000 1101 1001 1000 1000 0010 1101 1010 0010 0010 1010 0000(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 1 000 111 100 011 168(10) converted to signed binary in two's complement representation:

1 000 111 100 011 168(10) = 0000 0000 0000 0011 1000 1101 1001 1000 1000 0010 1101 1010 0010 0010 1010 0000

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