Convert 1 010 011 011 100 057 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 010 011 011 100 057(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 010 011 011 100 057 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;
  • 1 010 011 011 100 057 ÷ 2 = 505 005 505 550 028 + 1;
  • 505 005 505 550 028 ÷ 2 = 252 502 752 775 014 + 0;
  • 252 502 752 775 014 ÷ 2 = 126 251 376 387 507 + 0;
  • 126 251 376 387 507 ÷ 2 = 63 125 688 193 753 + 1;
  • 63 125 688 193 753 ÷ 2 = 31 562 844 096 876 + 1;
  • 31 562 844 096 876 ÷ 2 = 15 781 422 048 438 + 0;
  • 15 781 422 048 438 ÷ 2 = 7 890 711 024 219 + 0;
  • 7 890 711 024 219 ÷ 2 = 3 945 355 512 109 + 1;
  • 3 945 355 512 109 ÷ 2 = 1 972 677 756 054 + 1;
  • 1 972 677 756 054 ÷ 2 = 986 338 878 027 + 0;
  • 986 338 878 027 ÷ 2 = 493 169 439 013 + 1;
  • 493 169 439 013 ÷ 2 = 246 584 719 506 + 1;
  • 246 584 719 506 ÷ 2 = 123 292 359 753 + 0;
  • 123 292 359 753 ÷ 2 = 61 646 179 876 + 1;
  • 61 646 179 876 ÷ 2 = 30 823 089 938 + 0;
  • 30 823 089 938 ÷ 2 = 15 411 544 969 + 0;
  • 15 411 544 969 ÷ 2 = 7 705 772 484 + 1;
  • 7 705 772 484 ÷ 2 = 3 852 886 242 + 0;
  • 3 852 886 242 ÷ 2 = 1 926 443 121 + 0;
  • 1 926 443 121 ÷ 2 = 963 221 560 + 1;
  • 963 221 560 ÷ 2 = 481 610 780 + 0;
  • 481 610 780 ÷ 2 = 240 805 390 + 0;
  • 240 805 390 ÷ 2 = 120 402 695 + 0;
  • 120 402 695 ÷ 2 = 60 201 347 + 1;
  • 60 201 347 ÷ 2 = 30 100 673 + 1;
  • 30 100 673 ÷ 2 = 15 050 336 + 1;
  • 15 050 336 ÷ 2 = 7 525 168 + 0;
  • 7 525 168 ÷ 2 = 3 762 584 + 0;
  • 3 762 584 ÷ 2 = 1 881 292 + 0;
  • 1 881 292 ÷ 2 = 940 646 + 0;
  • 940 646 ÷ 2 = 470 323 + 0;
  • 470 323 ÷ 2 = 235 161 + 1;
  • 235 161 ÷ 2 = 117 580 + 1;
  • 117 580 ÷ 2 = 58 790 + 0;
  • 58 790 ÷ 2 = 29 395 + 0;
  • 29 395 ÷ 2 = 14 697 + 1;
  • 14 697 ÷ 2 = 7 348 + 1;
  • 7 348 ÷ 2 = 3 674 + 0;
  • 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.

1 010 011 011 100 057(10) = 11 1001 0110 1001 1001 1000 0011 1000 1001 0010 1101 1001 1001(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 1 010 011 011 100 057(10) converted to signed binary in two's complement representation:

1 010 011 011 100 057(10) = 0000 0000 0000 0011 1001 0110 1001 1001 1000 0011 1000 1001 0010 1101 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