Convert 1 011 000 100 110 278 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 011 000 100 110 278(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 011 000 100 110 278 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;
  • 1 011 000 100 110 278 ÷ 2 = 505 500 050 055 139 + 0;
  • 505 500 050 055 139 ÷ 2 = 252 750 025 027 569 + 1;
  • 252 750 025 027 569 ÷ 2 = 126 375 012 513 784 + 1;
  • 126 375 012 513 784 ÷ 2 = 63 187 506 256 892 + 0;
  • 63 187 506 256 892 ÷ 2 = 31 593 753 128 446 + 0;
  • 31 593 753 128 446 ÷ 2 = 15 796 876 564 223 + 0;
  • 15 796 876 564 223 ÷ 2 = 7 898 438 282 111 + 1;
  • 7 898 438 282 111 ÷ 2 = 3 949 219 141 055 + 1;
  • 3 949 219 141 055 ÷ 2 = 1 974 609 570 527 + 1;
  • 1 974 609 570 527 ÷ 2 = 987 304 785 263 + 1;
  • 987 304 785 263 ÷ 2 = 493 652 392 631 + 1;
  • 493 652 392 631 ÷ 2 = 246 826 196 315 + 1;
  • 246 826 196 315 ÷ 2 = 123 413 098 157 + 1;
  • 123 413 098 157 ÷ 2 = 61 706 549 078 + 1;
  • 61 706 549 078 ÷ 2 = 30 853 274 539 + 0;
  • 30 853 274 539 ÷ 2 = 15 426 637 269 + 1;
  • 15 426 637 269 ÷ 2 = 7 713 318 634 + 1;
  • 7 713 318 634 ÷ 2 = 3 856 659 317 + 0;
  • 3 856 659 317 ÷ 2 = 1 928 329 658 + 1;
  • 1 928 329 658 ÷ 2 = 964 164 829 + 0;
  • 964 164 829 ÷ 2 = 482 082 414 + 1;
  • 482 082 414 ÷ 2 = 241 041 207 + 0;
  • 241 041 207 ÷ 2 = 120 520 603 + 1;
  • 120 520 603 ÷ 2 = 60 260 301 + 1;
  • 60 260 301 ÷ 2 = 30 130 150 + 1;
  • 30 130 150 ÷ 2 = 15 065 075 + 0;
  • 15 065 075 ÷ 2 = 7 532 537 + 1;
  • 7 532 537 ÷ 2 = 3 766 268 + 1;
  • 3 766 268 ÷ 2 = 1 883 134 + 0;
  • 1 883 134 ÷ 2 = 941 567 + 0;
  • 941 567 ÷ 2 = 470 783 + 1;
  • 470 783 ÷ 2 = 235 391 + 1;
  • 235 391 ÷ 2 = 117 695 + 1;
  • 117 695 ÷ 2 = 58 847 + 1;
  • 58 847 ÷ 2 = 29 423 + 1;
  • 29 423 ÷ 2 = 14 711 + 1;
  • 14 711 ÷ 2 = 7 355 + 1;
  • 7 355 ÷ 2 = 3 677 + 1;
  • 3 677 ÷ 2 = 1 838 + 1;
  • 1 838 ÷ 2 = 919 + 0;
  • 919 ÷ 2 = 459 + 1;
  • 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.

1 011 000 100 110 278(10) = 11 1001 0111 0111 1111 1100 1101 1101 0101 1011 1111 1100 0110(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 1 011 000 100 110 278(10) converted to signed binary in two's complement representation:

1 011 000 100 110 278(10) = 0000 0000 0000 0011 1001 0111 0111 1111 1100 1101 1101 0101 1011 1111 1100 0110

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