Convert 1 011 110 000 009 591 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 011 110 000 009 591(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 011 110 000 009 591 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;
  • 1 011 110 000 009 591 ÷ 2 = 505 555 000 004 795 + 1;
  • 505 555 000 004 795 ÷ 2 = 252 777 500 002 397 + 1;
  • 252 777 500 002 397 ÷ 2 = 126 388 750 001 198 + 1;
  • 126 388 750 001 198 ÷ 2 = 63 194 375 000 599 + 0;
  • 63 194 375 000 599 ÷ 2 = 31 597 187 500 299 + 1;
  • 31 597 187 500 299 ÷ 2 = 15 798 593 750 149 + 1;
  • 15 798 593 750 149 ÷ 2 = 7 899 296 875 074 + 1;
  • 7 899 296 875 074 ÷ 2 = 3 949 648 437 537 + 0;
  • 3 949 648 437 537 ÷ 2 = 1 974 824 218 768 + 1;
  • 1 974 824 218 768 ÷ 2 = 987 412 109 384 + 0;
  • 987 412 109 384 ÷ 2 = 493 706 054 692 + 0;
  • 493 706 054 692 ÷ 2 = 246 853 027 346 + 0;
  • 246 853 027 346 ÷ 2 = 123 426 513 673 + 0;
  • 123 426 513 673 ÷ 2 = 61 713 256 836 + 1;
  • 61 713 256 836 ÷ 2 = 30 856 628 418 + 0;
  • 30 856 628 418 ÷ 2 = 15 428 314 209 + 0;
  • 15 428 314 209 ÷ 2 = 7 714 157 104 + 1;
  • 7 714 157 104 ÷ 2 = 3 857 078 552 + 0;
  • 3 857 078 552 ÷ 2 = 1 928 539 276 + 0;
  • 1 928 539 276 ÷ 2 = 964 269 638 + 0;
  • 964 269 638 ÷ 2 = 482 134 819 + 0;
  • 482 134 819 ÷ 2 = 241 067 409 + 1;
  • 241 067 409 ÷ 2 = 120 533 704 + 1;
  • 120 533 704 ÷ 2 = 60 266 852 + 0;
  • 60 266 852 ÷ 2 = 30 133 426 + 0;
  • 30 133 426 ÷ 2 = 15 066 713 + 0;
  • 15 066 713 ÷ 2 = 7 533 356 + 1;
  • 7 533 356 ÷ 2 = 3 766 678 + 0;
  • 3 766 678 ÷ 2 = 1 883 339 + 0;
  • 1 883 339 ÷ 2 = 941 669 + 1;
  • 941 669 ÷ 2 = 470 834 + 1;
  • 470 834 ÷ 2 = 235 417 + 0;
  • 235 417 ÷ 2 = 117 708 + 1;
  • 117 708 ÷ 2 = 58 854 + 0;
  • 58 854 ÷ 2 = 29 427 + 0;
  • 29 427 ÷ 2 = 14 713 + 1;
  • 14 713 ÷ 2 = 7 356 + 1;
  • 7 356 ÷ 2 = 3 678 + 0;
  • 3 678 ÷ 2 = 1 839 + 0;
  • 1 839 ÷ 2 = 919 + 1;
  • 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.

1 011 110 000 009 591(10) = 11 1001 0111 1001 1001 0110 0100 0110 0001 0010 0001 0111 0111(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 1 011 110 000 009 591(10) converted to signed binary in two's complement representation:

1 011 110 000 009 591(10) = 0000 0000 0000 0011 1001 0111 1001 1001 0110 0100 0110 0001 0010 0001 0111 0111

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