Convert 11 000 011 110 009 890 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 11 000 011 110 009 890(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
11 000 011 110 009 890 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;
  • 11 000 011 110 009 890 ÷ 2 = 5 500 005 555 004 945 + 0;
  • 5 500 005 555 004 945 ÷ 2 = 2 750 002 777 502 472 + 1;
  • 2 750 002 777 502 472 ÷ 2 = 1 375 001 388 751 236 + 0;
  • 1 375 001 388 751 236 ÷ 2 = 687 500 694 375 618 + 0;
  • 687 500 694 375 618 ÷ 2 = 343 750 347 187 809 + 0;
  • 343 750 347 187 809 ÷ 2 = 171 875 173 593 904 + 1;
  • 171 875 173 593 904 ÷ 2 = 85 937 586 796 952 + 0;
  • 85 937 586 796 952 ÷ 2 = 42 968 793 398 476 + 0;
  • 42 968 793 398 476 ÷ 2 = 21 484 396 699 238 + 0;
  • 21 484 396 699 238 ÷ 2 = 10 742 198 349 619 + 0;
  • 10 742 198 349 619 ÷ 2 = 5 371 099 174 809 + 1;
  • 5 371 099 174 809 ÷ 2 = 2 685 549 587 404 + 1;
  • 2 685 549 587 404 ÷ 2 = 1 342 774 793 702 + 0;
  • 1 342 774 793 702 ÷ 2 = 671 387 396 851 + 0;
  • 671 387 396 851 ÷ 2 = 335 693 698 425 + 1;
  • 335 693 698 425 ÷ 2 = 167 846 849 212 + 1;
  • 167 846 849 212 ÷ 2 = 83 923 424 606 + 0;
  • 83 923 424 606 ÷ 2 = 41 961 712 303 + 0;
  • 41 961 712 303 ÷ 2 = 20 980 856 151 + 1;
  • 20 980 856 151 ÷ 2 = 10 490 428 075 + 1;
  • 10 490 428 075 ÷ 2 = 5 245 214 037 + 1;
  • 5 245 214 037 ÷ 2 = 2 622 607 018 + 1;
  • 2 622 607 018 ÷ 2 = 1 311 303 509 + 0;
  • 1 311 303 509 ÷ 2 = 655 651 754 + 1;
  • 655 651 754 ÷ 2 = 327 825 877 + 0;
  • 327 825 877 ÷ 2 = 163 912 938 + 1;
  • 163 912 938 ÷ 2 = 81 956 469 + 0;
  • 81 956 469 ÷ 2 = 40 978 234 + 1;
  • 40 978 234 ÷ 2 = 20 489 117 + 0;
  • 20 489 117 ÷ 2 = 10 244 558 + 1;
  • 10 244 558 ÷ 2 = 5 122 279 + 0;
  • 5 122 279 ÷ 2 = 2 561 139 + 1;
  • 2 561 139 ÷ 2 = 1 280 569 + 1;
  • 1 280 569 ÷ 2 = 640 284 + 1;
  • 640 284 ÷ 2 = 320 142 + 0;
  • 320 142 ÷ 2 = 160 071 + 0;
  • 160 071 ÷ 2 = 80 035 + 1;
  • 80 035 ÷ 2 = 40 017 + 1;
  • 40 017 ÷ 2 = 20 008 + 1;
  • 20 008 ÷ 2 = 10 004 + 0;
  • 10 004 ÷ 2 = 5 002 + 0;
  • 5 002 ÷ 2 = 2 501 + 0;
  • 2 501 ÷ 2 = 1 250 + 1;
  • 1 250 ÷ 2 = 625 + 0;
  • 625 ÷ 2 = 312 + 1;
  • 312 ÷ 2 = 156 + 0;
  • 156 ÷ 2 = 78 + 0;
  • 78 ÷ 2 = 39 + 0;
  • 39 ÷ 2 = 19 + 1;
  • 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.

11 000 011 110 009 890(10) = 10 0111 0001 0100 0111 0011 1010 1010 1011 1100 1100 1100 0010 0010(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 54.

  • 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, 54,

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 11 000 011 110 009 890(10) converted to signed binary in two's complement representation:

11 000 011 110 009 890(10) = 0000 0000 0010 0111 0001 0100 0111 0011 1010 1010 1011 1100 1100 1100 0010 0010

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