Convert 110 011 111 001 097 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 110 011 111 001 097(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
110 011 111 001 097 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;
  • 110 011 111 001 097 ÷ 2 = 55 005 555 500 548 + 1;
  • 55 005 555 500 548 ÷ 2 = 27 502 777 750 274 + 0;
  • 27 502 777 750 274 ÷ 2 = 13 751 388 875 137 + 0;
  • 13 751 388 875 137 ÷ 2 = 6 875 694 437 568 + 1;
  • 6 875 694 437 568 ÷ 2 = 3 437 847 218 784 + 0;
  • 3 437 847 218 784 ÷ 2 = 1 718 923 609 392 + 0;
  • 1 718 923 609 392 ÷ 2 = 859 461 804 696 + 0;
  • 859 461 804 696 ÷ 2 = 429 730 902 348 + 0;
  • 429 730 902 348 ÷ 2 = 214 865 451 174 + 0;
  • 214 865 451 174 ÷ 2 = 107 432 725 587 + 0;
  • 107 432 725 587 ÷ 2 = 53 716 362 793 + 1;
  • 53 716 362 793 ÷ 2 = 26 858 181 396 + 1;
  • 26 858 181 396 ÷ 2 = 13 429 090 698 + 0;
  • 13 429 090 698 ÷ 2 = 6 714 545 349 + 0;
  • 6 714 545 349 ÷ 2 = 3 357 272 674 + 1;
  • 3 357 272 674 ÷ 2 = 1 678 636 337 + 0;
  • 1 678 636 337 ÷ 2 = 839 318 168 + 1;
  • 839 318 168 ÷ 2 = 419 659 084 + 0;
  • 419 659 084 ÷ 2 = 209 829 542 + 0;
  • 209 829 542 ÷ 2 = 104 914 771 + 0;
  • 104 914 771 ÷ 2 = 52 457 385 + 1;
  • 52 457 385 ÷ 2 = 26 228 692 + 1;
  • 26 228 692 ÷ 2 = 13 114 346 + 0;
  • 13 114 346 ÷ 2 = 6 557 173 + 0;
  • 6 557 173 ÷ 2 = 3 278 586 + 1;
  • 3 278 586 ÷ 2 = 1 639 293 + 0;
  • 1 639 293 ÷ 2 = 819 646 + 1;
  • 819 646 ÷ 2 = 409 823 + 0;
  • 409 823 ÷ 2 = 204 911 + 1;
  • 204 911 ÷ 2 = 102 455 + 1;
  • 102 455 ÷ 2 = 51 227 + 1;
  • 51 227 ÷ 2 = 25 613 + 1;
  • 25 613 ÷ 2 = 12 806 + 1;
  • 12 806 ÷ 2 = 6 403 + 0;
  • 6 403 ÷ 2 = 3 201 + 1;
  • 3 201 ÷ 2 = 1 600 + 1;
  • 1 600 ÷ 2 = 800 + 0;
  • 800 ÷ 2 = 400 + 0;
  • 400 ÷ 2 = 200 + 0;
  • 200 ÷ 2 = 100 + 0;
  • 100 ÷ 2 = 50 + 0;
  • 50 ÷ 2 = 25 + 0;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 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.

110 011 111 001 097(10) = 110 0100 0000 1101 1111 0101 0011 0001 0100 1100 0000 1001(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 47.

  • 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, 47,

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 110 011 111 001 097(10) converted to signed binary in two's complement representation:

110 011 111 001 097(10) = 0000 0000 0000 0000 0110 0100 0000 1101 1111 0101 0011 0001 0100 1100 0000 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