Convert 110 000 110 110 839 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 110 000 110 110 839(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
110 000 110 110 839 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;
  • 110 000 110 110 839 ÷ 2 = 55 000 055 055 419 + 1;
  • 55 000 055 055 419 ÷ 2 = 27 500 027 527 709 + 1;
  • 27 500 027 527 709 ÷ 2 = 13 750 013 763 854 + 1;
  • 13 750 013 763 854 ÷ 2 = 6 875 006 881 927 + 0;
  • 6 875 006 881 927 ÷ 2 = 3 437 503 440 963 + 1;
  • 3 437 503 440 963 ÷ 2 = 1 718 751 720 481 + 1;
  • 1 718 751 720 481 ÷ 2 = 859 375 860 240 + 1;
  • 859 375 860 240 ÷ 2 = 429 687 930 120 + 0;
  • 429 687 930 120 ÷ 2 = 214 843 965 060 + 0;
  • 214 843 965 060 ÷ 2 = 107 421 982 530 + 0;
  • 107 421 982 530 ÷ 2 = 53 710 991 265 + 0;
  • 53 710 991 265 ÷ 2 = 26 855 495 632 + 1;
  • 26 855 495 632 ÷ 2 = 13 427 747 816 + 0;
  • 13 427 747 816 ÷ 2 = 6 713 873 908 + 0;
  • 6 713 873 908 ÷ 2 = 3 356 936 954 + 0;
  • 3 356 936 954 ÷ 2 = 1 678 468 477 + 0;
  • 1 678 468 477 ÷ 2 = 839 234 238 + 1;
  • 839 234 238 ÷ 2 = 419 617 119 + 0;
  • 419 617 119 ÷ 2 = 209 808 559 + 1;
  • 209 808 559 ÷ 2 = 104 904 279 + 1;
  • 104 904 279 ÷ 2 = 52 452 139 + 1;
  • 52 452 139 ÷ 2 = 26 226 069 + 1;
  • 26 226 069 ÷ 2 = 13 113 034 + 1;
  • 13 113 034 ÷ 2 = 6 556 517 + 0;
  • 6 556 517 ÷ 2 = 3 278 258 + 1;
  • 3 278 258 ÷ 2 = 1 639 129 + 0;
  • 1 639 129 ÷ 2 = 819 564 + 1;
  • 819 564 ÷ 2 = 409 782 + 0;
  • 409 782 ÷ 2 = 204 891 + 0;
  • 204 891 ÷ 2 = 102 445 + 1;
  • 102 445 ÷ 2 = 51 222 + 1;
  • 51 222 ÷ 2 = 25 611 + 0;
  • 25 611 ÷ 2 = 12 805 + 1;
  • 12 805 ÷ 2 = 6 402 + 1;
  • 6 402 ÷ 2 = 3 201 + 0;
  • 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.

110 000 110 110 839(10) = 110 0100 0000 1011 0110 0101 0111 1101 0000 1000 0111 0111(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 110 000 110 110 839(10) converted to signed binary in two's complement representation:

110 000 110 110 839(10) = 0000 0000 0000 0000 0110 0100 0000 1011 0110 0101 0111 1101 0000 1000 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