Convert 1 111 000 000 001 075 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 111 000 000 001 075(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 111 000 000 001 075 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;
  • 1 111 000 000 001 075 ÷ 2 = 555 500 000 000 537 + 1;
  • 555 500 000 000 537 ÷ 2 = 277 750 000 000 268 + 1;
  • 277 750 000 000 268 ÷ 2 = 138 875 000 000 134 + 0;
  • 138 875 000 000 134 ÷ 2 = 69 437 500 000 067 + 0;
  • 69 437 500 000 067 ÷ 2 = 34 718 750 000 033 + 1;
  • 34 718 750 000 033 ÷ 2 = 17 359 375 000 016 + 1;
  • 17 359 375 000 016 ÷ 2 = 8 679 687 500 008 + 0;
  • 8 679 687 500 008 ÷ 2 = 4 339 843 750 004 + 0;
  • 4 339 843 750 004 ÷ 2 = 2 169 921 875 002 + 0;
  • 2 169 921 875 002 ÷ 2 = 1 084 960 937 501 + 0;
  • 1 084 960 937 501 ÷ 2 = 542 480 468 750 + 1;
  • 542 480 468 750 ÷ 2 = 271 240 234 375 + 0;
  • 271 240 234 375 ÷ 2 = 135 620 117 187 + 1;
  • 135 620 117 187 ÷ 2 = 67 810 058 593 + 1;
  • 67 810 058 593 ÷ 2 = 33 905 029 296 + 1;
  • 33 905 029 296 ÷ 2 = 16 952 514 648 + 0;
  • 16 952 514 648 ÷ 2 = 8 476 257 324 + 0;
  • 8 476 257 324 ÷ 2 = 4 238 128 662 + 0;
  • 4 238 128 662 ÷ 2 = 2 119 064 331 + 0;
  • 2 119 064 331 ÷ 2 = 1 059 532 165 + 1;
  • 1 059 532 165 ÷ 2 = 529 766 082 + 1;
  • 529 766 082 ÷ 2 = 264 883 041 + 0;
  • 264 883 041 ÷ 2 = 132 441 520 + 1;
  • 132 441 520 ÷ 2 = 66 220 760 + 0;
  • 66 220 760 ÷ 2 = 33 110 380 + 0;
  • 33 110 380 ÷ 2 = 16 555 190 + 0;
  • 16 555 190 ÷ 2 = 8 277 595 + 0;
  • 8 277 595 ÷ 2 = 4 138 797 + 1;
  • 4 138 797 ÷ 2 = 2 069 398 + 1;
  • 2 069 398 ÷ 2 = 1 034 699 + 0;
  • 1 034 699 ÷ 2 = 517 349 + 1;
  • 517 349 ÷ 2 = 258 674 + 1;
  • 258 674 ÷ 2 = 129 337 + 0;
  • 129 337 ÷ 2 = 64 668 + 1;
  • 64 668 ÷ 2 = 32 334 + 0;
  • 32 334 ÷ 2 = 16 167 + 0;
  • 16 167 ÷ 2 = 8 083 + 1;
  • 8 083 ÷ 2 = 4 041 + 1;
  • 4 041 ÷ 2 = 2 020 + 1;
  • 2 020 ÷ 2 = 1 010 + 0;
  • 1 010 ÷ 2 = 505 + 0;
  • 505 ÷ 2 = 252 + 1;
  • 252 ÷ 2 = 126 + 0;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 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.

1 111 000 000 001 075(10) = 11 1111 0010 0111 0010 1101 1000 0101 1000 0111 0100 0011 0011(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 1 111 000 000 001 075(10) converted to signed binary in two's complement representation:

1 111 000 000 001 075(10) = 0000 0000 0000 0011 1111 0010 0111 0010 1101 1000 0101 1000 0111 0100 0011 0011

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