Two's Complement: Integer ↗ Binary: 1 011 000 000 000 061 Convert the Integer Number to a Signed Binary in Two's Complement Representation. Write the Base Ten Decimal System Number as a Binary Code (Written in Base Two)

Signed integer number 1 011 000 000 000 061(10) converted and written as a signed binary in two's complement representation (base 2) = ?

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;
  • 1 011 000 000 000 061 ÷ 2 = 505 500 000 000 030 + 1;
  • 505 500 000 000 030 ÷ 2 = 252 750 000 000 015 + 0;
  • 252 750 000 000 015 ÷ 2 = 126 375 000 000 007 + 1;
  • 126 375 000 000 007 ÷ 2 = 63 187 500 000 003 + 1;
  • 63 187 500 000 003 ÷ 2 = 31 593 750 000 001 + 1;
  • 31 593 750 000 001 ÷ 2 = 15 796 875 000 000 + 1;
  • 15 796 875 000 000 ÷ 2 = 7 898 437 500 000 + 0;
  • 7 898 437 500 000 ÷ 2 = 3 949 218 750 000 + 0;
  • 3 949 218 750 000 ÷ 2 = 1 974 609 375 000 + 0;
  • 1 974 609 375 000 ÷ 2 = 987 304 687 500 + 0;
  • 987 304 687 500 ÷ 2 = 493 652 343 750 + 0;
  • 493 652 343 750 ÷ 2 = 246 826 171 875 + 0;
  • 246 826 171 875 ÷ 2 = 123 413 085 937 + 1;
  • 123 413 085 937 ÷ 2 = 61 706 542 968 + 1;
  • 61 706 542 968 ÷ 2 = 30 853 271 484 + 0;
  • 30 853 271 484 ÷ 2 = 15 426 635 742 + 0;
  • 15 426 635 742 ÷ 2 = 7 713 317 871 + 0;
  • 7 713 317 871 ÷ 2 = 3 856 658 935 + 1;
  • 3 856 658 935 ÷ 2 = 1 928 329 467 + 1;
  • 1 928 329 467 ÷ 2 = 964 164 733 + 1;
  • 964 164 733 ÷ 2 = 482 082 366 + 1;
  • 482 082 366 ÷ 2 = 241 041 183 + 0;
  • 241 041 183 ÷ 2 = 120 520 591 + 1;
  • 120 520 591 ÷ 2 = 60 260 295 + 1;
  • 60 260 295 ÷ 2 = 30 130 147 + 1;
  • 30 130 147 ÷ 2 = 15 065 073 + 1;
  • 15 065 073 ÷ 2 = 7 532 536 + 1;
  • 7 532 536 ÷ 2 = 3 766 268 + 0;
  • 3 766 268 ÷ 2 = 1 883 134 + 0;
  • 1 883 134 ÷ 2 = 941 567 + 0;
  • 941 567 ÷ 2 = 470 783 + 1;
  • 470 783 ÷ 2 = 235 391 + 1;
  • 235 391 ÷ 2 = 117 695 + 1;
  • 117 695 ÷ 2 = 58 847 + 1;
  • 58 847 ÷ 2 = 29 423 + 1;
  • 29 423 ÷ 2 = 14 711 + 1;
  • 14 711 ÷ 2 = 7 355 + 1;
  • 7 355 ÷ 2 = 3 677 + 1;
  • 3 677 ÷ 2 = 1 838 + 1;
  • 1 838 ÷ 2 = 919 + 0;
  • 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.


1 011 000 000 000 061(10) = 11 1001 0111 0111 1111 1100 0111 1101 1110 0011 0000 0011 1101(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.


Number 1 011 000 000 000 061(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in two's complement representation:

1 011 000 000 000 061(10) = 0000 0000 0000 0011 1001 0111 0111 1111 1100 0111 1101 1110 0011 0000 0011 1101

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

More operations on integers written as signed binary numbers in two's complement representation:

» Signed integer number 1 011 000 000 000 060 converted from decimal system (from base 10) and written as a signed binary in two's complement representation (written in base 2) = ?

» Signed integer number 1 011 000 000 000 062 converted from decimal system (from base 10) and written as a signed binary in two's complement representation (written in base 2) = ?

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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