Two's Complement: Integer ↗ Binary: -19 531 235 312 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 -19 531 235 312(10) converted and written as a signed binary in two's complement representation (base 2) = ?

1. Start with the positive version of the number:

|-19 531 235 312| = 19 531 235 312

2. 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;
  • 19 531 235 312 ÷ 2 = 9 765 617 656 + 0;
  • 9 765 617 656 ÷ 2 = 4 882 808 828 + 0;
  • 4 882 808 828 ÷ 2 = 2 441 404 414 + 0;
  • 2 441 404 414 ÷ 2 = 1 220 702 207 + 0;
  • 1 220 702 207 ÷ 2 = 610 351 103 + 1;
  • 610 351 103 ÷ 2 = 305 175 551 + 1;
  • 305 175 551 ÷ 2 = 152 587 775 + 1;
  • 152 587 775 ÷ 2 = 76 293 887 + 1;
  • 76 293 887 ÷ 2 = 38 146 943 + 1;
  • 38 146 943 ÷ 2 = 19 073 471 + 1;
  • 19 073 471 ÷ 2 = 9 536 735 + 1;
  • 9 536 735 ÷ 2 = 4 768 367 + 1;
  • 4 768 367 ÷ 2 = 2 384 183 + 1;
  • 2 384 183 ÷ 2 = 1 192 091 + 1;
  • 1 192 091 ÷ 2 = 596 045 + 1;
  • 596 045 ÷ 2 = 298 022 + 1;
  • 298 022 ÷ 2 = 149 011 + 0;
  • 149 011 ÷ 2 = 74 505 + 1;
  • 74 505 ÷ 2 = 37 252 + 1;
  • 37 252 ÷ 2 = 18 626 + 0;
  • 18 626 ÷ 2 = 9 313 + 0;
  • 9 313 ÷ 2 = 4 656 + 1;
  • 4 656 ÷ 2 = 2 328 + 0;
  • 2 328 ÷ 2 = 1 164 + 0;
  • 1 164 ÷ 2 = 582 + 0;
  • 582 ÷ 2 = 291 + 0;
  • 291 ÷ 2 = 145 + 1;
  • 145 ÷ 2 = 72 + 1;
  • 72 ÷ 2 = 36 + 0;
  • 36 ÷ 2 = 18 + 0;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

3. Construct the base 2 representation of the positive number:

Take all the remainders starting from the bottom of the list constructed above.


19 531 235 312(10) = 100 1000 1100 0010 0110 1111 1111 1111 0000(2)


4. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 35.

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, 35,

3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)

=== is: 64.


5. 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.


19 531 235 312(10) = 0000 0000 0000 0000 0000 0000 0000 0100 1000 1100 0010 0110 1111 1111 1111 0000


6. Get the negative integer number representation. Part 1:

To write the negative integer number on 64 bits (8 Bytes),

as a signed binary in one's complement representation,

... replace all the bits on 0 with 1s and all the bits set on 1 with 0s.


Reverse the digits, flip the digits:

Replace the bits set on 0 with 1s and the bits set on 1 with 0s.

!(0000 0000 0000 0000 0000 0000 0000 0100 1000 1100 0010 0110 1111 1111 1111 0000)

= 1111 1111 1111 1111 1111 1111 1111 1011 0111 0011 1101 1001 0000 0000 0000 1111


7. Get the negative integer number representation. Part 2:

To write the negative integer number on 64 bits (8 Bytes),

as a signed binary in two's complement representation,

add 1 to the number calculated above

1111 1111 1111 1111 1111 1111 1111 1011 0111 0011 1101 1001 0000 0000 0000 1111

(to the signed binary in one's complement representation)


Binary addition carries on a value of 2:

0 + 0 = 0

0 + 1 = 1

1 + 1 = 10

1 + 10 = 11

1 + 11 = 100


Add 1 to the number calculated above
(to the signed binary number in one's complement representation):

-19 531 235 312 =

1111 1111 1111 1111 1111 1111 1111 1011 0111 0011 1101 1001 0000 0000 0000 1111 + 1


Number -19 531 235 312(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:

-19 531 235 312(10) = 1111 1111 1111 1111 1111 1111 1111 1011 0111 0011 1101 1001 0000 0000 0001 0000

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 -19 531 235 313 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 -19 531 235 311 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