Two's Complement: Integer ↗ Binary: -40 000 112 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 -40 000 112(10) converted and written as a signed binary in two's complement representation (base 2) = ?

1. Start with the positive version of the number:

|-40 000 112| = 40 000 112

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;
  • 40 000 112 ÷ 2 = 20 000 056 + 0;
  • 20 000 056 ÷ 2 = 10 000 028 + 0;
  • 10 000 028 ÷ 2 = 5 000 014 + 0;
  • 5 000 014 ÷ 2 = 2 500 007 + 0;
  • 2 500 007 ÷ 2 = 1 250 003 + 1;
  • 1 250 003 ÷ 2 = 625 001 + 1;
  • 625 001 ÷ 2 = 312 500 + 1;
  • 312 500 ÷ 2 = 156 250 + 0;
  • 156 250 ÷ 2 = 78 125 + 0;
  • 78 125 ÷ 2 = 39 062 + 1;
  • 39 062 ÷ 2 = 19 531 + 0;
  • 19 531 ÷ 2 = 9 765 + 1;
  • 9 765 ÷ 2 = 4 882 + 1;
  • 4 882 ÷ 2 = 2 441 + 0;
  • 2 441 ÷ 2 = 1 220 + 1;
  • 1 220 ÷ 2 = 610 + 0;
  • 610 ÷ 2 = 305 + 0;
  • 305 ÷ 2 = 152 + 1;
  • 152 ÷ 2 = 76 + 0;
  • 76 ÷ 2 = 38 + 0;
  • 38 ÷ 2 = 19 + 0;
  • 19 ÷ 2 = 9 + 1;
  • 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.


40 000 112(10) = 10 0110 0010 0101 1010 0111 0000(2)


4. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 26.


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, 26,

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


=== is: 32.


5. Get the positive binary computer representation on 32 bits (4 Bytes):

If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 32.


40 000 112(10) = 0000 0010 0110 0010 0101 1010 0111 0000


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

To write the negative integer number on 32 bits (4 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 0010 0110 0010 0101 1010 0111 0000)


= 1111 1101 1001 1101 1010 0101 1000 1111


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

To write the negative integer number on 32 bits (4 Bytes),

as a signed binary in two's complement representation,


add 1 to the number calculated above

1111 1101 1001 1101 1010 0101 1000 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):

-40 000 112 =

1111 1101 1001 1101 1010 0101 1000 1111 + 1


Number -40 000 112(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:

-40 000 112(10) = 1111 1101 1001 1101 1010 0101 1001 0000

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