Two's Complement: Integer ↗ Binary: 10 000 010 184 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 10 000 010 184(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;
  • 10 000 010 184 ÷ 2 = 5 000 005 092 + 0;
  • 5 000 005 092 ÷ 2 = 2 500 002 546 + 0;
  • 2 500 002 546 ÷ 2 = 1 250 001 273 + 0;
  • 1 250 001 273 ÷ 2 = 625 000 636 + 1;
  • 625 000 636 ÷ 2 = 312 500 318 + 0;
  • 312 500 318 ÷ 2 = 156 250 159 + 0;
  • 156 250 159 ÷ 2 = 78 125 079 + 1;
  • 78 125 079 ÷ 2 = 39 062 539 + 1;
  • 39 062 539 ÷ 2 = 19 531 269 + 1;
  • 19 531 269 ÷ 2 = 9 765 634 + 1;
  • 9 765 634 ÷ 2 = 4 882 817 + 0;
  • 4 882 817 ÷ 2 = 2 441 408 + 1;
  • 2 441 408 ÷ 2 = 1 220 704 + 0;
  • 1 220 704 ÷ 2 = 610 352 + 0;
  • 610 352 ÷ 2 = 305 176 + 0;
  • 305 176 ÷ 2 = 152 588 + 0;
  • 152 588 ÷ 2 = 76 294 + 0;
  • 76 294 ÷ 2 = 38 147 + 0;
  • 38 147 ÷ 2 = 19 073 + 1;
  • 19 073 ÷ 2 = 9 536 + 1;
  • 9 536 ÷ 2 = 4 768 + 0;
  • 4 768 ÷ 2 = 2 384 + 0;
  • 2 384 ÷ 2 = 1 192 + 0;
  • 1 192 ÷ 2 = 596 + 0;
  • 596 ÷ 2 = 298 + 0;
  • 298 ÷ 2 = 149 + 0;
  • 149 ÷ 2 = 74 + 1;
  • 74 ÷ 2 = 37 + 0;
  • 37 ÷ 2 = 18 + 1;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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.


10 000 010 184(10) = 10 0101 0100 0000 1100 0000 1011 1100 1000(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 34.

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, 34,

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 10 000 010 184(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:

10 000 010 184(10) = 0000 0000 0000 0000 0000 0000 0000 0010 0101 0100 0000 1100 0000 1011 1100 1000

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 10 000 010 183 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 10 000 010 185 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