Two's Complement: Integer ↗ Binary: 1 111 001 010 138 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 1 111 001 010 138(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;
  • 1 111 001 010 138 ÷ 2 = 555 500 505 069 + 0;
  • 555 500 505 069 ÷ 2 = 277 750 252 534 + 1;
  • 277 750 252 534 ÷ 2 = 138 875 126 267 + 0;
  • 138 875 126 267 ÷ 2 = 69 437 563 133 + 1;
  • 69 437 563 133 ÷ 2 = 34 718 781 566 + 1;
  • 34 718 781 566 ÷ 2 = 17 359 390 783 + 0;
  • 17 359 390 783 ÷ 2 = 8 679 695 391 + 1;
  • 8 679 695 391 ÷ 2 = 4 339 847 695 + 1;
  • 4 339 847 695 ÷ 2 = 2 169 923 847 + 1;
  • 2 169 923 847 ÷ 2 = 1 084 961 923 + 1;
  • 1 084 961 923 ÷ 2 = 542 480 961 + 1;
  • 542 480 961 ÷ 2 = 271 240 480 + 1;
  • 271 240 480 ÷ 2 = 135 620 240 + 0;
  • 135 620 240 ÷ 2 = 67 810 120 + 0;
  • 67 810 120 ÷ 2 = 33 905 060 + 0;
  • 33 905 060 ÷ 2 = 16 952 530 + 0;
  • 16 952 530 ÷ 2 = 8 476 265 + 0;
  • 8 476 265 ÷ 2 = 4 238 132 + 1;
  • 4 238 132 ÷ 2 = 2 119 066 + 0;
  • 2 119 066 ÷ 2 = 1 059 533 + 0;
  • 1 059 533 ÷ 2 = 529 766 + 1;
  • 529 766 ÷ 2 = 264 883 + 0;
  • 264 883 ÷ 2 = 132 441 + 1;
  • 132 441 ÷ 2 = 66 220 + 1;
  • 66 220 ÷ 2 = 33 110 + 0;
  • 33 110 ÷ 2 = 16 555 + 0;
  • 16 555 ÷ 2 = 8 277 + 1;
  • 8 277 ÷ 2 = 4 138 + 1;
  • 4 138 ÷ 2 = 2 069 + 0;
  • 2 069 ÷ 2 = 1 034 + 1;
  • 1 034 ÷ 2 = 517 + 0;
  • 517 ÷ 2 = 258 + 1;
  • 258 ÷ 2 = 129 + 0;
  • 129 ÷ 2 = 64 + 1;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 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.


1 111 001 010 138(10) = 1 0000 0010 1010 1100 1101 0010 0000 1111 1101 1010(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 41.

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, 41,

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 1 111 001 010 138(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:

1 111 001 010 138(10) = 0000 0000 0000 0000 0000 0001 0000 0010 1010 1100 1101 0010 0000 1111 1101 1010

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 1 111 001 010 137 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 1 111 001 010 139 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