Two's Complement: Integer ↗ Binary: 1 010 000 011 000 144 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 1 010 000 011 000 144(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;
  • 1 010 000 011 000 144 ÷ 2 = 505 000 005 500 072 + 0;
  • 505 000 005 500 072 ÷ 2 = 252 500 002 750 036 + 0;
  • 252 500 002 750 036 ÷ 2 = 126 250 001 375 018 + 0;
  • 126 250 001 375 018 ÷ 2 = 63 125 000 687 509 + 0;
  • 63 125 000 687 509 ÷ 2 = 31 562 500 343 754 + 1;
  • 31 562 500 343 754 ÷ 2 = 15 781 250 171 877 + 0;
  • 15 781 250 171 877 ÷ 2 = 7 890 625 085 938 + 1;
  • 7 890 625 085 938 ÷ 2 = 3 945 312 542 969 + 0;
  • 3 945 312 542 969 ÷ 2 = 1 972 656 271 484 + 1;
  • 1 972 656 271 484 ÷ 2 = 986 328 135 742 + 0;
  • 986 328 135 742 ÷ 2 = 493 164 067 871 + 0;
  • 493 164 067 871 ÷ 2 = 246 582 033 935 + 1;
  • 246 582 033 935 ÷ 2 = 123 291 016 967 + 1;
  • 123 291 016 967 ÷ 2 = 61 645 508 483 + 1;
  • 61 645 508 483 ÷ 2 = 30 822 754 241 + 1;
  • 30 822 754 241 ÷ 2 = 15 411 377 120 + 1;
  • 15 411 377 120 ÷ 2 = 7 705 688 560 + 0;
  • 7 705 688 560 ÷ 2 = 3 852 844 280 + 0;
  • 3 852 844 280 ÷ 2 = 1 926 422 140 + 0;
  • 1 926 422 140 ÷ 2 = 963 211 070 + 0;
  • 963 211 070 ÷ 2 = 481 605 535 + 0;
  • 481 605 535 ÷ 2 = 240 802 767 + 1;
  • 240 802 767 ÷ 2 = 120 401 383 + 1;
  • 120 401 383 ÷ 2 = 60 200 691 + 1;
  • 60 200 691 ÷ 2 = 30 100 345 + 1;
  • 30 100 345 ÷ 2 = 15 050 172 + 1;
  • 15 050 172 ÷ 2 = 7 525 086 + 0;
  • 7 525 086 ÷ 2 = 3 762 543 + 0;
  • 3 762 543 ÷ 2 = 1 881 271 + 1;
  • 1 881 271 ÷ 2 = 940 635 + 1;
  • 940 635 ÷ 2 = 470 317 + 1;
  • 470 317 ÷ 2 = 235 158 + 1;
  • 235 158 ÷ 2 = 117 579 + 0;
  • 117 579 ÷ 2 = 58 789 + 1;
  • 58 789 ÷ 2 = 29 394 + 1;
  • 29 394 ÷ 2 = 14 697 + 0;
  • 14 697 ÷ 2 = 7 348 + 1;
  • 7 348 ÷ 2 = 3 674 + 0;
  • 3 674 ÷ 2 = 1 837 + 0;
  • 1 837 ÷ 2 = 918 + 1;
  • 918 ÷ 2 = 459 + 0;
  • 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.


1 010 000 011 000 144(10) = 11 1001 0110 1001 0110 1111 0011 1110 0000 1111 1001 0101 0000(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 1 010 000 011 000 144(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:

1 010 000 011 000 144(10) = 0000 0000 0000 0011 1001 0110 1001 0110 1111 0011 1110 0000 1111 1001 0101 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 1 010 000 011 000 143 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 1 010 000 011 000 145 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