Two's Complement: Integer ↗ Binary: 100 011 011 110 149 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 100 011 011 110 149(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;
  • 100 011 011 110 149 ÷ 2 = 50 005 505 555 074 + 1;
  • 50 005 505 555 074 ÷ 2 = 25 002 752 777 537 + 0;
  • 25 002 752 777 537 ÷ 2 = 12 501 376 388 768 + 1;
  • 12 501 376 388 768 ÷ 2 = 6 250 688 194 384 + 0;
  • 6 250 688 194 384 ÷ 2 = 3 125 344 097 192 + 0;
  • 3 125 344 097 192 ÷ 2 = 1 562 672 048 596 + 0;
  • 1 562 672 048 596 ÷ 2 = 781 336 024 298 + 0;
  • 781 336 024 298 ÷ 2 = 390 668 012 149 + 0;
  • 390 668 012 149 ÷ 2 = 195 334 006 074 + 1;
  • 195 334 006 074 ÷ 2 = 97 667 003 037 + 0;
  • 97 667 003 037 ÷ 2 = 48 833 501 518 + 1;
  • 48 833 501 518 ÷ 2 = 24 416 750 759 + 0;
  • 24 416 750 759 ÷ 2 = 12 208 375 379 + 1;
  • 12 208 375 379 ÷ 2 = 6 104 187 689 + 1;
  • 6 104 187 689 ÷ 2 = 3 052 093 844 + 1;
  • 3 052 093 844 ÷ 2 = 1 526 046 922 + 0;
  • 1 526 046 922 ÷ 2 = 763 023 461 + 0;
  • 763 023 461 ÷ 2 = 381 511 730 + 1;
  • 381 511 730 ÷ 2 = 190 755 865 + 0;
  • 190 755 865 ÷ 2 = 95 377 932 + 1;
  • 95 377 932 ÷ 2 = 47 688 966 + 0;
  • 47 688 966 ÷ 2 = 23 844 483 + 0;
  • 23 844 483 ÷ 2 = 11 922 241 + 1;
  • 11 922 241 ÷ 2 = 5 961 120 + 1;
  • 5 961 120 ÷ 2 = 2 980 560 + 0;
  • 2 980 560 ÷ 2 = 1 490 280 + 0;
  • 1 490 280 ÷ 2 = 745 140 + 0;
  • 745 140 ÷ 2 = 372 570 + 0;
  • 372 570 ÷ 2 = 186 285 + 0;
  • 186 285 ÷ 2 = 93 142 + 1;
  • 93 142 ÷ 2 = 46 571 + 0;
  • 46 571 ÷ 2 = 23 285 + 1;
  • 23 285 ÷ 2 = 11 642 + 1;
  • 11 642 ÷ 2 = 5 821 + 0;
  • 5 821 ÷ 2 = 2 910 + 1;
  • 2 910 ÷ 2 = 1 455 + 0;
  • 1 455 ÷ 2 = 727 + 1;
  • 727 ÷ 2 = 363 + 1;
  • 363 ÷ 2 = 181 + 1;
  • 181 ÷ 2 = 90 + 1;
  • 90 ÷ 2 = 45 + 0;
  • 45 ÷ 2 = 22 + 1;
  • 22 ÷ 2 = 11 + 0;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 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.


100 011 011 110 149(10) = 101 1010 1111 0101 1010 0000 1100 1010 0111 0101 0000 0101(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 47.

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, 47,

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 100 011 011 110 149(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:

100 011 011 110 149(10) = 0000 0000 0000 0000 0101 1010 1111 0101 1010 0000 1100 1010 0111 0101 0000 0101

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 100 011 011 110 148 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 100 011 011 110 150 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