Two's Complement: Integer ↗ Binary: 100 000 000 000 000 125 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 100 000 000 000 000 125(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;
  • 100 000 000 000 000 125 ÷ 2 = 50 000 000 000 000 062 + 1;
  • 50 000 000 000 000 062 ÷ 2 = 25 000 000 000 000 031 + 0;
  • 25 000 000 000 000 031 ÷ 2 = 12 500 000 000 000 015 + 1;
  • 12 500 000 000 000 015 ÷ 2 = 6 250 000 000 000 007 + 1;
  • 6 250 000 000 000 007 ÷ 2 = 3 125 000 000 000 003 + 1;
  • 3 125 000 000 000 003 ÷ 2 = 1 562 500 000 000 001 + 1;
  • 1 562 500 000 000 001 ÷ 2 = 781 250 000 000 000 + 1;
  • 781 250 000 000 000 ÷ 2 = 390 625 000 000 000 + 0;
  • 390 625 000 000 000 ÷ 2 = 195 312 500 000 000 + 0;
  • 195 312 500 000 000 ÷ 2 = 97 656 250 000 000 + 0;
  • 97 656 250 000 000 ÷ 2 = 48 828 125 000 000 + 0;
  • 48 828 125 000 000 ÷ 2 = 24 414 062 500 000 + 0;
  • 24 414 062 500 000 ÷ 2 = 12 207 031 250 000 + 0;
  • 12 207 031 250 000 ÷ 2 = 6 103 515 625 000 + 0;
  • 6 103 515 625 000 ÷ 2 = 3 051 757 812 500 + 0;
  • 3 051 757 812 500 ÷ 2 = 1 525 878 906 250 + 0;
  • 1 525 878 906 250 ÷ 2 = 762 939 453 125 + 0;
  • 762 939 453 125 ÷ 2 = 381 469 726 562 + 1;
  • 381 469 726 562 ÷ 2 = 190 734 863 281 + 0;
  • 190 734 863 281 ÷ 2 = 95 367 431 640 + 1;
  • 95 367 431 640 ÷ 2 = 47 683 715 820 + 0;
  • 47 683 715 820 ÷ 2 = 23 841 857 910 + 0;
  • 23 841 857 910 ÷ 2 = 11 920 928 955 + 0;
  • 11 920 928 955 ÷ 2 = 5 960 464 477 + 1;
  • 5 960 464 477 ÷ 2 = 2 980 232 238 + 1;
  • 2 980 232 238 ÷ 2 = 1 490 116 119 + 0;
  • 1 490 116 119 ÷ 2 = 745 058 059 + 1;
  • 745 058 059 ÷ 2 = 372 529 029 + 1;
  • 372 529 029 ÷ 2 = 186 264 514 + 1;
  • 186 264 514 ÷ 2 = 93 132 257 + 0;
  • 93 132 257 ÷ 2 = 46 566 128 + 1;
  • 46 566 128 ÷ 2 = 23 283 064 + 0;
  • 23 283 064 ÷ 2 = 11 641 532 + 0;
  • 11 641 532 ÷ 2 = 5 820 766 + 0;
  • 5 820 766 ÷ 2 = 2 910 383 + 0;
  • 2 910 383 ÷ 2 = 1 455 191 + 1;
  • 1 455 191 ÷ 2 = 727 595 + 1;
  • 727 595 ÷ 2 = 363 797 + 1;
  • 363 797 ÷ 2 = 181 898 + 1;
  • 181 898 ÷ 2 = 90 949 + 0;
  • 90 949 ÷ 2 = 45 474 + 1;
  • 45 474 ÷ 2 = 22 737 + 0;
  • 22 737 ÷ 2 = 11 368 + 1;
  • 11 368 ÷ 2 = 5 684 + 0;
  • 5 684 ÷ 2 = 2 842 + 0;
  • 2 842 ÷ 2 = 1 421 + 0;
  • 1 421 ÷ 2 = 710 + 1;
  • 710 ÷ 2 = 355 + 0;
  • 355 ÷ 2 = 177 + 1;
  • 177 ÷ 2 = 88 + 1;
  • 88 ÷ 2 = 44 + 0;
  • 44 ÷ 2 = 22 + 0;
  • 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.


100 000 000 000 000 125(10) = 1 0110 0011 0100 0101 0111 1000 0101 1101 1000 1010 0000 0000 0111 1101(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 57.

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, 57,

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 100 000 000 000 000 125(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:

100 000 000 000 000 125(10) = 0000 0001 0110 0011 0100 0101 0111 1000 0101 1101 1000 1010 0000 0000 0111 1101

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 100 000 000 000 000 124 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 100 000 000 000 000 126 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