Convert 121 212 121 211 903 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 121 212 121 211 903(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
121 212 121 211 903 to a signed binary in two's (2's) complement representation?

  • A signed integer, written in base ten, or a decimal system number, is a number written using the digits 0 through 9 and the sign, which can be positive (+) or negative (-). If positive, the sign is usually not written. A number written in base two, or binary, is a number written using only the digits 0 and 1.

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;
  • 121 212 121 211 903 ÷ 2 = 60 606 060 605 951 + 1;
  • 60 606 060 605 951 ÷ 2 = 30 303 030 302 975 + 1;
  • 30 303 030 302 975 ÷ 2 = 15 151 515 151 487 + 1;
  • 15 151 515 151 487 ÷ 2 = 7 575 757 575 743 + 1;
  • 7 575 757 575 743 ÷ 2 = 3 787 878 787 871 + 1;
  • 3 787 878 787 871 ÷ 2 = 1 893 939 393 935 + 1;
  • 1 893 939 393 935 ÷ 2 = 946 969 696 967 + 1;
  • 946 969 696 967 ÷ 2 = 473 484 848 483 + 1;
  • 473 484 848 483 ÷ 2 = 236 742 424 241 + 1;
  • 236 742 424 241 ÷ 2 = 118 371 212 120 + 1;
  • 118 371 212 120 ÷ 2 = 59 185 606 060 + 0;
  • 59 185 606 060 ÷ 2 = 29 592 803 030 + 0;
  • 29 592 803 030 ÷ 2 = 14 796 401 515 + 0;
  • 14 796 401 515 ÷ 2 = 7 398 200 757 + 1;
  • 7 398 200 757 ÷ 2 = 3 699 100 378 + 1;
  • 3 699 100 378 ÷ 2 = 1 849 550 189 + 0;
  • 1 849 550 189 ÷ 2 = 924 775 094 + 1;
  • 924 775 094 ÷ 2 = 462 387 547 + 0;
  • 462 387 547 ÷ 2 = 231 193 773 + 1;
  • 231 193 773 ÷ 2 = 115 596 886 + 1;
  • 115 596 886 ÷ 2 = 57 798 443 + 0;
  • 57 798 443 ÷ 2 = 28 899 221 + 1;
  • 28 899 221 ÷ 2 = 14 449 610 + 1;
  • 14 449 610 ÷ 2 = 7 224 805 + 0;
  • 7 224 805 ÷ 2 = 3 612 402 + 1;
  • 3 612 402 ÷ 2 = 1 806 201 + 0;
  • 1 806 201 ÷ 2 = 903 100 + 1;
  • 903 100 ÷ 2 = 451 550 + 0;
  • 451 550 ÷ 2 = 225 775 + 0;
  • 225 775 ÷ 2 = 112 887 + 1;
  • 112 887 ÷ 2 = 56 443 + 1;
  • 56 443 ÷ 2 = 28 221 + 1;
  • 28 221 ÷ 2 = 14 110 + 1;
  • 14 110 ÷ 2 = 7 055 + 0;
  • 7 055 ÷ 2 = 3 527 + 1;
  • 3 527 ÷ 2 = 1 763 + 1;
  • 1 763 ÷ 2 = 881 + 1;
  • 881 ÷ 2 = 440 + 1;
  • 440 ÷ 2 = 220 + 0;
  • 220 ÷ 2 = 110 + 0;
  • 110 ÷ 2 = 55 + 0;
  • 55 ÷ 2 = 27 + 1;
  • 27 ÷ 2 = 13 + 1;
  • 13 ÷ 2 = 6 + 1;
  • 6 ÷ 2 = 3 + 0;
  • 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.

121 212 121 211 903(10) = 110 1110 0011 1101 1110 0101 0110 1101 0110 0011 1111 1111(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.


Decimal Number 121 212 121 211 903(10) converted to signed binary in two's complement representation:

121 212 121 211 903(10) = 0000 0000 0000 0000 0110 1110 0011 1101 1110 0101 0110 1101 0110 0011 1111 1111

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

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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