Convert 2 157 529 056 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 2 157 529 056(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
2 157 529 056 to a signed binary in one's (1'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.

Stop when you get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 2 157 529 056 ÷ 2 = 1 078 764 528 + 0;
  • 1 078 764 528 ÷ 2 = 539 382 264 + 0;
  • 539 382 264 ÷ 2 = 269 691 132 + 0;
  • 269 691 132 ÷ 2 = 134 845 566 + 0;
  • 134 845 566 ÷ 2 = 67 422 783 + 0;
  • 67 422 783 ÷ 2 = 33 711 391 + 1;
  • 33 711 391 ÷ 2 = 16 855 695 + 1;
  • 16 855 695 ÷ 2 = 8 427 847 + 1;
  • 8 427 847 ÷ 2 = 4 213 923 + 1;
  • 4 213 923 ÷ 2 = 2 106 961 + 1;
  • 2 106 961 ÷ 2 = 1 053 480 + 1;
  • 1 053 480 ÷ 2 = 526 740 + 0;
  • 526 740 ÷ 2 = 263 370 + 0;
  • 263 370 ÷ 2 = 131 685 + 0;
  • 131 685 ÷ 2 = 65 842 + 1;
  • 65 842 ÷ 2 = 32 921 + 0;
  • 32 921 ÷ 2 = 16 460 + 1;
  • 16 460 ÷ 2 = 8 230 + 0;
  • 8 230 ÷ 2 = 4 115 + 0;
  • 4 115 ÷ 2 = 2 057 + 1;
  • 2 057 ÷ 2 = 1 028 + 1;
  • 1 028 ÷ 2 = 514 + 0;
  • 514 ÷ 2 = 257 + 0;
  • 257 ÷ 2 = 128 + 1;
  • 128 ÷ 2 = 64 + 0;
  • 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.

2 157 529 056(10) = 1000 0000 1001 1001 0100 0111 1110 0000(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 32.

  • 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, 32,

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 2 157 529 056(10) converted to signed binary in one's complement representation:

2 157 529 056(10) = 0000 0000 0000 0000 0000 0000 0000 0000 1000 0000 1001 1001 0100 0111 1110 0000

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

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How to convert signed integers from the decimal system to signed binary in one's complement representation

Follow the steps below to convert a signed base 10 integer number to signed binary in one'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 that is to be converted to binary, keeping track of each remainder, until we get a quotient that is equal to 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, fill in '0' bits in front (to the left) of the base 2 number calculated above, up to the right length; this way the first bit (leftmost) will always 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 '1's and all '1' bits with '0's.

Example: convert the negative number -49 from the decimal system (base ten) to signed binary one's complement:

  • 1. Start with the positive version of the number: |-49| = 49
  • 2. Divide repeatedly 49 by 2, keeping track of each remainder:
    • division = quotient + remainder
    • 49 ÷ 2 = 24 + 1
    • 24 ÷ 2 = 12 + 0
    • 12 ÷ 2 = 6 + 0
    • 6 ÷ 2 = 3 + 0
    • 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:
    49(10) = 11 0001(2)
  • 4. The actual bit length of base 2 representation is 6, so the positive binary computer representation of a signed binary will take in this case 8 bits (the least power of 2 that is larger than 6) - add '0's in front of the base 2 number, up to the required length:
    49(10) = 0011 0001(2)
  • 5. To get the negative integer number representation in signed binary one's complement, replace all '0' bits with '1's and all '1' bits with '0's:
    -49(10) = 1100 1110
  • Number -49(10), signed integer, converted from the decimal system (base 10) to signed binary in one's complement representation = 1100 1110