Convert 10 000 000 000 000 215 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 10 000 000 000 000 215(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
10 000 000 000 000 215 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;
  • 10 000 000 000 000 215 ÷ 2 = 5 000 000 000 000 107 + 1;
  • 5 000 000 000 000 107 ÷ 2 = 2 500 000 000 000 053 + 1;
  • 2 500 000 000 000 053 ÷ 2 = 1 250 000 000 000 026 + 1;
  • 1 250 000 000 000 026 ÷ 2 = 625 000 000 000 013 + 0;
  • 625 000 000 000 013 ÷ 2 = 312 500 000 000 006 + 1;
  • 312 500 000 000 006 ÷ 2 = 156 250 000 000 003 + 0;
  • 156 250 000 000 003 ÷ 2 = 78 125 000 000 001 + 1;
  • 78 125 000 000 001 ÷ 2 = 39 062 500 000 000 + 1;
  • 39 062 500 000 000 ÷ 2 = 19 531 250 000 000 + 0;
  • 19 531 250 000 000 ÷ 2 = 9 765 625 000 000 + 0;
  • 9 765 625 000 000 ÷ 2 = 4 882 812 500 000 + 0;
  • 4 882 812 500 000 ÷ 2 = 2 441 406 250 000 + 0;
  • 2 441 406 250 000 ÷ 2 = 1 220 703 125 000 + 0;
  • 1 220 703 125 000 ÷ 2 = 610 351 562 500 + 0;
  • 610 351 562 500 ÷ 2 = 305 175 781 250 + 0;
  • 305 175 781 250 ÷ 2 = 152 587 890 625 + 0;
  • 152 587 890 625 ÷ 2 = 76 293 945 312 + 1;
  • 76 293 945 312 ÷ 2 = 38 146 972 656 + 0;
  • 38 146 972 656 ÷ 2 = 19 073 486 328 + 0;
  • 19 073 486 328 ÷ 2 = 9 536 743 164 + 0;
  • 9 536 743 164 ÷ 2 = 4 768 371 582 + 0;
  • 4 768 371 582 ÷ 2 = 2 384 185 791 + 0;
  • 2 384 185 791 ÷ 2 = 1 192 092 895 + 1;
  • 1 192 092 895 ÷ 2 = 596 046 447 + 1;
  • 596 046 447 ÷ 2 = 298 023 223 + 1;
  • 298 023 223 ÷ 2 = 149 011 611 + 1;
  • 149 011 611 ÷ 2 = 74 505 805 + 1;
  • 74 505 805 ÷ 2 = 37 252 902 + 1;
  • 37 252 902 ÷ 2 = 18 626 451 + 0;
  • 18 626 451 ÷ 2 = 9 313 225 + 1;
  • 9 313 225 ÷ 2 = 4 656 612 + 1;
  • 4 656 612 ÷ 2 = 2 328 306 + 0;
  • 2 328 306 ÷ 2 = 1 164 153 + 0;
  • 1 164 153 ÷ 2 = 582 076 + 1;
  • 582 076 ÷ 2 = 291 038 + 0;
  • 291 038 ÷ 2 = 145 519 + 0;
  • 145 519 ÷ 2 = 72 759 + 1;
  • 72 759 ÷ 2 = 36 379 + 1;
  • 36 379 ÷ 2 = 18 189 + 1;
  • 18 189 ÷ 2 = 9 094 + 1;
  • 9 094 ÷ 2 = 4 547 + 0;
  • 4 547 ÷ 2 = 2 273 + 1;
  • 2 273 ÷ 2 = 1 136 + 1;
  • 1 136 ÷ 2 = 568 + 0;
  • 568 ÷ 2 = 284 + 0;
  • 284 ÷ 2 = 142 + 0;
  • 142 ÷ 2 = 71 + 0;
  • 71 ÷ 2 = 35 + 1;
  • 35 ÷ 2 = 17 + 1;
  • 17 ÷ 2 = 8 + 1;
  • 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.

10 000 000 000 000 215(10) = 10 0011 1000 0110 1111 0010 0110 1111 1100 0001 0000 0000 1101 0111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 54.

  • 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, 54,

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 10 000 000 000 000 215(10) converted to signed binary in one's complement representation:

10 000 000 000 000 215(10) = 0000 0000 0010 0011 1000 0110 1111 0010 0110 1111 1100 0001 0000 0000 1101 0111

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