Convert 1 111 110 011 111 051 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 1 111 110 011 111 051(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
1 111 110 011 111 051 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;
  • 1 111 110 011 111 051 ÷ 2 = 555 555 005 555 525 + 1;
  • 555 555 005 555 525 ÷ 2 = 277 777 502 777 762 + 1;
  • 277 777 502 777 762 ÷ 2 = 138 888 751 388 881 + 0;
  • 138 888 751 388 881 ÷ 2 = 69 444 375 694 440 + 1;
  • 69 444 375 694 440 ÷ 2 = 34 722 187 847 220 + 0;
  • 34 722 187 847 220 ÷ 2 = 17 361 093 923 610 + 0;
  • 17 361 093 923 610 ÷ 2 = 8 680 546 961 805 + 0;
  • 8 680 546 961 805 ÷ 2 = 4 340 273 480 902 + 1;
  • 4 340 273 480 902 ÷ 2 = 2 170 136 740 451 + 0;
  • 2 170 136 740 451 ÷ 2 = 1 085 068 370 225 + 1;
  • 1 085 068 370 225 ÷ 2 = 542 534 185 112 + 1;
  • 542 534 185 112 ÷ 2 = 271 267 092 556 + 0;
  • 271 267 092 556 ÷ 2 = 135 633 546 278 + 0;
  • 135 633 546 278 ÷ 2 = 67 816 773 139 + 0;
  • 67 816 773 139 ÷ 2 = 33 908 386 569 + 1;
  • 33 908 386 569 ÷ 2 = 16 954 193 284 + 1;
  • 16 954 193 284 ÷ 2 = 8 477 096 642 + 0;
  • 8 477 096 642 ÷ 2 = 4 238 548 321 + 0;
  • 4 238 548 321 ÷ 2 = 2 119 274 160 + 1;
  • 2 119 274 160 ÷ 2 = 1 059 637 080 + 0;
  • 1 059 637 080 ÷ 2 = 529 818 540 + 0;
  • 529 818 540 ÷ 2 = 264 909 270 + 0;
  • 264 909 270 ÷ 2 = 132 454 635 + 0;
  • 132 454 635 ÷ 2 = 66 227 317 + 1;
  • 66 227 317 ÷ 2 = 33 113 658 + 1;
  • 33 113 658 ÷ 2 = 16 556 829 + 0;
  • 16 556 829 ÷ 2 = 8 278 414 + 1;
  • 8 278 414 ÷ 2 = 4 139 207 + 0;
  • 4 139 207 ÷ 2 = 2 069 603 + 1;
  • 2 069 603 ÷ 2 = 1 034 801 + 1;
  • 1 034 801 ÷ 2 = 517 400 + 1;
  • 517 400 ÷ 2 = 258 700 + 0;
  • 258 700 ÷ 2 = 129 350 + 0;
  • 129 350 ÷ 2 = 64 675 + 0;
  • 64 675 ÷ 2 = 32 337 + 1;
  • 32 337 ÷ 2 = 16 168 + 1;
  • 16 168 ÷ 2 = 8 084 + 0;
  • 8 084 ÷ 2 = 4 042 + 0;
  • 4 042 ÷ 2 = 2 021 + 0;
  • 2 021 ÷ 2 = 1 010 + 1;
  • 1 010 ÷ 2 = 505 + 0;
  • 505 ÷ 2 = 252 + 1;
  • 252 ÷ 2 = 126 + 0;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 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.

1 111 110 011 111 051(10) = 11 1111 0010 1000 1100 0111 0101 1000 0100 1100 0110 1000 1011(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 50.

  • 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, 50,

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 1 111 110 011 111 051(10) converted to signed binary in one's complement representation:

1 111 110 011 111 051(10) = 0000 0000 0000 0011 1111 0010 1000 1100 0111 0101 1000 0100 1100 0110 1000 1011

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