Convert 1 010 111 111 115 to a signed binary in one's complement representation, from a base 10 decimal system signed integer number

How to convert a signed integer in decimal system (in base 10):
1 010 111 111 115(10)
to a signed binary one's complement representation

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;
  • 1 010 111 111 115 ÷ 2 = 505 055 555 557 + 1;
  • 505 055 555 557 ÷ 2 = 252 527 777 778 + 1;
  • 252 527 777 778 ÷ 2 = 126 263 888 889 + 0;
  • 126 263 888 889 ÷ 2 = 63 131 944 444 + 1;
  • 63 131 944 444 ÷ 2 = 31 565 972 222 + 0;
  • 31 565 972 222 ÷ 2 = 15 782 986 111 + 0;
  • 15 782 986 111 ÷ 2 = 7 891 493 055 + 1;
  • 7 891 493 055 ÷ 2 = 3 945 746 527 + 1;
  • 3 945 746 527 ÷ 2 = 1 972 873 263 + 1;
  • 1 972 873 263 ÷ 2 = 986 436 631 + 1;
  • 986 436 631 ÷ 2 = 493 218 315 + 1;
  • 493 218 315 ÷ 2 = 246 609 157 + 1;
  • 246 609 157 ÷ 2 = 123 304 578 + 1;
  • 123 304 578 ÷ 2 = 61 652 289 + 0;
  • 61 652 289 ÷ 2 = 30 826 144 + 1;
  • 30 826 144 ÷ 2 = 15 413 072 + 0;
  • 15 413 072 ÷ 2 = 7 706 536 + 0;
  • 7 706 536 ÷ 2 = 3 853 268 + 0;
  • 3 853 268 ÷ 2 = 1 926 634 + 0;
  • 1 926 634 ÷ 2 = 963 317 + 0;
  • 963 317 ÷ 2 = 481 658 + 1;
  • 481 658 ÷ 2 = 240 829 + 0;
  • 240 829 ÷ 2 = 120 414 + 1;
  • 120 414 ÷ 2 = 60 207 + 0;
  • 60 207 ÷ 2 = 30 103 + 1;
  • 30 103 ÷ 2 = 15 051 + 1;
  • 15 051 ÷ 2 = 7 525 + 1;
  • 7 525 ÷ 2 = 3 762 + 1;
  • 3 762 ÷ 2 = 1 881 + 0;
  • 1 881 ÷ 2 = 940 + 1;
  • 940 ÷ 2 = 470 + 0;
  • 470 ÷ 2 = 235 + 0;
  • 235 ÷ 2 = 117 + 1;
  • 117 ÷ 2 = 58 + 1;
  • 58 ÷ 2 = 29 + 0;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 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.

1 010 111 111 115(10) = 1110 1011 0010 1111 0101 0000 0101 1111 1100 1011(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 40.

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; ...

First bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.

The least number that is:


a power of 2


and is larger than the actual length, 40,


so that the first bit (leftmost) could be zero


(we deal with a positive number at this moment)


is: 64.


4. 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:

1 010 111 111 115(10) = 0000 0000 0000 0000 0000 0000 1110 1011 0010 1111 0101 0000 0101 1111 1100 1011


Conclusion:

Number 1 010 111 111 115, a signed integer, converted from decimal system (base 10) to a signed binary one's complement representation:

1 010 111 111 115(10) = 0000 0000 0000 0000 0000 0000 1110 1011 0010 1111 0101 0000 0101 1111 1100 1011

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


More operations of this kind:

1 010 111 111 114 = ? | 1 010 111 111 116 = ?


Convert signed integer numbers from the decimal system (base ten) to signed binary one's complement representation

How to convert a base 10 signed integer number to signed binary in one's complement representation:

1) Divide the positive version of number repeatedly by 2, keeping track of each remainder, till getting a quotient that is 0.

2) Construct the base 2 representation by taking the previously calculated remainders starting from the last remainder up to the first one, in that order.

3) Construct the positive binary computer representation so that the first bit is 0.

4) Only if the initial number is negative, switch all the bits from 0 to 1 and from 1 to 0 (reversing the digits).

Latest signed integer numbers converted from decimal system to signed binary in one's complement representation

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