Base ten decimal system signed integer number 111 111 111 111 111 converted to signed binary in one's complement representation

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

1. Divide the number repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:

  • division = quotient + remainder;
  • 111 111 111 111 111 ÷ 2 = 55 555 555 555 555 + 1;
  • 55 555 555 555 555 ÷ 2 = 27 777 777 777 777 + 1;
  • 27 777 777 777 777 ÷ 2 = 13 888 888 888 888 + 1;
  • 13 888 888 888 888 ÷ 2 = 6 944 444 444 444 + 0;
  • 6 944 444 444 444 ÷ 2 = 3 472 222 222 222 + 0;
  • 3 472 222 222 222 ÷ 2 = 1 736 111 111 111 + 0;
  • 1 736 111 111 111 ÷ 2 = 868 055 555 555 + 1;
  • 868 055 555 555 ÷ 2 = 434 027 777 777 + 1;
  • 434 027 777 777 ÷ 2 = 217 013 888 888 + 1;
  • 217 013 888 888 ÷ 2 = 108 506 944 444 + 0;
  • 108 506 944 444 ÷ 2 = 54 253 472 222 + 0;
  • 54 253 472 222 ÷ 2 = 27 126 736 111 + 0;
  • 27 126 736 111 ÷ 2 = 13 563 368 055 + 1;
  • 13 563 368 055 ÷ 2 = 6 781 684 027 + 1;
  • 6 781 684 027 ÷ 2 = 3 390 842 013 + 1;
  • 3 390 842 013 ÷ 2 = 1 695 421 006 + 1;
  • 1 695 421 006 ÷ 2 = 847 710 503 + 0;
  • 847 710 503 ÷ 2 = 423 855 251 + 1;
  • 423 855 251 ÷ 2 = 211 927 625 + 1;
  • 211 927 625 ÷ 2 = 105 963 812 + 1;
  • 105 963 812 ÷ 2 = 52 981 906 + 0;
  • 52 981 906 ÷ 2 = 26 490 953 + 0;
  • 26 490 953 ÷ 2 = 13 245 476 + 1;
  • 13 245 476 ÷ 2 = 6 622 738 + 0;
  • 6 622 738 ÷ 2 = 3 311 369 + 0;
  • 3 311 369 ÷ 2 = 1 655 684 + 1;
  • 1 655 684 ÷ 2 = 827 842 + 0;
  • 827 842 ÷ 2 = 413 921 + 0;
  • 413 921 ÷ 2 = 206 960 + 1;
  • 206 960 ÷ 2 = 103 480 + 0;
  • 103 480 ÷ 2 = 51 740 + 0;
  • 51 740 ÷ 2 = 25 870 + 0;
  • 25 870 ÷ 2 = 12 935 + 0;
  • 12 935 ÷ 2 = 6 467 + 1;
  • 6 467 ÷ 2 = 3 233 + 1;
  • 3 233 ÷ 2 = 1 616 + 1;
  • 1 616 ÷ 2 = 808 + 0;
  • 808 ÷ 2 = 404 + 0;
  • 404 ÷ 2 = 202 + 0;
  • 202 ÷ 2 = 101 + 0;
  • 101 ÷ 2 = 50 + 1;
  • 50 ÷ 2 = 25 + 0;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above:

111 111 111 111 111(10) = 110 0101 0000 1110 0001 0010 0100 1110 1111 0001 1100 0111(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; ...

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 so that the first bit (leftmost) could be zero 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:

111 111 111 111 111(10) = 0000 0000 0000 0000 0110 0101 0000 1110 0001 0010 0100 1110 1111 0001 1100 0111

Conclusion:

Number 111 111 111 111 111, a signed integer, converted from decimal system (base 10) to a signed binary one's complement representation:
111 111 111 111 111(10) = 0000 0000 0000 0000 0110 0101 0000 1110 0001 0010 0100 1110 1111 0001 1100 0111

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

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

How to convert a base ten 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 we get a quotient that is zero.

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 zero.

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

Latest signed integers 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