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;
- 1 010 111 111 219 ÷ 2 = 505 055 555 609 + 1;
- 505 055 555 609 ÷ 2 = 252 527 777 804 + 1;
- 252 527 777 804 ÷ 2 = 126 263 888 902 + 0;
- 126 263 888 902 ÷ 2 = 63 131 944 451 + 0;
- 63 131 944 451 ÷ 2 = 31 565 972 225 + 1;
- 31 565 972 225 ÷ 2 = 15 782 986 112 + 1;
- 15 782 986 112 ÷ 2 = 7 891 493 056 + 0;
- 7 891 493 056 ÷ 2 = 3 945 746 528 + 0;
- 3 945 746 528 ÷ 2 = 1 972 873 264 + 0;
- 1 972 873 264 ÷ 2 = 986 436 632 + 0;
- 986 436 632 ÷ 2 = 493 218 316 + 0;
- 493 218 316 ÷ 2 = 246 609 158 + 0;
- 246 609 158 ÷ 2 = 123 304 579 + 0;
- 123 304 579 ÷ 2 = 61 652 289 + 1;
- 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.
1 010 111 111 219(10) = 1110 1011 0010 1111 0101 0000 0110 0000 0011 0011(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; ...
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, 40,
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.
Number 1 010 111 111 219(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in one's complement representation:
1 010 111 111 219(10) = 0000 0000 0000 0000 0000 0000 1110 1011 0010 1111 0101 0000 0110 0000 0011 0011
Spaces were used to group digits: for binary, by 4, for decimal, by 3.