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;
- 1 000 100 100 109 989 ÷ 2 = 500 050 050 054 994 + 1;
- 500 050 050 054 994 ÷ 2 = 250 025 025 027 497 + 0;
- 250 025 025 027 497 ÷ 2 = 125 012 512 513 748 + 1;
- 125 012 512 513 748 ÷ 2 = 62 506 256 256 874 + 0;
- 62 506 256 256 874 ÷ 2 = 31 253 128 128 437 + 0;
- 31 253 128 128 437 ÷ 2 = 15 626 564 064 218 + 1;
- 15 626 564 064 218 ÷ 2 = 7 813 282 032 109 + 0;
- 7 813 282 032 109 ÷ 2 = 3 906 641 016 054 + 1;
- 3 906 641 016 054 ÷ 2 = 1 953 320 508 027 + 0;
- 1 953 320 508 027 ÷ 2 = 976 660 254 013 + 1;
- 976 660 254 013 ÷ 2 = 488 330 127 006 + 1;
- 488 330 127 006 ÷ 2 = 244 165 063 503 + 0;
- 244 165 063 503 ÷ 2 = 122 082 531 751 + 1;
- 122 082 531 751 ÷ 2 = 61 041 265 875 + 1;
- 61 041 265 875 ÷ 2 = 30 520 632 937 + 1;
- 30 520 632 937 ÷ 2 = 15 260 316 468 + 1;
- 15 260 316 468 ÷ 2 = 7 630 158 234 + 0;
- 7 630 158 234 ÷ 2 = 3 815 079 117 + 0;
- 3 815 079 117 ÷ 2 = 1 907 539 558 + 1;
- 1 907 539 558 ÷ 2 = 953 769 779 + 0;
- 953 769 779 ÷ 2 = 476 884 889 + 1;
- 476 884 889 ÷ 2 = 238 442 444 + 1;
- 238 442 444 ÷ 2 = 119 221 222 + 0;
- 119 221 222 ÷ 2 = 59 610 611 + 0;
- 59 610 611 ÷ 2 = 29 805 305 + 1;
- 29 805 305 ÷ 2 = 14 902 652 + 1;
- 14 902 652 ÷ 2 = 7 451 326 + 0;
- 7 451 326 ÷ 2 = 3 725 663 + 0;
- 3 725 663 ÷ 2 = 1 862 831 + 1;
- 1 862 831 ÷ 2 = 931 415 + 1;
- 931 415 ÷ 2 = 465 707 + 1;
- 465 707 ÷ 2 = 232 853 + 1;
- 232 853 ÷ 2 = 116 426 + 1;
- 116 426 ÷ 2 = 58 213 + 0;
- 58 213 ÷ 2 = 29 106 + 1;
- 29 106 ÷ 2 = 14 553 + 0;
- 14 553 ÷ 2 = 7 276 + 1;
- 7 276 ÷ 2 = 3 638 + 0;
- 3 638 ÷ 2 = 1 819 + 0;
- 1 819 ÷ 2 = 909 + 1;
- 909 ÷ 2 = 454 + 1;
- 454 ÷ 2 = 227 + 0;
- 227 ÷ 2 = 113 + 1;
- 113 ÷ 2 = 56 + 1;
- 56 ÷ 2 = 28 + 0;
- 28 ÷ 2 = 14 + 0;
- 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.
1 000 100 100 109 989(10) = 11 1000 1101 1001 0101 1111 0011 0011 0100 1111 0110 1010 0101(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.
Number 1 000 100 100 109 989(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:
1 000 100 100 109 989(10) = 0000 0000 0000 0011 1000 1101 1001 0101 1111 0011 0011 0100 1111 0110 1010 0101
Spaces were used to group digits: for binary, by 4, for decimal, by 3.