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;
- 100 010 001 000 027 ÷ 2 = 50 005 000 500 013 + 1;
- 50 005 000 500 013 ÷ 2 = 25 002 500 250 006 + 1;
- 25 002 500 250 006 ÷ 2 = 12 501 250 125 003 + 0;
- 12 501 250 125 003 ÷ 2 = 6 250 625 062 501 + 1;
- 6 250 625 062 501 ÷ 2 = 3 125 312 531 250 + 1;
- 3 125 312 531 250 ÷ 2 = 1 562 656 265 625 + 0;
- 1 562 656 265 625 ÷ 2 = 781 328 132 812 + 1;
- 781 328 132 812 ÷ 2 = 390 664 066 406 + 0;
- 390 664 066 406 ÷ 2 = 195 332 033 203 + 0;
- 195 332 033 203 ÷ 2 = 97 666 016 601 + 1;
- 97 666 016 601 ÷ 2 = 48 833 008 300 + 1;
- 48 833 008 300 ÷ 2 = 24 416 504 150 + 0;
- 24 416 504 150 ÷ 2 = 12 208 252 075 + 0;
- 12 208 252 075 ÷ 2 = 6 104 126 037 + 1;
- 6 104 126 037 ÷ 2 = 3 052 063 018 + 1;
- 3 052 063 018 ÷ 2 = 1 526 031 509 + 0;
- 1 526 031 509 ÷ 2 = 763 015 754 + 1;
- 763 015 754 ÷ 2 = 381 507 877 + 0;
- 381 507 877 ÷ 2 = 190 753 938 + 1;
- 190 753 938 ÷ 2 = 95 376 969 + 0;
- 95 376 969 ÷ 2 = 47 688 484 + 1;
- 47 688 484 ÷ 2 = 23 844 242 + 0;
- 23 844 242 ÷ 2 = 11 922 121 + 0;
- 11 922 121 ÷ 2 = 5 961 060 + 1;
- 5 961 060 ÷ 2 = 2 980 530 + 0;
- 2 980 530 ÷ 2 = 1 490 265 + 0;
- 1 490 265 ÷ 2 = 745 132 + 1;
- 745 132 ÷ 2 = 372 566 + 0;
- 372 566 ÷ 2 = 186 283 + 0;
- 186 283 ÷ 2 = 93 141 + 1;
- 93 141 ÷ 2 = 46 570 + 1;
- 46 570 ÷ 2 = 23 285 + 0;
- 23 285 ÷ 2 = 11 642 + 1;
- 11 642 ÷ 2 = 5 821 + 0;
- 5 821 ÷ 2 = 2 910 + 1;
- 2 910 ÷ 2 = 1 455 + 0;
- 1 455 ÷ 2 = 727 + 1;
- 727 ÷ 2 = 363 + 1;
- 363 ÷ 2 = 181 + 1;
- 181 ÷ 2 = 90 + 1;
- 90 ÷ 2 = 45 + 0;
- 45 ÷ 2 = 22 + 1;
- 22 ÷ 2 = 11 + 0;
- 11 ÷ 2 = 5 + 1;
- 5 ÷ 2 = 2 + 1;
- 2 ÷ 2 = 1 + 0;
- 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.
100 010 001 000 027(10) = 101 1010 1111 0101 0110 0100 1001 0101 0110 0110 0101 1011(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; ...
The first bit (the leftmost) is reserved for 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, 47,
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 100 010 001 000 027(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
100 010 001 000 027(10) = 0000 0000 0000 0000 0101 1010 1111 0101 0110 0100 1001 0101 0110 0110 0101 1011
Spaces were used to group digits: for binary, by 4, for decimal, by 3.