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;
- 100 001 100 100 001 ÷ 2 = 50 000 550 050 000 + 1;
- 50 000 550 050 000 ÷ 2 = 25 000 275 025 000 + 0;
- 25 000 275 025 000 ÷ 2 = 12 500 137 512 500 + 0;
- 12 500 137 512 500 ÷ 2 = 6 250 068 756 250 + 0;
- 6 250 068 756 250 ÷ 2 = 3 125 034 378 125 + 0;
- 3 125 034 378 125 ÷ 2 = 1 562 517 189 062 + 1;
- 1 562 517 189 062 ÷ 2 = 781 258 594 531 + 0;
- 781 258 594 531 ÷ 2 = 390 629 297 265 + 1;
- 390 629 297 265 ÷ 2 = 195 314 648 632 + 1;
- 195 314 648 632 ÷ 2 = 97 657 324 316 + 0;
- 97 657 324 316 ÷ 2 = 48 828 662 158 + 0;
- 48 828 662 158 ÷ 2 = 24 414 331 079 + 0;
- 24 414 331 079 ÷ 2 = 12 207 165 539 + 1;
- 12 207 165 539 ÷ 2 = 6 103 582 769 + 1;
- 6 103 582 769 ÷ 2 = 3 051 791 384 + 1;
- 3 051 791 384 ÷ 2 = 1 525 895 692 + 0;
- 1 525 895 692 ÷ 2 = 762 947 846 + 0;
- 762 947 846 ÷ 2 = 381 473 923 + 0;
- 381 473 923 ÷ 2 = 190 736 961 + 1;
- 190 736 961 ÷ 2 = 95 368 480 + 1;
- 95 368 480 ÷ 2 = 47 684 240 + 0;
- 47 684 240 ÷ 2 = 23 842 120 + 0;
- 23 842 120 ÷ 2 = 11 921 060 + 0;
- 11 921 060 ÷ 2 = 5 960 530 + 0;
- 5 960 530 ÷ 2 = 2 980 265 + 0;
- 2 980 265 ÷ 2 = 1 490 132 + 1;
- 1 490 132 ÷ 2 = 745 066 + 0;
- 745 066 ÷ 2 = 372 533 + 0;
- 372 533 ÷ 2 = 186 266 + 1;
- 186 266 ÷ 2 = 93 133 + 0;
- 93 133 ÷ 2 = 46 566 + 1;
- 46 566 ÷ 2 = 23 283 + 0;
- 23 283 ÷ 2 = 11 641 + 1;
- 11 641 ÷ 2 = 5 820 + 1;
- 5 820 ÷ 2 = 2 910 + 0;
- 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.
100 001 100 100 001(10) = 101 1010 1111 0011 0101 0010 0000 1100 0111 0001 1010 0001(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 100 001 100 100 001(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
100 001 100 100 001(10) = 0000 0000 0000 0000 0101 1010 1111 0011 0101 0010 0000 1100 0111 0001 1010 0001
Spaces were used to group digits: for binary, by 4, for decimal, by 3.