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;
- 10 101 010 100 101 104 ÷ 2 = 5 050 505 050 050 552 + 0;
- 5 050 505 050 050 552 ÷ 2 = 2 525 252 525 025 276 + 0;
- 2 525 252 525 025 276 ÷ 2 = 1 262 626 262 512 638 + 0;
- 1 262 626 262 512 638 ÷ 2 = 631 313 131 256 319 + 0;
- 631 313 131 256 319 ÷ 2 = 315 656 565 628 159 + 1;
- 315 656 565 628 159 ÷ 2 = 157 828 282 814 079 + 1;
- 157 828 282 814 079 ÷ 2 = 78 914 141 407 039 + 1;
- 78 914 141 407 039 ÷ 2 = 39 457 070 703 519 + 1;
- 39 457 070 703 519 ÷ 2 = 19 728 535 351 759 + 1;
- 19 728 535 351 759 ÷ 2 = 9 864 267 675 879 + 1;
- 9 864 267 675 879 ÷ 2 = 4 932 133 837 939 + 1;
- 4 932 133 837 939 ÷ 2 = 2 466 066 918 969 + 1;
- 2 466 066 918 969 ÷ 2 = 1 233 033 459 484 + 1;
- 1 233 033 459 484 ÷ 2 = 616 516 729 742 + 0;
- 616 516 729 742 ÷ 2 = 308 258 364 871 + 0;
- 308 258 364 871 ÷ 2 = 154 129 182 435 + 1;
- 154 129 182 435 ÷ 2 = 77 064 591 217 + 1;
- 77 064 591 217 ÷ 2 = 38 532 295 608 + 1;
- 38 532 295 608 ÷ 2 = 19 266 147 804 + 0;
- 19 266 147 804 ÷ 2 = 9 633 073 902 + 0;
- 9 633 073 902 ÷ 2 = 4 816 536 951 + 0;
- 4 816 536 951 ÷ 2 = 2 408 268 475 + 1;
- 2 408 268 475 ÷ 2 = 1 204 134 237 + 1;
- 1 204 134 237 ÷ 2 = 602 067 118 + 1;
- 602 067 118 ÷ 2 = 301 033 559 + 0;
- 301 033 559 ÷ 2 = 150 516 779 + 1;
- 150 516 779 ÷ 2 = 75 258 389 + 1;
- 75 258 389 ÷ 2 = 37 629 194 + 1;
- 37 629 194 ÷ 2 = 18 814 597 + 0;
- 18 814 597 ÷ 2 = 9 407 298 + 1;
- 9 407 298 ÷ 2 = 4 703 649 + 0;
- 4 703 649 ÷ 2 = 2 351 824 + 1;
- 2 351 824 ÷ 2 = 1 175 912 + 0;
- 1 175 912 ÷ 2 = 587 956 + 0;
- 587 956 ÷ 2 = 293 978 + 0;
- 293 978 ÷ 2 = 146 989 + 0;
- 146 989 ÷ 2 = 73 494 + 1;
- 73 494 ÷ 2 = 36 747 + 0;
- 36 747 ÷ 2 = 18 373 + 1;
- 18 373 ÷ 2 = 9 186 + 1;
- 9 186 ÷ 2 = 4 593 + 0;
- 4 593 ÷ 2 = 2 296 + 1;
- 2 296 ÷ 2 = 1 148 + 0;
- 1 148 ÷ 2 = 574 + 0;
- 574 ÷ 2 = 287 + 0;
- 287 ÷ 2 = 143 + 1;
- 143 ÷ 2 = 71 + 1;
- 71 ÷ 2 = 35 + 1;
- 35 ÷ 2 = 17 + 1;
- 17 ÷ 2 = 8 + 1;
- 8 ÷ 2 = 4 + 0;
- 4 ÷ 2 = 2 + 0;
- 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.
10 101 010 100 101 104(10) = 10 0011 1110 0010 1101 0000 1010 1110 1110 0011 1001 1111 1111 0000(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 54.
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, 54,
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 10 101 010 100 101 104(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
10 101 010 100 101 104(10) = 0000 0000 0010 0011 1110 0010 1101 0000 1010 1110 1110 0011 1001 1111 1111 0000
Spaces were used to group digits: for binary, by 4, for decimal, by 3.