2. 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;
- 9 185 794 525 094 739 980 ÷ 2 = 4 592 897 262 547 369 990 + 0;
- 4 592 897 262 547 369 990 ÷ 2 = 2 296 448 631 273 684 995 + 0;
- 2 296 448 631 273 684 995 ÷ 2 = 1 148 224 315 636 842 497 + 1;
- 1 148 224 315 636 842 497 ÷ 2 = 574 112 157 818 421 248 + 1;
- 574 112 157 818 421 248 ÷ 2 = 287 056 078 909 210 624 + 0;
- 287 056 078 909 210 624 ÷ 2 = 143 528 039 454 605 312 + 0;
- 143 528 039 454 605 312 ÷ 2 = 71 764 019 727 302 656 + 0;
- 71 764 019 727 302 656 ÷ 2 = 35 882 009 863 651 328 + 0;
- 35 882 009 863 651 328 ÷ 2 = 17 941 004 931 825 664 + 0;
- 17 941 004 931 825 664 ÷ 2 = 8 970 502 465 912 832 + 0;
- 8 970 502 465 912 832 ÷ 2 = 4 485 251 232 956 416 + 0;
- 4 485 251 232 956 416 ÷ 2 = 2 242 625 616 478 208 + 0;
- 2 242 625 616 478 208 ÷ 2 = 1 121 312 808 239 104 + 0;
- 1 121 312 808 239 104 ÷ 2 = 560 656 404 119 552 + 0;
- 560 656 404 119 552 ÷ 2 = 280 328 202 059 776 + 0;
- 280 328 202 059 776 ÷ 2 = 140 164 101 029 888 + 0;
- 140 164 101 029 888 ÷ 2 = 70 082 050 514 944 + 0;
- 70 082 050 514 944 ÷ 2 = 35 041 025 257 472 + 0;
- 35 041 025 257 472 ÷ 2 = 17 520 512 628 736 + 0;
- 17 520 512 628 736 ÷ 2 = 8 760 256 314 368 + 0;
- 8 760 256 314 368 ÷ 2 = 4 380 128 157 184 + 0;
- 4 380 128 157 184 ÷ 2 = 2 190 064 078 592 + 0;
- 2 190 064 078 592 ÷ 2 = 1 095 032 039 296 + 0;
- 1 095 032 039 296 ÷ 2 = 547 516 019 648 + 0;
- 547 516 019 648 ÷ 2 = 273 758 009 824 + 0;
- 273 758 009 824 ÷ 2 = 136 879 004 912 + 0;
- 136 879 004 912 ÷ 2 = 68 439 502 456 + 0;
- 68 439 502 456 ÷ 2 = 34 219 751 228 + 0;
- 34 219 751 228 ÷ 2 = 17 109 875 614 + 0;
- 17 109 875 614 ÷ 2 = 8 554 937 807 + 0;
- 8 554 937 807 ÷ 2 = 4 277 468 903 + 1;
- 4 277 468 903 ÷ 2 = 2 138 734 451 + 1;
- 2 138 734 451 ÷ 2 = 1 069 367 225 + 1;
- 1 069 367 225 ÷ 2 = 534 683 612 + 1;
- 534 683 612 ÷ 2 = 267 341 806 + 0;
- 267 341 806 ÷ 2 = 133 670 903 + 0;
- 133 670 903 ÷ 2 = 66 835 451 + 1;
- 66 835 451 ÷ 2 = 33 417 725 + 1;
- 33 417 725 ÷ 2 = 16 708 862 + 1;
- 16 708 862 ÷ 2 = 8 354 431 + 0;
- 8 354 431 ÷ 2 = 4 177 215 + 1;
- 4 177 215 ÷ 2 = 2 088 607 + 1;
- 2 088 607 ÷ 2 = 1 044 303 + 1;
- 1 044 303 ÷ 2 = 522 151 + 1;
- 522 151 ÷ 2 = 261 075 + 1;
- 261 075 ÷ 2 = 130 537 + 1;
- 130 537 ÷ 2 = 65 268 + 1;
- 65 268 ÷ 2 = 32 634 + 0;
- 32 634 ÷ 2 = 16 317 + 0;
- 16 317 ÷ 2 = 8 158 + 1;
- 8 158 ÷ 2 = 4 079 + 0;
- 4 079 ÷ 2 = 2 039 + 1;
- 2 039 ÷ 2 = 1 019 + 1;
- 1 019 ÷ 2 = 509 + 1;
- 509 ÷ 2 = 254 + 1;
- 254 ÷ 2 = 127 + 0;
- 127 ÷ 2 = 63 + 1;
- 63 ÷ 2 = 31 + 1;
- 31 ÷ 2 = 15 + 1;
- 15 ÷ 2 = 7 + 1;
- 7 ÷ 2 = 3 + 1;
- 3 ÷ 2 = 1 + 1;
- 1 ÷ 2 = 0 + 1;
3. Construct the base 2 representation of the positive number:
Take all the remainders starting from the bottom of the list constructed above.
9 185 794 525 094 739 980(10) = 111 1111 0111 1010 0111 1111 0111 0011 1100 0000 0000 0000 0000 0000 0000 1100(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 63.
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, 63,
3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)
=== is: 64.
5. 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:
9 185 794 525 094 739 980(10) = 0111 1111 0111 1010 0111 1111 0111 0011 1100 0000 0000 0000 0000 0000 0000 1100
6. Get the negative integer number representation:
To get the negative integer number representation on 64 bits (8 Bytes),
... change the first bit (the leftmost), from 0 to 1...
Number -9 185 794 525 094 739 980(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
-9 185 794 525 094 739 980(10) = 1111 1111 0111 1010 0111 1111 0111 0011 1100 0000 0000 0000 0000 0000 0000 1100
Spaces were used to group digits: for binary, by 4, for decimal, by 3.