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;
- 8 157 989 233 041 780 782 ÷ 2 = 4 078 994 616 520 890 391 + 0;
- 4 078 994 616 520 890 391 ÷ 2 = 2 039 497 308 260 445 195 + 1;
- 2 039 497 308 260 445 195 ÷ 2 = 1 019 748 654 130 222 597 + 1;
- 1 019 748 654 130 222 597 ÷ 2 = 509 874 327 065 111 298 + 1;
- 509 874 327 065 111 298 ÷ 2 = 254 937 163 532 555 649 + 0;
- 254 937 163 532 555 649 ÷ 2 = 127 468 581 766 277 824 + 1;
- 127 468 581 766 277 824 ÷ 2 = 63 734 290 883 138 912 + 0;
- 63 734 290 883 138 912 ÷ 2 = 31 867 145 441 569 456 + 0;
- 31 867 145 441 569 456 ÷ 2 = 15 933 572 720 784 728 + 0;
- 15 933 572 720 784 728 ÷ 2 = 7 966 786 360 392 364 + 0;
- 7 966 786 360 392 364 ÷ 2 = 3 983 393 180 196 182 + 0;
- 3 983 393 180 196 182 ÷ 2 = 1 991 696 590 098 091 + 0;
- 1 991 696 590 098 091 ÷ 2 = 995 848 295 049 045 + 1;
- 995 848 295 049 045 ÷ 2 = 497 924 147 524 522 + 1;
- 497 924 147 524 522 ÷ 2 = 248 962 073 762 261 + 0;
- 248 962 073 762 261 ÷ 2 = 124 481 036 881 130 + 1;
- 124 481 036 881 130 ÷ 2 = 62 240 518 440 565 + 0;
- 62 240 518 440 565 ÷ 2 = 31 120 259 220 282 + 1;
- 31 120 259 220 282 ÷ 2 = 15 560 129 610 141 + 0;
- 15 560 129 610 141 ÷ 2 = 7 780 064 805 070 + 1;
- 7 780 064 805 070 ÷ 2 = 3 890 032 402 535 + 0;
- 3 890 032 402 535 ÷ 2 = 1 945 016 201 267 + 1;
- 1 945 016 201 267 ÷ 2 = 972 508 100 633 + 1;
- 972 508 100 633 ÷ 2 = 486 254 050 316 + 1;
- 486 254 050 316 ÷ 2 = 243 127 025 158 + 0;
- 243 127 025 158 ÷ 2 = 121 563 512 579 + 0;
- 121 563 512 579 ÷ 2 = 60 781 756 289 + 1;
- 60 781 756 289 ÷ 2 = 30 390 878 144 + 1;
- 30 390 878 144 ÷ 2 = 15 195 439 072 + 0;
- 15 195 439 072 ÷ 2 = 7 597 719 536 + 0;
- 7 597 719 536 ÷ 2 = 3 798 859 768 + 0;
- 3 798 859 768 ÷ 2 = 1 899 429 884 + 0;
- 1 899 429 884 ÷ 2 = 949 714 942 + 0;
- 949 714 942 ÷ 2 = 474 857 471 + 0;
- 474 857 471 ÷ 2 = 237 428 735 + 1;
- 237 428 735 ÷ 2 = 118 714 367 + 1;
- 118 714 367 ÷ 2 = 59 357 183 + 1;
- 59 357 183 ÷ 2 = 29 678 591 + 1;
- 29 678 591 ÷ 2 = 14 839 295 + 1;
- 14 839 295 ÷ 2 = 7 419 647 + 1;
- 7 419 647 ÷ 2 = 3 709 823 + 1;
- 3 709 823 ÷ 2 = 1 854 911 + 1;
- 1 854 911 ÷ 2 = 927 455 + 1;
- 927 455 ÷ 2 = 463 727 + 1;
- 463 727 ÷ 2 = 231 863 + 1;
- 231 863 ÷ 2 = 115 931 + 1;
- 115 931 ÷ 2 = 57 965 + 1;
- 57 965 ÷ 2 = 28 982 + 1;
- 28 982 ÷ 2 = 14 491 + 0;
- 14 491 ÷ 2 = 7 245 + 1;
- 7 245 ÷ 2 = 3 622 + 1;
- 3 622 ÷ 2 = 1 811 + 0;
- 1 811 ÷ 2 = 905 + 1;
- 905 ÷ 2 = 452 + 1;
- 452 ÷ 2 = 226 + 0;
- 226 ÷ 2 = 113 + 0;
- 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;
3. Construct the base 2 representation of the positive number:
Take all the remainders starting from the bottom of the list constructed above.
8 157 989 233 041 780 782(10) = 111 0001 0011 0110 1111 1111 1111 1100 0000 1100 1110 1010 1011 0000 0010 1110(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:
8 157 989 233 041 780 782(10) = 0111 0001 0011 0110 1111 1111 1111 1100 0000 1100 1110 1010 1011 0000 0010 1110
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 -8 157 989 233 041 780 782(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
-8 157 989 233 041 780 782(10) = 1111 0001 0011 0110 1111 1111 1111 1100 0000 1100 1110 1010 1011 0000 0010 1110
Spaces were used to group digits: for binary, by 4, for decimal, by 3.