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;
- 8 157 989 228 746 813 600 ÷ 2 = 4 078 994 614 373 406 800 + 0;
- 4 078 994 614 373 406 800 ÷ 2 = 2 039 497 307 186 703 400 + 0;
- 2 039 497 307 186 703 400 ÷ 2 = 1 019 748 653 593 351 700 + 0;
- 1 019 748 653 593 351 700 ÷ 2 = 509 874 326 796 675 850 + 0;
- 509 874 326 796 675 850 ÷ 2 = 254 937 163 398 337 925 + 0;
- 254 937 163 398 337 925 ÷ 2 = 127 468 581 699 168 962 + 1;
- 127 468 581 699 168 962 ÷ 2 = 63 734 290 849 584 481 + 0;
- 63 734 290 849 584 481 ÷ 2 = 31 867 145 424 792 240 + 1;
- 31 867 145 424 792 240 ÷ 2 = 15 933 572 712 396 120 + 0;
- 15 933 572 712 396 120 ÷ 2 = 7 966 786 356 198 060 + 0;
- 7 966 786 356 198 060 ÷ 2 = 3 983 393 178 099 030 + 0;
- 3 983 393 178 099 030 ÷ 2 = 1 991 696 589 049 515 + 0;
- 1 991 696 589 049 515 ÷ 2 = 995 848 294 524 757 + 1;
- 995 848 294 524 757 ÷ 2 = 497 924 147 262 378 + 1;
- 497 924 147 262 378 ÷ 2 = 248 962 073 631 189 + 0;
- 248 962 073 631 189 ÷ 2 = 124 481 036 815 594 + 1;
- 124 481 036 815 594 ÷ 2 = 62 240 518 407 797 + 0;
- 62 240 518 407 797 ÷ 2 = 31 120 259 203 898 + 1;
- 31 120 259 203 898 ÷ 2 = 15 560 129 601 949 + 0;
- 15 560 129 601 949 ÷ 2 = 7 780 064 800 974 + 1;
- 7 780 064 800 974 ÷ 2 = 3 890 032 400 487 + 0;
- 3 890 032 400 487 ÷ 2 = 1 945 016 200 243 + 1;
- 1 945 016 200 243 ÷ 2 = 972 508 100 121 + 1;
- 972 508 100 121 ÷ 2 = 486 254 050 060 + 1;
- 486 254 050 060 ÷ 2 = 243 127 025 030 + 0;
- 243 127 025 030 ÷ 2 = 121 563 512 515 + 0;
- 121 563 512 515 ÷ 2 = 60 781 756 257 + 1;
- 60 781 756 257 ÷ 2 = 30 390 878 128 + 1;
- 30 390 878 128 ÷ 2 = 15 195 439 064 + 0;
- 15 195 439 064 ÷ 2 = 7 597 719 532 + 0;
- 7 597 719 532 ÷ 2 = 3 798 859 766 + 0;
- 3 798 859 766 ÷ 2 = 1 899 429 883 + 0;
- 1 899 429 883 ÷ 2 = 949 714 941 + 1;
- 949 714 941 ÷ 2 = 474 857 470 + 1;
- 474 857 470 ÷ 2 = 237 428 735 + 0;
- 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.
8 157 989 228 746 813 600(10) = 111 0001 0011 0110 1111 1111 1111 1011 0000 1100 1110 1010 1011 0000 1010 0000(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:
8 157 989 228 746 813 600(10) = 0111 0001 0011 0110 1111 1111 1111 1011 0000 1100 1110 1010 1011 0000 1010 0000
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 -8 157 989 228 746 813 600(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
-8 157 989 228 746 813 600(10) = 1111 0001 0011 0110 1111 1111 1111 1011 0000 1100 1110 1010 1011 0000 1010 0000
Spaces were used to group digits: for binary, by 4, for decimal, by 3.