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;
- 2 991 006 890 006 229 ÷ 2 = 1 495 503 445 003 114 + 1;
- 1 495 503 445 003 114 ÷ 2 = 747 751 722 501 557 + 0;
- 747 751 722 501 557 ÷ 2 = 373 875 861 250 778 + 1;
- 373 875 861 250 778 ÷ 2 = 186 937 930 625 389 + 0;
- 186 937 930 625 389 ÷ 2 = 93 468 965 312 694 + 1;
- 93 468 965 312 694 ÷ 2 = 46 734 482 656 347 + 0;
- 46 734 482 656 347 ÷ 2 = 23 367 241 328 173 + 1;
- 23 367 241 328 173 ÷ 2 = 11 683 620 664 086 + 1;
- 11 683 620 664 086 ÷ 2 = 5 841 810 332 043 + 0;
- 5 841 810 332 043 ÷ 2 = 2 920 905 166 021 + 1;
- 2 920 905 166 021 ÷ 2 = 1 460 452 583 010 + 1;
- 1 460 452 583 010 ÷ 2 = 730 226 291 505 + 0;
- 730 226 291 505 ÷ 2 = 365 113 145 752 + 1;
- 365 113 145 752 ÷ 2 = 182 556 572 876 + 0;
- 182 556 572 876 ÷ 2 = 91 278 286 438 + 0;
- 91 278 286 438 ÷ 2 = 45 639 143 219 + 0;
- 45 639 143 219 ÷ 2 = 22 819 571 609 + 1;
- 22 819 571 609 ÷ 2 = 11 409 785 804 + 1;
- 11 409 785 804 ÷ 2 = 5 704 892 902 + 0;
- 5 704 892 902 ÷ 2 = 2 852 446 451 + 0;
- 2 852 446 451 ÷ 2 = 1 426 223 225 + 1;
- 1 426 223 225 ÷ 2 = 713 111 612 + 1;
- 713 111 612 ÷ 2 = 356 555 806 + 0;
- 356 555 806 ÷ 2 = 178 277 903 + 0;
- 178 277 903 ÷ 2 = 89 138 951 + 1;
- 89 138 951 ÷ 2 = 44 569 475 + 1;
- 44 569 475 ÷ 2 = 22 284 737 + 1;
- 22 284 737 ÷ 2 = 11 142 368 + 1;
- 11 142 368 ÷ 2 = 5 571 184 + 0;
- 5 571 184 ÷ 2 = 2 785 592 + 0;
- 2 785 592 ÷ 2 = 1 392 796 + 0;
- 1 392 796 ÷ 2 = 696 398 + 0;
- 696 398 ÷ 2 = 348 199 + 0;
- 348 199 ÷ 2 = 174 099 + 1;
- 174 099 ÷ 2 = 87 049 + 1;
- 87 049 ÷ 2 = 43 524 + 1;
- 43 524 ÷ 2 = 21 762 + 0;
- 21 762 ÷ 2 = 10 881 + 0;
- 10 881 ÷ 2 = 5 440 + 1;
- 5 440 ÷ 2 = 2 720 + 0;
- 2 720 ÷ 2 = 1 360 + 0;
- 1 360 ÷ 2 = 680 + 0;
- 680 ÷ 2 = 340 + 0;
- 340 ÷ 2 = 170 + 0;
- 170 ÷ 2 = 85 + 0;
- 85 ÷ 2 = 42 + 1;
- 42 ÷ 2 = 21 + 0;
- 21 ÷ 2 = 10 + 1;
- 10 ÷ 2 = 5 + 0;
- 5 ÷ 2 = 2 + 1;
- 2 ÷ 2 = 1 + 0;
- 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.
2 991 006 890 006 229(10) = 1010 1010 0000 0100 1110 0000 1111 0011 0011 0001 0110 1101 0101(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 52.
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, 52,
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:
2 991 006 890 006 229(10) = 0000 0000 0000 1010 1010 0000 0100 1110 0000 1111 0011 0011 0001 0110 1101 0101
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 -2 991 006 890 006 229(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
-2 991 006 890 006 229(10) = 1000 0000 0000 1010 1010 0000 0100 1110 0000 1111 0011 0011 0001 0110 1101 0101
Spaces were used to group digits: for binary, by 4, for decimal, by 3.