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;
- 478 868 574 082 066 864 ÷ 2 = 239 434 287 041 033 432 + 0;
- 239 434 287 041 033 432 ÷ 2 = 119 717 143 520 516 716 + 0;
- 119 717 143 520 516 716 ÷ 2 = 59 858 571 760 258 358 + 0;
- 59 858 571 760 258 358 ÷ 2 = 29 929 285 880 129 179 + 0;
- 29 929 285 880 129 179 ÷ 2 = 14 964 642 940 064 589 + 1;
- 14 964 642 940 064 589 ÷ 2 = 7 482 321 470 032 294 + 1;
- 7 482 321 470 032 294 ÷ 2 = 3 741 160 735 016 147 + 0;
- 3 741 160 735 016 147 ÷ 2 = 1 870 580 367 508 073 + 1;
- 1 870 580 367 508 073 ÷ 2 = 935 290 183 754 036 + 1;
- 935 290 183 754 036 ÷ 2 = 467 645 091 877 018 + 0;
- 467 645 091 877 018 ÷ 2 = 233 822 545 938 509 + 0;
- 233 822 545 938 509 ÷ 2 = 116 911 272 969 254 + 1;
- 116 911 272 969 254 ÷ 2 = 58 455 636 484 627 + 0;
- 58 455 636 484 627 ÷ 2 = 29 227 818 242 313 + 1;
- 29 227 818 242 313 ÷ 2 = 14 613 909 121 156 + 1;
- 14 613 909 121 156 ÷ 2 = 7 306 954 560 578 + 0;
- 7 306 954 560 578 ÷ 2 = 3 653 477 280 289 + 0;
- 3 653 477 280 289 ÷ 2 = 1 826 738 640 144 + 1;
- 1 826 738 640 144 ÷ 2 = 913 369 320 072 + 0;
- 913 369 320 072 ÷ 2 = 456 684 660 036 + 0;
- 456 684 660 036 ÷ 2 = 228 342 330 018 + 0;
- 228 342 330 018 ÷ 2 = 114 171 165 009 + 0;
- 114 171 165 009 ÷ 2 = 57 085 582 504 + 1;
- 57 085 582 504 ÷ 2 = 28 542 791 252 + 0;
- 28 542 791 252 ÷ 2 = 14 271 395 626 + 0;
- 14 271 395 626 ÷ 2 = 7 135 697 813 + 0;
- 7 135 697 813 ÷ 2 = 3 567 848 906 + 1;
- 3 567 848 906 ÷ 2 = 1 783 924 453 + 0;
- 1 783 924 453 ÷ 2 = 891 962 226 + 1;
- 891 962 226 ÷ 2 = 445 981 113 + 0;
- 445 981 113 ÷ 2 = 222 990 556 + 1;
- 222 990 556 ÷ 2 = 111 495 278 + 0;
- 111 495 278 ÷ 2 = 55 747 639 + 0;
- 55 747 639 ÷ 2 = 27 873 819 + 1;
- 27 873 819 ÷ 2 = 13 936 909 + 1;
- 13 936 909 ÷ 2 = 6 968 454 + 1;
- 6 968 454 ÷ 2 = 3 484 227 + 0;
- 3 484 227 ÷ 2 = 1 742 113 + 1;
- 1 742 113 ÷ 2 = 871 056 + 1;
- 871 056 ÷ 2 = 435 528 + 0;
- 435 528 ÷ 2 = 217 764 + 0;
- 217 764 ÷ 2 = 108 882 + 0;
- 108 882 ÷ 2 = 54 441 + 0;
- 54 441 ÷ 2 = 27 220 + 1;
- 27 220 ÷ 2 = 13 610 + 0;
- 13 610 ÷ 2 = 6 805 + 0;
- 6 805 ÷ 2 = 3 402 + 1;
- 3 402 ÷ 2 = 1 701 + 0;
- 1 701 ÷ 2 = 850 + 1;
- 850 ÷ 2 = 425 + 0;
- 425 ÷ 2 = 212 + 1;
- 212 ÷ 2 = 106 + 0;
- 106 ÷ 2 = 53 + 0;
- 53 ÷ 2 = 26 + 1;
- 26 ÷ 2 = 13 + 0;
- 13 ÷ 2 = 6 + 1;
- 6 ÷ 2 = 3 + 0;
- 3 ÷ 2 = 1 + 1;
- 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.
478 868 574 082 066 864(10) = 110 1010 0101 0100 1000 0110 1110 0101 0100 0100 0010 0110 1001 1011 0000(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 59.
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) indicates 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, 59,
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 478 868 574 082 066 864(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in two's complement representation:
478 868 574 082 066 864(10) = 0000 0110 1010 0101 0100 1000 0110 1110 0101 0100 0100 0010 0110 1001 1011 0000
Spaces were used to group digits: for binary, by 4, for decimal, by 3.