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;
- 84 688 897 866 054 ÷ 2 = 42 344 448 933 027 + 0;
- 42 344 448 933 027 ÷ 2 = 21 172 224 466 513 + 1;
- 21 172 224 466 513 ÷ 2 = 10 586 112 233 256 + 1;
- 10 586 112 233 256 ÷ 2 = 5 293 056 116 628 + 0;
- 5 293 056 116 628 ÷ 2 = 2 646 528 058 314 + 0;
- 2 646 528 058 314 ÷ 2 = 1 323 264 029 157 + 0;
- 1 323 264 029 157 ÷ 2 = 661 632 014 578 + 1;
- 661 632 014 578 ÷ 2 = 330 816 007 289 + 0;
- 330 816 007 289 ÷ 2 = 165 408 003 644 + 1;
- 165 408 003 644 ÷ 2 = 82 704 001 822 + 0;
- 82 704 001 822 ÷ 2 = 41 352 000 911 + 0;
- 41 352 000 911 ÷ 2 = 20 676 000 455 + 1;
- 20 676 000 455 ÷ 2 = 10 338 000 227 + 1;
- 10 338 000 227 ÷ 2 = 5 169 000 113 + 1;
- 5 169 000 113 ÷ 2 = 2 584 500 056 + 1;
- 2 584 500 056 ÷ 2 = 1 292 250 028 + 0;
- 1 292 250 028 ÷ 2 = 646 125 014 + 0;
- 646 125 014 ÷ 2 = 323 062 507 + 0;
- 323 062 507 ÷ 2 = 161 531 253 + 1;
- 161 531 253 ÷ 2 = 80 765 626 + 1;
- 80 765 626 ÷ 2 = 40 382 813 + 0;
- 40 382 813 ÷ 2 = 20 191 406 + 1;
- 20 191 406 ÷ 2 = 10 095 703 + 0;
- 10 095 703 ÷ 2 = 5 047 851 + 1;
- 5 047 851 ÷ 2 = 2 523 925 + 1;
- 2 523 925 ÷ 2 = 1 261 962 + 1;
- 1 261 962 ÷ 2 = 630 981 + 0;
- 630 981 ÷ 2 = 315 490 + 1;
- 315 490 ÷ 2 = 157 745 + 0;
- 157 745 ÷ 2 = 78 872 + 1;
- 78 872 ÷ 2 = 39 436 + 0;
- 39 436 ÷ 2 = 19 718 + 0;
- 19 718 ÷ 2 = 9 859 + 0;
- 9 859 ÷ 2 = 4 929 + 1;
- 4 929 ÷ 2 = 2 464 + 1;
- 2 464 ÷ 2 = 1 232 + 0;
- 1 232 ÷ 2 = 616 + 0;
- 616 ÷ 2 = 308 + 0;
- 308 ÷ 2 = 154 + 0;
- 154 ÷ 2 = 77 + 0;
- 77 ÷ 2 = 38 + 1;
- 38 ÷ 2 = 19 + 0;
- 19 ÷ 2 = 9 + 1;
- 9 ÷ 2 = 4 + 1;
- 4 ÷ 2 = 2 + 0;
- 2 ÷ 2 = 1 + 0;
- 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.
84 688 897 866 054(10) = 100 1101 0000 0110 0010 1011 1010 1100 0111 1001 0100 0110(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 47.
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, 47,
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 84 688 897 866 054(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:
84 688 897 866 054(10) = 0000 0000 0000 0000 0100 1101 0000 0110 0010 1011 1010 1100 0111 1001 0100 0110
Spaces were used to group digits: for binary, by 4, for decimal, by 3.