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;
- 1 000 100 010 010 047 ÷ 2 = 500 050 005 005 023 + 1;
- 500 050 005 005 023 ÷ 2 = 250 025 002 502 511 + 1;
- 250 025 002 502 511 ÷ 2 = 125 012 501 251 255 + 1;
- 125 012 501 251 255 ÷ 2 = 62 506 250 625 627 + 1;
- 62 506 250 625 627 ÷ 2 = 31 253 125 312 813 + 1;
- 31 253 125 312 813 ÷ 2 = 15 626 562 656 406 + 1;
- 15 626 562 656 406 ÷ 2 = 7 813 281 328 203 + 0;
- 7 813 281 328 203 ÷ 2 = 3 906 640 664 101 + 1;
- 3 906 640 664 101 ÷ 2 = 1 953 320 332 050 + 1;
- 1 953 320 332 050 ÷ 2 = 976 660 166 025 + 0;
- 976 660 166 025 ÷ 2 = 488 330 083 012 + 1;
- 488 330 083 012 ÷ 2 = 244 165 041 506 + 0;
- 244 165 041 506 ÷ 2 = 122 082 520 753 + 0;
- 122 082 520 753 ÷ 2 = 61 041 260 376 + 1;
- 61 041 260 376 ÷ 2 = 30 520 630 188 + 0;
- 30 520 630 188 ÷ 2 = 15 260 315 094 + 0;
- 15 260 315 094 ÷ 2 = 7 630 157 547 + 0;
- 7 630 157 547 ÷ 2 = 3 815 078 773 + 1;
- 3 815 078 773 ÷ 2 = 1 907 539 386 + 1;
- 1 907 539 386 ÷ 2 = 953 769 693 + 0;
- 953 769 693 ÷ 2 = 476 884 846 + 1;
- 476 884 846 ÷ 2 = 238 442 423 + 0;
- 238 442 423 ÷ 2 = 119 221 211 + 1;
- 119 221 211 ÷ 2 = 59 610 605 + 1;
- 59 610 605 ÷ 2 = 29 805 302 + 1;
- 29 805 302 ÷ 2 = 14 902 651 + 0;
- 14 902 651 ÷ 2 = 7 451 325 + 1;
- 7 451 325 ÷ 2 = 3 725 662 + 1;
- 3 725 662 ÷ 2 = 1 862 831 + 0;
- 1 862 831 ÷ 2 = 931 415 + 1;
- 931 415 ÷ 2 = 465 707 + 1;
- 465 707 ÷ 2 = 232 853 + 1;
- 232 853 ÷ 2 = 116 426 + 1;
- 116 426 ÷ 2 = 58 213 + 0;
- 58 213 ÷ 2 = 29 106 + 1;
- 29 106 ÷ 2 = 14 553 + 0;
- 14 553 ÷ 2 = 7 276 + 1;
- 7 276 ÷ 2 = 3 638 + 0;
- 3 638 ÷ 2 = 1 819 + 0;
- 1 819 ÷ 2 = 909 + 1;
- 909 ÷ 2 = 454 + 1;
- 454 ÷ 2 = 227 + 0;
- 227 ÷ 2 = 113 + 1;
- 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;
2. Construct the base 2 representation of the positive number:
Take all the remainders starting from the bottom of the list constructed above.
1 000 100 010 010 047(10) = 11 1000 1101 1001 0101 1110 1101 1101 0110 0010 0101 1011 1111(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 50.
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, 50,
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 1 000 100 010 010 047(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
1 000 100 010 010 047(10) = 0000 0000 0000 0011 1000 1101 1001 0101 1110 1101 1101 0110 0010 0101 1011 1111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.