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;
- 1 000 000 001 110 070 ÷ 2 = 500 000 000 555 035 + 0;
- 500 000 000 555 035 ÷ 2 = 250 000 000 277 517 + 1;
- 250 000 000 277 517 ÷ 2 = 125 000 000 138 758 + 1;
- 125 000 000 138 758 ÷ 2 = 62 500 000 069 379 + 0;
- 62 500 000 069 379 ÷ 2 = 31 250 000 034 689 + 1;
- 31 250 000 034 689 ÷ 2 = 15 625 000 017 344 + 1;
- 15 625 000 017 344 ÷ 2 = 7 812 500 008 672 + 0;
- 7 812 500 008 672 ÷ 2 = 3 906 250 004 336 + 0;
- 3 906 250 004 336 ÷ 2 = 1 953 125 002 168 + 0;
- 1 953 125 002 168 ÷ 2 = 976 562 501 084 + 0;
- 976 562 501 084 ÷ 2 = 488 281 250 542 + 0;
- 488 281 250 542 ÷ 2 = 244 140 625 271 + 0;
- 244 140 625 271 ÷ 2 = 122 070 312 635 + 1;
- 122 070 312 635 ÷ 2 = 61 035 156 317 + 1;
- 61 035 156 317 ÷ 2 = 30 517 578 158 + 1;
- 30 517 578 158 ÷ 2 = 15 258 789 079 + 0;
- 15 258 789 079 ÷ 2 = 7 629 394 539 + 1;
- 7 629 394 539 ÷ 2 = 3 814 697 269 + 1;
- 3 814 697 269 ÷ 2 = 1 907 348 634 + 1;
- 1 907 348 634 ÷ 2 = 953 674 317 + 0;
- 953 674 317 ÷ 2 = 476 837 158 + 1;
- 476 837 158 ÷ 2 = 238 418 579 + 0;
- 238 418 579 ÷ 2 = 119 209 289 + 1;
- 119 209 289 ÷ 2 = 59 604 644 + 1;
- 59 604 644 ÷ 2 = 29 802 322 + 0;
- 29 802 322 ÷ 2 = 14 901 161 + 0;
- 14 901 161 ÷ 2 = 7 450 580 + 1;
- 7 450 580 ÷ 2 = 3 725 290 + 0;
- 3 725 290 ÷ 2 = 1 862 645 + 0;
- 1 862 645 ÷ 2 = 931 322 + 1;
- 931 322 ÷ 2 = 465 661 + 0;
- 465 661 ÷ 2 = 232 830 + 1;
- 232 830 ÷ 2 = 116 415 + 0;
- 116 415 ÷ 2 = 58 207 + 1;
- 58 207 ÷ 2 = 29 103 + 1;
- 29 103 ÷ 2 = 14 551 + 1;
- 14 551 ÷ 2 = 7 275 + 1;
- 7 275 ÷ 2 = 3 637 + 1;
- 3 637 ÷ 2 = 1 818 + 1;
- 1 818 ÷ 2 = 909 + 0;
- 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.
1 000 000 001 110 070(10) = 11 1000 1101 0111 1110 1010 0100 1101 0111 0111 0000 0011 0110(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 1 000 000 001 110 070(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
1 000 000 001 110 070(10) = 0000 0000 0000 0011 1000 1101 0111 1110 1010 0100 1101 0111 0111 0000 0011 0110
Spaces were used to group digits: for binary, by 4, for decimal, by 3.