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;
- 1 111 000 110 000 119 ÷ 2 = 555 500 055 000 059 + 1;
- 555 500 055 000 059 ÷ 2 = 277 750 027 500 029 + 1;
- 277 750 027 500 029 ÷ 2 = 138 875 013 750 014 + 1;
- 138 875 013 750 014 ÷ 2 = 69 437 506 875 007 + 0;
- 69 437 506 875 007 ÷ 2 = 34 718 753 437 503 + 1;
- 34 718 753 437 503 ÷ 2 = 17 359 376 718 751 + 1;
- 17 359 376 718 751 ÷ 2 = 8 679 688 359 375 + 1;
- 8 679 688 359 375 ÷ 2 = 4 339 844 179 687 + 1;
- 4 339 844 179 687 ÷ 2 = 2 169 922 089 843 + 1;
- 2 169 922 089 843 ÷ 2 = 1 084 961 044 921 + 1;
- 1 084 961 044 921 ÷ 2 = 542 480 522 460 + 1;
- 542 480 522 460 ÷ 2 = 271 240 261 230 + 0;
- 271 240 261 230 ÷ 2 = 135 620 130 615 + 0;
- 135 620 130 615 ÷ 2 = 67 810 065 307 + 1;
- 67 810 065 307 ÷ 2 = 33 905 032 653 + 1;
- 33 905 032 653 ÷ 2 = 16 952 516 326 + 1;
- 16 952 516 326 ÷ 2 = 8 476 258 163 + 0;
- 8 476 258 163 ÷ 2 = 4 238 129 081 + 1;
- 4 238 129 081 ÷ 2 = 2 119 064 540 + 1;
- 2 119 064 540 ÷ 2 = 1 059 532 270 + 0;
- 1 059 532 270 ÷ 2 = 529 766 135 + 0;
- 529 766 135 ÷ 2 = 264 883 067 + 1;
- 264 883 067 ÷ 2 = 132 441 533 + 1;
- 132 441 533 ÷ 2 = 66 220 766 + 1;
- 66 220 766 ÷ 2 = 33 110 383 + 0;
- 33 110 383 ÷ 2 = 16 555 191 + 1;
- 16 555 191 ÷ 2 = 8 277 595 + 1;
- 8 277 595 ÷ 2 = 4 138 797 + 1;
- 4 138 797 ÷ 2 = 2 069 398 + 1;
- 2 069 398 ÷ 2 = 1 034 699 + 0;
- 1 034 699 ÷ 2 = 517 349 + 1;
- 517 349 ÷ 2 = 258 674 + 1;
- 258 674 ÷ 2 = 129 337 + 0;
- 129 337 ÷ 2 = 64 668 + 1;
- 64 668 ÷ 2 = 32 334 + 0;
- 32 334 ÷ 2 = 16 167 + 0;
- 16 167 ÷ 2 = 8 083 + 1;
- 8 083 ÷ 2 = 4 041 + 1;
- 4 041 ÷ 2 = 2 020 + 1;
- 2 020 ÷ 2 = 1 010 + 0;
- 1 010 ÷ 2 = 505 + 0;
- 505 ÷ 2 = 252 + 1;
- 252 ÷ 2 = 126 + 0;
- 126 ÷ 2 = 63 + 0;
- 63 ÷ 2 = 31 + 1;
- 31 ÷ 2 = 15 + 1;
- 15 ÷ 2 = 7 + 1;
- 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.
1 111 000 110 000 119(10) = 11 1111 0010 0111 0010 1101 1110 1110 0110 1110 0111 1111 0111(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 1 111 000 110 000 119(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
1 111 000 110 000 119(10) = 0000 0000 0000 0011 1111 0010 0111 0010 1101 1110 1110 0110 1110 0111 1111 0111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.