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;
- 11 111 110 001 010 017 ÷ 2 = 5 555 555 000 505 008 + 1;
- 5 555 555 000 505 008 ÷ 2 = 2 777 777 500 252 504 + 0;
- 2 777 777 500 252 504 ÷ 2 = 1 388 888 750 126 252 + 0;
- 1 388 888 750 126 252 ÷ 2 = 694 444 375 063 126 + 0;
- 694 444 375 063 126 ÷ 2 = 347 222 187 531 563 + 0;
- 347 222 187 531 563 ÷ 2 = 173 611 093 765 781 + 1;
- 173 611 093 765 781 ÷ 2 = 86 805 546 882 890 + 1;
- 86 805 546 882 890 ÷ 2 = 43 402 773 441 445 + 0;
- 43 402 773 441 445 ÷ 2 = 21 701 386 720 722 + 1;
- 21 701 386 720 722 ÷ 2 = 10 850 693 360 361 + 0;
- 10 850 693 360 361 ÷ 2 = 5 425 346 680 180 + 1;
- 5 425 346 680 180 ÷ 2 = 2 712 673 340 090 + 0;
- 2 712 673 340 090 ÷ 2 = 1 356 336 670 045 + 0;
- 1 356 336 670 045 ÷ 2 = 678 168 335 022 + 1;
- 678 168 335 022 ÷ 2 = 339 084 167 511 + 0;
- 339 084 167 511 ÷ 2 = 169 542 083 755 + 1;
- 169 542 083 755 ÷ 2 = 84 771 041 877 + 1;
- 84 771 041 877 ÷ 2 = 42 385 520 938 + 1;
- 42 385 520 938 ÷ 2 = 21 192 760 469 + 0;
- 21 192 760 469 ÷ 2 = 10 596 380 234 + 1;
- 10 596 380 234 ÷ 2 = 5 298 190 117 + 0;
- 5 298 190 117 ÷ 2 = 2 649 095 058 + 1;
- 2 649 095 058 ÷ 2 = 1 324 547 529 + 0;
- 1 324 547 529 ÷ 2 = 662 273 764 + 1;
- 662 273 764 ÷ 2 = 331 136 882 + 0;
- 331 136 882 ÷ 2 = 165 568 441 + 0;
- 165 568 441 ÷ 2 = 82 784 220 + 1;
- 82 784 220 ÷ 2 = 41 392 110 + 0;
- 41 392 110 ÷ 2 = 20 696 055 + 0;
- 20 696 055 ÷ 2 = 10 348 027 + 1;
- 10 348 027 ÷ 2 = 5 174 013 + 1;
- 5 174 013 ÷ 2 = 2 587 006 + 1;
- 2 587 006 ÷ 2 = 1 293 503 + 0;
- 1 293 503 ÷ 2 = 646 751 + 1;
- 646 751 ÷ 2 = 323 375 + 1;
- 323 375 ÷ 2 = 161 687 + 1;
- 161 687 ÷ 2 = 80 843 + 1;
- 80 843 ÷ 2 = 40 421 + 1;
- 40 421 ÷ 2 = 20 210 + 1;
- 20 210 ÷ 2 = 10 105 + 0;
- 10 105 ÷ 2 = 5 052 + 1;
- 5 052 ÷ 2 = 2 526 + 0;
- 2 526 ÷ 2 = 1 263 + 0;
- 1 263 ÷ 2 = 631 + 1;
- 631 ÷ 2 = 315 + 1;
- 315 ÷ 2 = 157 + 1;
- 157 ÷ 2 = 78 + 1;
- 78 ÷ 2 = 39 + 0;
- 39 ÷ 2 = 19 + 1;
- 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.
11 111 110 001 010 017(10) = 10 0111 0111 1001 0111 1110 1110 0100 1010 1011 1010 0101 0110 0001(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 54.
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, 54,
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 11 111 110 001 010 017(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
11 111 110 001 010 017(10) = 0000 0000 0010 0111 0111 1001 0111 1110 1110 0100 1010 1011 1010 0101 0110 0001
Spaces were used to group digits: for binary, by 4, for decimal, by 3.