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;
- 1 111 111 111 111 208 ÷ 2 = 555 555 555 555 604 + 0;
- 555 555 555 555 604 ÷ 2 = 277 777 777 777 802 + 0;
- 277 777 777 777 802 ÷ 2 = 138 888 888 888 901 + 0;
- 138 888 888 888 901 ÷ 2 = 69 444 444 444 450 + 1;
- 69 444 444 444 450 ÷ 2 = 34 722 222 222 225 + 0;
- 34 722 222 222 225 ÷ 2 = 17 361 111 111 112 + 1;
- 17 361 111 111 112 ÷ 2 = 8 680 555 555 556 + 0;
- 8 680 555 555 556 ÷ 2 = 4 340 277 777 778 + 0;
- 4 340 277 777 778 ÷ 2 = 2 170 138 888 889 + 0;
- 2 170 138 888 889 ÷ 2 = 1 085 069 444 444 + 1;
- 1 085 069 444 444 ÷ 2 = 542 534 722 222 + 0;
- 542 534 722 222 ÷ 2 = 271 267 361 111 + 0;
- 271 267 361 111 ÷ 2 = 135 633 680 555 + 1;
- 135 633 680 555 ÷ 2 = 67 816 840 277 + 1;
- 67 816 840 277 ÷ 2 = 33 908 420 138 + 1;
- 33 908 420 138 ÷ 2 = 16 954 210 069 + 0;
- 16 954 210 069 ÷ 2 = 8 477 105 034 + 1;
- 8 477 105 034 ÷ 2 = 4 238 552 517 + 0;
- 4 238 552 517 ÷ 2 = 2 119 276 258 + 1;
- 2 119 276 258 ÷ 2 = 1 059 638 129 + 0;
- 1 059 638 129 ÷ 2 = 529 819 064 + 1;
- 529 819 064 ÷ 2 = 264 909 532 + 0;
- 264 909 532 ÷ 2 = 132 454 766 + 0;
- 132 454 766 ÷ 2 = 66 227 383 + 0;
- 66 227 383 ÷ 2 = 33 113 691 + 1;
- 33 113 691 ÷ 2 = 16 556 845 + 1;
- 16 556 845 ÷ 2 = 8 278 422 + 1;
- 8 278 422 ÷ 2 = 4 139 211 + 0;
- 4 139 211 ÷ 2 = 2 069 605 + 1;
- 2 069 605 ÷ 2 = 1 034 802 + 1;
- 1 034 802 ÷ 2 = 517 401 + 0;
- 517 401 ÷ 2 = 258 700 + 1;
- 258 700 ÷ 2 = 129 350 + 0;
- 129 350 ÷ 2 = 64 675 + 0;
- 64 675 ÷ 2 = 32 337 + 1;
- 32 337 ÷ 2 = 16 168 + 1;
- 16 168 ÷ 2 = 8 084 + 0;
- 8 084 ÷ 2 = 4 042 + 0;
- 4 042 ÷ 2 = 2 021 + 0;
- 2 021 ÷ 2 = 1 010 + 1;
- 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.
1 111 111 111 111 208(10) = 11 1111 0010 1000 1100 1011 0111 0001 0101 0111 0010 0010 1000(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) 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, 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 1 111 111 111 111 208(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in one's complement representation:
1 111 111 111 111 208(10) = 0000 0000 0000 0011 1111 0010 1000 1100 1011 0111 0001 0101 0111 0010 0010 1000
Spaces were used to group digits: for binary, by 4, for decimal, by 3.