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;
- 1 111 110 011 109 937 ÷ 2 = 555 555 005 554 968 + 1;
- 555 555 005 554 968 ÷ 2 = 277 777 502 777 484 + 0;
- 277 777 502 777 484 ÷ 2 = 138 888 751 388 742 + 0;
- 138 888 751 388 742 ÷ 2 = 69 444 375 694 371 + 0;
- 69 444 375 694 371 ÷ 2 = 34 722 187 847 185 + 1;
- 34 722 187 847 185 ÷ 2 = 17 361 093 923 592 + 1;
- 17 361 093 923 592 ÷ 2 = 8 680 546 961 796 + 0;
- 8 680 546 961 796 ÷ 2 = 4 340 273 480 898 + 0;
- 4 340 273 480 898 ÷ 2 = 2 170 136 740 449 + 0;
- 2 170 136 740 449 ÷ 2 = 1 085 068 370 224 + 1;
- 1 085 068 370 224 ÷ 2 = 542 534 185 112 + 0;
- 542 534 185 112 ÷ 2 = 271 267 092 556 + 0;
- 271 267 092 556 ÷ 2 = 135 633 546 278 + 0;
- 135 633 546 278 ÷ 2 = 67 816 773 139 + 0;
- 67 816 773 139 ÷ 2 = 33 908 386 569 + 1;
- 33 908 386 569 ÷ 2 = 16 954 193 284 + 1;
- 16 954 193 284 ÷ 2 = 8 477 096 642 + 0;
- 8 477 096 642 ÷ 2 = 4 238 548 321 + 0;
- 4 238 548 321 ÷ 2 = 2 119 274 160 + 1;
- 2 119 274 160 ÷ 2 = 1 059 637 080 + 0;
- 1 059 637 080 ÷ 2 = 529 818 540 + 0;
- 529 818 540 ÷ 2 = 264 909 270 + 0;
- 264 909 270 ÷ 2 = 132 454 635 + 0;
- 132 454 635 ÷ 2 = 66 227 317 + 1;
- 66 227 317 ÷ 2 = 33 113 658 + 1;
- 33 113 658 ÷ 2 = 16 556 829 + 0;
- 16 556 829 ÷ 2 = 8 278 414 + 1;
- 8 278 414 ÷ 2 = 4 139 207 + 0;
- 4 139 207 ÷ 2 = 2 069 603 + 1;
- 2 069 603 ÷ 2 = 1 034 801 + 1;
- 1 034 801 ÷ 2 = 517 400 + 1;
- 517 400 ÷ 2 = 258 700 + 0;
- 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.
1 111 110 011 109 937(10) = 11 1111 0010 1000 1100 0111 0101 1000 0100 1100 0010 0011 0001(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 1 111 110 011 109 937(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:
1 111 110 011 109 937(10) = 0000 0000 0000 0011 1111 0010 1000 1100 0111 0101 1000 0100 1100 0010 0011 0001
Spaces were used to group digits: for binary, by 4, for decimal, by 3.