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;
- 110 100 001 110 107 ÷ 2 = 55 050 000 555 053 + 1;
- 55 050 000 555 053 ÷ 2 = 27 525 000 277 526 + 1;
- 27 525 000 277 526 ÷ 2 = 13 762 500 138 763 + 0;
- 13 762 500 138 763 ÷ 2 = 6 881 250 069 381 + 1;
- 6 881 250 069 381 ÷ 2 = 3 440 625 034 690 + 1;
- 3 440 625 034 690 ÷ 2 = 1 720 312 517 345 + 0;
- 1 720 312 517 345 ÷ 2 = 860 156 258 672 + 1;
- 860 156 258 672 ÷ 2 = 430 078 129 336 + 0;
- 430 078 129 336 ÷ 2 = 215 039 064 668 + 0;
- 215 039 064 668 ÷ 2 = 107 519 532 334 + 0;
- 107 519 532 334 ÷ 2 = 53 759 766 167 + 0;
- 53 759 766 167 ÷ 2 = 26 879 883 083 + 1;
- 26 879 883 083 ÷ 2 = 13 439 941 541 + 1;
- 13 439 941 541 ÷ 2 = 6 719 970 770 + 1;
- 6 719 970 770 ÷ 2 = 3 359 985 385 + 0;
- 3 359 985 385 ÷ 2 = 1 679 992 692 + 1;
- 1 679 992 692 ÷ 2 = 839 996 346 + 0;
- 839 996 346 ÷ 2 = 419 998 173 + 0;
- 419 998 173 ÷ 2 = 209 999 086 + 1;
- 209 999 086 ÷ 2 = 104 999 543 + 0;
- 104 999 543 ÷ 2 = 52 499 771 + 1;
- 52 499 771 ÷ 2 = 26 249 885 + 1;
- 26 249 885 ÷ 2 = 13 124 942 + 1;
- 13 124 942 ÷ 2 = 6 562 471 + 0;
- 6 562 471 ÷ 2 = 3 281 235 + 1;
- 3 281 235 ÷ 2 = 1 640 617 + 1;
- 1 640 617 ÷ 2 = 820 308 + 1;
- 820 308 ÷ 2 = 410 154 + 0;
- 410 154 ÷ 2 = 205 077 + 0;
- 205 077 ÷ 2 = 102 538 + 1;
- 102 538 ÷ 2 = 51 269 + 0;
- 51 269 ÷ 2 = 25 634 + 1;
- 25 634 ÷ 2 = 12 817 + 0;
- 12 817 ÷ 2 = 6 408 + 1;
- 6 408 ÷ 2 = 3 204 + 0;
- 3 204 ÷ 2 = 1 602 + 0;
- 1 602 ÷ 2 = 801 + 0;
- 801 ÷ 2 = 400 + 1;
- 400 ÷ 2 = 200 + 0;
- 200 ÷ 2 = 100 + 0;
- 100 ÷ 2 = 50 + 0;
- 50 ÷ 2 = 25 + 0;
- 25 ÷ 2 = 12 + 1;
- 12 ÷ 2 = 6 + 0;
- 6 ÷ 2 = 3 + 0;
- 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.
110 100 001 110 107(10) = 110 0100 0010 0010 1010 0111 0111 0100 1011 1000 0101 1011(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 47.
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, 47,
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 110 100 001 110 107(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
110 100 001 110 107(10) = 0000 0000 0000 0000 0110 0100 0010 0010 1010 0111 0111 0100 1011 1000 0101 1011
Spaces were used to group digits: for binary, by 4, for decimal, by 3.