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;
- 100 000 010 066 ÷ 2 = 50 000 005 033 + 0;
- 50 000 005 033 ÷ 2 = 25 000 002 516 + 1;
- 25 000 002 516 ÷ 2 = 12 500 001 258 + 0;
- 12 500 001 258 ÷ 2 = 6 250 000 629 + 0;
- 6 250 000 629 ÷ 2 = 3 125 000 314 + 1;
- 3 125 000 314 ÷ 2 = 1 562 500 157 + 0;
- 1 562 500 157 ÷ 2 = 781 250 078 + 1;
- 781 250 078 ÷ 2 = 390 625 039 + 0;
- 390 625 039 ÷ 2 = 195 312 519 + 1;
- 195 312 519 ÷ 2 = 97 656 259 + 1;
- 97 656 259 ÷ 2 = 48 828 129 + 1;
- 48 828 129 ÷ 2 = 24 414 064 + 1;
- 24 414 064 ÷ 2 = 12 207 032 + 0;
- 12 207 032 ÷ 2 = 6 103 516 + 0;
- 6 103 516 ÷ 2 = 3 051 758 + 0;
- 3 051 758 ÷ 2 = 1 525 879 + 0;
- 1 525 879 ÷ 2 = 762 939 + 1;
- 762 939 ÷ 2 = 381 469 + 1;
- 381 469 ÷ 2 = 190 734 + 1;
- 190 734 ÷ 2 = 95 367 + 0;
- 95 367 ÷ 2 = 47 683 + 1;
- 47 683 ÷ 2 = 23 841 + 1;
- 23 841 ÷ 2 = 11 920 + 1;
- 11 920 ÷ 2 = 5 960 + 0;
- 5 960 ÷ 2 = 2 980 + 0;
- 2 980 ÷ 2 = 1 490 + 0;
- 1 490 ÷ 2 = 745 + 0;
- 745 ÷ 2 = 372 + 1;
- 372 ÷ 2 = 186 + 0;
- 186 ÷ 2 = 93 + 0;
- 93 ÷ 2 = 46 + 1;
- 46 ÷ 2 = 23 + 0;
- 23 ÷ 2 = 11 + 1;
- 11 ÷ 2 = 5 + 1;
- 5 ÷ 2 = 2 + 1;
- 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.
100 000 010 066(10) = 1 0111 0100 1000 0111 0111 0000 1111 0101 0010(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 37.
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, 37,
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 100 000 010 066(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
100 000 010 066(10) = 0000 0000 0000 0000 0000 0000 0001 0111 0100 1000 0111 0111 0000 1111 0101 0010
Spaces were used to group digits: for binary, by 4, for decimal, by 3.