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;
- 101 111 100 002 ÷ 2 = 50 555 550 001 + 0;
- 50 555 550 001 ÷ 2 = 25 277 775 000 + 1;
- 25 277 775 000 ÷ 2 = 12 638 887 500 + 0;
- 12 638 887 500 ÷ 2 = 6 319 443 750 + 0;
- 6 319 443 750 ÷ 2 = 3 159 721 875 + 0;
- 3 159 721 875 ÷ 2 = 1 579 860 937 + 1;
- 1 579 860 937 ÷ 2 = 789 930 468 + 1;
- 789 930 468 ÷ 2 = 394 965 234 + 0;
- 394 965 234 ÷ 2 = 197 482 617 + 0;
- 197 482 617 ÷ 2 = 98 741 308 + 1;
- 98 741 308 ÷ 2 = 49 370 654 + 0;
- 49 370 654 ÷ 2 = 24 685 327 + 0;
- 24 685 327 ÷ 2 = 12 342 663 + 1;
- 12 342 663 ÷ 2 = 6 171 331 + 1;
- 6 171 331 ÷ 2 = 3 085 665 + 1;
- 3 085 665 ÷ 2 = 1 542 832 + 1;
- 1 542 832 ÷ 2 = 771 416 + 0;
- 771 416 ÷ 2 = 385 708 + 0;
- 385 708 ÷ 2 = 192 854 + 0;
- 192 854 ÷ 2 = 96 427 + 0;
- 96 427 ÷ 2 = 48 213 + 1;
- 48 213 ÷ 2 = 24 106 + 1;
- 24 106 ÷ 2 = 12 053 + 0;
- 12 053 ÷ 2 = 6 026 + 1;
- 6 026 ÷ 2 = 3 013 + 0;
- 3 013 ÷ 2 = 1 506 + 1;
- 1 506 ÷ 2 = 753 + 0;
- 753 ÷ 2 = 376 + 1;
- 376 ÷ 2 = 188 + 0;
- 188 ÷ 2 = 94 + 0;
- 94 ÷ 2 = 47 + 0;
- 47 ÷ 2 = 23 + 1;
- 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.
101 111 100 002(10) = 1 0111 1000 1010 1011 0000 1111 0010 0110 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 101 111 100 002(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
101 111 100 002(10) = 0000 0000 0000 0000 0000 0000 0001 0111 1000 1010 1011 0000 1111 0010 0110 0010
Spaces were used to group digits: for binary, by 4, for decimal, by 3.