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;
- 2 500 000 015 ÷ 2 = 1 250 000 007 + 1;
- 1 250 000 007 ÷ 2 = 625 000 003 + 1;
- 625 000 003 ÷ 2 = 312 500 001 + 1;
- 312 500 001 ÷ 2 = 156 250 000 + 1;
- 156 250 000 ÷ 2 = 78 125 000 + 0;
- 78 125 000 ÷ 2 = 39 062 500 + 0;
- 39 062 500 ÷ 2 = 19 531 250 + 0;
- 19 531 250 ÷ 2 = 9 765 625 + 0;
- 9 765 625 ÷ 2 = 4 882 812 + 1;
- 4 882 812 ÷ 2 = 2 441 406 + 0;
- 2 441 406 ÷ 2 = 1 220 703 + 0;
- 1 220 703 ÷ 2 = 610 351 + 1;
- 610 351 ÷ 2 = 305 175 + 1;
- 305 175 ÷ 2 = 152 587 + 1;
- 152 587 ÷ 2 = 76 293 + 1;
- 76 293 ÷ 2 = 38 146 + 1;
- 38 146 ÷ 2 = 19 073 + 0;
- 19 073 ÷ 2 = 9 536 + 1;
- 9 536 ÷ 2 = 4 768 + 0;
- 4 768 ÷ 2 = 2 384 + 0;
- 2 384 ÷ 2 = 1 192 + 0;
- 1 192 ÷ 2 = 596 + 0;
- 596 ÷ 2 = 298 + 0;
- 298 ÷ 2 = 149 + 0;
- 149 ÷ 2 = 74 + 1;
- 74 ÷ 2 = 37 + 0;
- 37 ÷ 2 = 18 + 1;
- 18 ÷ 2 = 9 + 0;
- 9 ÷ 2 = 4 + 1;
- 4 ÷ 2 = 2 + 0;
- 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.
2 500 000 015(10) = 1001 0101 0000 0010 1111 1001 0000 1111(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 32.
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, 32,
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 2 500 000 015(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
2 500 000 015(10) = 0000 0000 0000 0000 0000 0000 0000 0000 1001 0101 0000 0010 1111 1001 0000 1111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.