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;
- 314 159 265 350 ÷ 2 = 157 079 632 675 + 0;
- 157 079 632 675 ÷ 2 = 78 539 816 337 + 1;
- 78 539 816 337 ÷ 2 = 39 269 908 168 + 1;
- 39 269 908 168 ÷ 2 = 19 634 954 084 + 0;
- 19 634 954 084 ÷ 2 = 9 817 477 042 + 0;
- 9 817 477 042 ÷ 2 = 4 908 738 521 + 0;
- 4 908 738 521 ÷ 2 = 2 454 369 260 + 1;
- 2 454 369 260 ÷ 2 = 1 227 184 630 + 0;
- 1 227 184 630 ÷ 2 = 613 592 315 + 0;
- 613 592 315 ÷ 2 = 306 796 157 + 1;
- 306 796 157 ÷ 2 = 153 398 078 + 1;
- 153 398 078 ÷ 2 = 76 699 039 + 0;
- 76 699 039 ÷ 2 = 38 349 519 + 1;
- 38 349 519 ÷ 2 = 19 174 759 + 1;
- 19 174 759 ÷ 2 = 9 587 379 + 1;
- 9 587 379 ÷ 2 = 4 793 689 + 1;
- 4 793 689 ÷ 2 = 2 396 844 + 1;
- 2 396 844 ÷ 2 = 1 198 422 + 0;
- 1 198 422 ÷ 2 = 599 211 + 0;
- 599 211 ÷ 2 = 299 605 + 1;
- 299 605 ÷ 2 = 149 802 + 1;
- 149 802 ÷ 2 = 74 901 + 0;
- 74 901 ÷ 2 = 37 450 + 1;
- 37 450 ÷ 2 = 18 725 + 0;
- 18 725 ÷ 2 = 9 362 + 1;
- 9 362 ÷ 2 = 4 681 + 0;
- 4 681 ÷ 2 = 2 340 + 1;
- 2 340 ÷ 2 = 1 170 + 0;
- 1 170 ÷ 2 = 585 + 0;
- 585 ÷ 2 = 292 + 1;
- 292 ÷ 2 = 146 + 0;
- 146 ÷ 2 = 73 + 0;
- 73 ÷ 2 = 36 + 1;
- 36 ÷ 2 = 18 + 0;
- 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.
314 159 265 350(10) = 100 1001 0010 0101 0101 1001 1111 0110 0100 0110(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 39.
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, 39,
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 314 159 265 350(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
314 159 265 350(10) = 0000 0000 0000 0000 0000 0000 0100 1001 0010 0101 0101 1001 1111 0110 0100 0110
Spaces were used to group digits: for binary, by 4, for decimal, by 3.