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;
- 11 111 010 000 025 ÷ 2 = 5 555 505 000 012 + 1;
- 5 555 505 000 012 ÷ 2 = 2 777 752 500 006 + 0;
- 2 777 752 500 006 ÷ 2 = 1 388 876 250 003 + 0;
- 1 388 876 250 003 ÷ 2 = 694 438 125 001 + 1;
- 694 438 125 001 ÷ 2 = 347 219 062 500 + 1;
- 347 219 062 500 ÷ 2 = 173 609 531 250 + 0;
- 173 609 531 250 ÷ 2 = 86 804 765 625 + 0;
- 86 804 765 625 ÷ 2 = 43 402 382 812 + 1;
- 43 402 382 812 ÷ 2 = 21 701 191 406 + 0;
- 21 701 191 406 ÷ 2 = 10 850 595 703 + 0;
- 10 850 595 703 ÷ 2 = 5 425 297 851 + 1;
- 5 425 297 851 ÷ 2 = 2 712 648 925 + 1;
- 2 712 648 925 ÷ 2 = 1 356 324 462 + 1;
- 1 356 324 462 ÷ 2 = 678 162 231 + 0;
- 678 162 231 ÷ 2 = 339 081 115 + 1;
- 339 081 115 ÷ 2 = 169 540 557 + 1;
- 169 540 557 ÷ 2 = 84 770 278 + 1;
- 84 770 278 ÷ 2 = 42 385 139 + 0;
- 42 385 139 ÷ 2 = 21 192 569 + 1;
- 21 192 569 ÷ 2 = 10 596 284 + 1;
- 10 596 284 ÷ 2 = 5 298 142 + 0;
- 5 298 142 ÷ 2 = 2 649 071 + 0;
- 2 649 071 ÷ 2 = 1 324 535 + 1;
- 1 324 535 ÷ 2 = 662 267 + 1;
- 662 267 ÷ 2 = 331 133 + 1;
- 331 133 ÷ 2 = 165 566 + 1;
- 165 566 ÷ 2 = 82 783 + 0;
- 82 783 ÷ 2 = 41 391 + 1;
- 41 391 ÷ 2 = 20 695 + 1;
- 20 695 ÷ 2 = 10 347 + 1;
- 10 347 ÷ 2 = 5 173 + 1;
- 5 173 ÷ 2 = 2 586 + 1;
- 2 586 ÷ 2 = 1 293 + 0;
- 1 293 ÷ 2 = 646 + 1;
- 646 ÷ 2 = 323 + 0;
- 323 ÷ 2 = 161 + 1;
- 161 ÷ 2 = 80 + 1;
- 80 ÷ 2 = 40 + 0;
- 40 ÷ 2 = 20 + 0;
- 20 ÷ 2 = 10 + 0;
- 10 ÷ 2 = 5 + 0;
- 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.
11 111 010 000 025(10) = 1010 0001 1010 1111 1011 1100 1101 1101 1100 1001 1001(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 44.
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, 44,
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 11 111 010 000 025(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
11 111 010 000 025(10) = 0000 0000 0000 0000 0000 1010 0001 1010 1111 1011 1100 1101 1101 1100 1001 1001
Spaces were used to group digits: for binary, by 4, for decimal, by 3.