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;
- 1 101 110 111 100 085 ÷ 2 = 550 555 055 550 042 + 1;
- 550 555 055 550 042 ÷ 2 = 275 277 527 775 021 + 0;
- 275 277 527 775 021 ÷ 2 = 137 638 763 887 510 + 1;
- 137 638 763 887 510 ÷ 2 = 68 819 381 943 755 + 0;
- 68 819 381 943 755 ÷ 2 = 34 409 690 971 877 + 1;
- 34 409 690 971 877 ÷ 2 = 17 204 845 485 938 + 1;
- 17 204 845 485 938 ÷ 2 = 8 602 422 742 969 + 0;
- 8 602 422 742 969 ÷ 2 = 4 301 211 371 484 + 1;
- 4 301 211 371 484 ÷ 2 = 2 150 605 685 742 + 0;
- 2 150 605 685 742 ÷ 2 = 1 075 302 842 871 + 0;
- 1 075 302 842 871 ÷ 2 = 537 651 421 435 + 1;
- 537 651 421 435 ÷ 2 = 268 825 710 717 + 1;
- 268 825 710 717 ÷ 2 = 134 412 855 358 + 1;
- 134 412 855 358 ÷ 2 = 67 206 427 679 + 0;
- 67 206 427 679 ÷ 2 = 33 603 213 839 + 1;
- 33 603 213 839 ÷ 2 = 16 801 606 919 + 1;
- 16 801 606 919 ÷ 2 = 8 400 803 459 + 1;
- 8 400 803 459 ÷ 2 = 4 200 401 729 + 1;
- 4 200 401 729 ÷ 2 = 2 100 200 864 + 1;
- 2 100 200 864 ÷ 2 = 1 050 100 432 + 0;
- 1 050 100 432 ÷ 2 = 525 050 216 + 0;
- 525 050 216 ÷ 2 = 262 525 108 + 0;
- 262 525 108 ÷ 2 = 131 262 554 + 0;
- 131 262 554 ÷ 2 = 65 631 277 + 0;
- 65 631 277 ÷ 2 = 32 815 638 + 1;
- 32 815 638 ÷ 2 = 16 407 819 + 0;
- 16 407 819 ÷ 2 = 8 203 909 + 1;
- 8 203 909 ÷ 2 = 4 101 954 + 1;
- 4 101 954 ÷ 2 = 2 050 977 + 0;
- 2 050 977 ÷ 2 = 1 025 488 + 1;
- 1 025 488 ÷ 2 = 512 744 + 0;
- 512 744 ÷ 2 = 256 372 + 0;
- 256 372 ÷ 2 = 128 186 + 0;
- 128 186 ÷ 2 = 64 093 + 0;
- 64 093 ÷ 2 = 32 046 + 1;
- 32 046 ÷ 2 = 16 023 + 0;
- 16 023 ÷ 2 = 8 011 + 1;
- 8 011 ÷ 2 = 4 005 + 1;
- 4 005 ÷ 2 = 2 002 + 1;
- 2 002 ÷ 2 = 1 001 + 0;
- 1 001 ÷ 2 = 500 + 1;
- 500 ÷ 2 = 250 + 0;
- 250 ÷ 2 = 125 + 0;
- 125 ÷ 2 = 62 + 1;
- 62 ÷ 2 = 31 + 0;
- 31 ÷ 2 = 15 + 1;
- 15 ÷ 2 = 7 + 1;
- 7 ÷ 2 = 3 + 1;
- 3 ÷ 2 = 1 + 1;
- 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.
1 101 110 111 100 085(10) = 11 1110 1001 0111 0100 0010 1101 0000 0111 1101 1100 1011 0101(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 50.
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) indicates 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, 50,
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 1 101 110 111 100 085(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in two's complement representation:
1 101 110 111 100 085(10) = 0000 0000 0000 0011 1110 1001 0111 0100 0010 1101 0000 0111 1101 1100 1011 0101
Spaces were used to group digits: for binary, by 4, for decimal, by 3.