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;
- 1 001 000 010 101 080 ÷ 2 = 500 500 005 050 540 + 0;
- 500 500 005 050 540 ÷ 2 = 250 250 002 525 270 + 0;
- 250 250 002 525 270 ÷ 2 = 125 125 001 262 635 + 0;
- 125 125 001 262 635 ÷ 2 = 62 562 500 631 317 + 1;
- 62 562 500 631 317 ÷ 2 = 31 281 250 315 658 + 1;
- 31 281 250 315 658 ÷ 2 = 15 640 625 157 829 + 0;
- 15 640 625 157 829 ÷ 2 = 7 820 312 578 914 + 1;
- 7 820 312 578 914 ÷ 2 = 3 910 156 289 457 + 0;
- 3 910 156 289 457 ÷ 2 = 1 955 078 144 728 + 1;
- 1 955 078 144 728 ÷ 2 = 977 539 072 364 + 0;
- 977 539 072 364 ÷ 2 = 488 769 536 182 + 0;
- 488 769 536 182 ÷ 2 = 244 384 768 091 + 0;
- 244 384 768 091 ÷ 2 = 122 192 384 045 + 1;
- 122 192 384 045 ÷ 2 = 61 096 192 022 + 1;
- 61 096 192 022 ÷ 2 = 30 548 096 011 + 0;
- 30 548 096 011 ÷ 2 = 15 274 048 005 + 1;
- 15 274 048 005 ÷ 2 = 7 637 024 002 + 1;
- 7 637 024 002 ÷ 2 = 3 818 512 001 + 0;
- 3 818 512 001 ÷ 2 = 1 909 256 000 + 1;
- 1 909 256 000 ÷ 2 = 954 628 000 + 0;
- 954 628 000 ÷ 2 = 477 314 000 + 0;
- 477 314 000 ÷ 2 = 238 657 000 + 0;
- 238 657 000 ÷ 2 = 119 328 500 + 0;
- 119 328 500 ÷ 2 = 59 664 250 + 0;
- 59 664 250 ÷ 2 = 29 832 125 + 0;
- 29 832 125 ÷ 2 = 14 916 062 + 1;
- 14 916 062 ÷ 2 = 7 458 031 + 0;
- 7 458 031 ÷ 2 = 3 729 015 + 1;
- 3 729 015 ÷ 2 = 1 864 507 + 1;
- 1 864 507 ÷ 2 = 932 253 + 1;
- 932 253 ÷ 2 = 466 126 + 1;
- 466 126 ÷ 2 = 233 063 + 0;
- 233 063 ÷ 2 = 116 531 + 1;
- 116 531 ÷ 2 = 58 265 + 1;
- 58 265 ÷ 2 = 29 132 + 1;
- 29 132 ÷ 2 = 14 566 + 0;
- 14 566 ÷ 2 = 7 283 + 0;
- 7 283 ÷ 2 = 3 641 + 1;
- 3 641 ÷ 2 = 1 820 + 1;
- 1 820 ÷ 2 = 910 + 0;
- 910 ÷ 2 = 455 + 0;
- 455 ÷ 2 = 227 + 1;
- 227 ÷ 2 = 113 + 1;
- 113 ÷ 2 = 56 + 1;
- 56 ÷ 2 = 28 + 0;
- 28 ÷ 2 = 14 + 0;
- 14 ÷ 2 = 7 + 0;
- 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.
1 001 000 010 101 080(10) = 11 1000 1110 0110 0111 0111 1010 0000 0101 1011 0001 0101 1000(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 1 001 000 010 101 080(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:
1 001 000 010 101 080(10) = 0000 0000 0000 0011 1000 1110 0110 0111 0111 1010 0000 0101 1011 0001 0101 1000
Spaces were used to group digits: for binary, by 4, for decimal, by 3.