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;
- 899 999 999 993 ÷ 2 = 449 999 999 996 + 1;
- 449 999 999 996 ÷ 2 = 224 999 999 998 + 0;
- 224 999 999 998 ÷ 2 = 112 499 999 999 + 0;
- 112 499 999 999 ÷ 2 = 56 249 999 999 + 1;
- 56 249 999 999 ÷ 2 = 28 124 999 999 + 1;
- 28 124 999 999 ÷ 2 = 14 062 499 999 + 1;
- 14 062 499 999 ÷ 2 = 7 031 249 999 + 1;
- 7 031 249 999 ÷ 2 = 3 515 624 999 + 1;
- 3 515 624 999 ÷ 2 = 1 757 812 499 + 1;
- 1 757 812 499 ÷ 2 = 878 906 249 + 1;
- 878 906 249 ÷ 2 = 439 453 124 + 1;
- 439 453 124 ÷ 2 = 219 726 562 + 0;
- 219 726 562 ÷ 2 = 109 863 281 + 0;
- 109 863 281 ÷ 2 = 54 931 640 + 1;
- 54 931 640 ÷ 2 = 27 465 820 + 0;
- 27 465 820 ÷ 2 = 13 732 910 + 0;
- 13 732 910 ÷ 2 = 6 866 455 + 0;
- 6 866 455 ÷ 2 = 3 433 227 + 1;
- 3 433 227 ÷ 2 = 1 716 613 + 1;
- 1 716 613 ÷ 2 = 858 306 + 1;
- 858 306 ÷ 2 = 429 153 + 0;
- 429 153 ÷ 2 = 214 576 + 1;
- 214 576 ÷ 2 = 107 288 + 0;
- 107 288 ÷ 2 = 53 644 + 0;
- 53 644 ÷ 2 = 26 822 + 0;
- 26 822 ÷ 2 = 13 411 + 0;
- 13 411 ÷ 2 = 6 705 + 1;
- 6 705 ÷ 2 = 3 352 + 1;
- 3 352 ÷ 2 = 1 676 + 0;
- 1 676 ÷ 2 = 838 + 0;
- 838 ÷ 2 = 419 + 0;
- 419 ÷ 2 = 209 + 1;
- 209 ÷ 2 = 104 + 1;
- 104 ÷ 2 = 52 + 0;
- 52 ÷ 2 = 26 + 0;
- 26 ÷ 2 = 13 + 0;
- 13 ÷ 2 = 6 + 1;
- 6 ÷ 2 = 3 + 0;
- 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.
899 999 999 993(10) = 1101 0001 1000 1100 0010 1110 0010 0111 1111 1001(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 40.
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, 40,
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 899 999 999 993(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:
899 999 999 993(10) = 0000 0000 0000 0000 0000 0000 1101 0001 1000 1100 0010 1110 0010 0111 1111 1001
Spaces were used to group digits: for binary, by 4, for decimal, by 3.