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;
- 100 000 000 002 ÷ 2 = 50 000 000 001 + 0;
- 50 000 000 001 ÷ 2 = 25 000 000 000 + 1;
- 25 000 000 000 ÷ 2 = 12 500 000 000 + 0;
- 12 500 000 000 ÷ 2 = 6 250 000 000 + 0;
- 6 250 000 000 ÷ 2 = 3 125 000 000 + 0;
- 3 125 000 000 ÷ 2 = 1 562 500 000 + 0;
- 1 562 500 000 ÷ 2 = 781 250 000 + 0;
- 781 250 000 ÷ 2 = 390 625 000 + 0;
- 390 625 000 ÷ 2 = 195 312 500 + 0;
- 195 312 500 ÷ 2 = 97 656 250 + 0;
- 97 656 250 ÷ 2 = 48 828 125 + 0;
- 48 828 125 ÷ 2 = 24 414 062 + 1;
- 24 414 062 ÷ 2 = 12 207 031 + 0;
- 12 207 031 ÷ 2 = 6 103 515 + 1;
- 6 103 515 ÷ 2 = 3 051 757 + 1;
- 3 051 757 ÷ 2 = 1 525 878 + 1;
- 1 525 878 ÷ 2 = 762 939 + 0;
- 762 939 ÷ 2 = 381 469 + 1;
- 381 469 ÷ 2 = 190 734 + 1;
- 190 734 ÷ 2 = 95 367 + 0;
- 95 367 ÷ 2 = 47 683 + 1;
- 47 683 ÷ 2 = 23 841 + 1;
- 23 841 ÷ 2 = 11 920 + 1;
- 11 920 ÷ 2 = 5 960 + 0;
- 5 960 ÷ 2 = 2 980 + 0;
- 2 980 ÷ 2 = 1 490 + 0;
- 1 490 ÷ 2 = 745 + 0;
- 745 ÷ 2 = 372 + 1;
- 372 ÷ 2 = 186 + 0;
- 186 ÷ 2 = 93 + 0;
- 93 ÷ 2 = 46 + 1;
- 46 ÷ 2 = 23 + 0;
- 23 ÷ 2 = 11 + 1;
- 11 ÷ 2 = 5 + 1;
- 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.
100 000 000 002(10) = 1 0111 0100 1000 0111 0110 1110 1000 0000 0010(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 37.
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, 37,
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 100 000 000 002(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
100 000 000 002(10) = 0000 0000 0000 0000 0000 0000 0001 0111 0100 1000 0111 0110 1110 1000 0000 0010
Spaces were used to group digits: for binary, by 4, for decimal, by 3.