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;
- 100 111 159 ÷ 2 = 50 055 579 + 1;
- 50 055 579 ÷ 2 = 25 027 789 + 1;
- 25 027 789 ÷ 2 = 12 513 894 + 1;
- 12 513 894 ÷ 2 = 6 256 947 + 0;
- 6 256 947 ÷ 2 = 3 128 473 + 1;
- 3 128 473 ÷ 2 = 1 564 236 + 1;
- 1 564 236 ÷ 2 = 782 118 + 0;
- 782 118 ÷ 2 = 391 059 + 0;
- 391 059 ÷ 2 = 195 529 + 1;
- 195 529 ÷ 2 = 97 764 + 1;
- 97 764 ÷ 2 = 48 882 + 0;
- 48 882 ÷ 2 = 24 441 + 0;
- 24 441 ÷ 2 = 12 220 + 1;
- 12 220 ÷ 2 = 6 110 + 0;
- 6 110 ÷ 2 = 3 055 + 0;
- 3 055 ÷ 2 = 1 527 + 1;
- 1 527 ÷ 2 = 763 + 1;
- 763 ÷ 2 = 381 + 1;
- 381 ÷ 2 = 190 + 1;
- 190 ÷ 2 = 95 + 0;
- 95 ÷ 2 = 47 + 1;
- 47 ÷ 2 = 23 + 1;
- 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.
100 111 159(10) = 101 1111 0111 1001 0011 0011 0111(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 27.
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, 27,
3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)
=== is: 32.
4. Get the positive binary computer representation on 32 bits (4 Bytes):
If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 32.
Number 100 111 159(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in one's complement representation:
100 111 159(10) = 0000 0101 1111 0111 1001 0011 0011 0111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.