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;
- 42 941 671 ÷ 2 = 21 470 835 + 1;
- 21 470 835 ÷ 2 = 10 735 417 + 1;
- 10 735 417 ÷ 2 = 5 367 708 + 1;
- 5 367 708 ÷ 2 = 2 683 854 + 0;
- 2 683 854 ÷ 2 = 1 341 927 + 0;
- 1 341 927 ÷ 2 = 670 963 + 1;
- 670 963 ÷ 2 = 335 481 + 1;
- 335 481 ÷ 2 = 167 740 + 1;
- 167 740 ÷ 2 = 83 870 + 0;
- 83 870 ÷ 2 = 41 935 + 0;
- 41 935 ÷ 2 = 20 967 + 1;
- 20 967 ÷ 2 = 10 483 + 1;
- 10 483 ÷ 2 = 5 241 + 1;
- 5 241 ÷ 2 = 2 620 + 1;
- 2 620 ÷ 2 = 1 310 + 0;
- 1 310 ÷ 2 = 655 + 0;
- 655 ÷ 2 = 327 + 1;
- 327 ÷ 2 = 163 + 1;
- 163 ÷ 2 = 81 + 1;
- 81 ÷ 2 = 40 + 1;
- 40 ÷ 2 = 20 + 0;
- 20 ÷ 2 = 10 + 0;
- 10 ÷ 2 = 5 + 0;
- 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.
42 941 671(10) = 10 1000 1111 0011 1100 1110 0111(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 26.
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, 26,
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 42 941 671(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:
42 941 671(10) = 0000 0010 1000 1111 0011 1100 1110 0111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.