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;
- 47 154 667 ÷ 2 = 23 577 333 + 1;
- 23 577 333 ÷ 2 = 11 788 666 + 1;
- 11 788 666 ÷ 2 = 5 894 333 + 0;
- 5 894 333 ÷ 2 = 2 947 166 + 1;
- 2 947 166 ÷ 2 = 1 473 583 + 0;
- 1 473 583 ÷ 2 = 736 791 + 1;
- 736 791 ÷ 2 = 368 395 + 1;
- 368 395 ÷ 2 = 184 197 + 1;
- 184 197 ÷ 2 = 92 098 + 1;
- 92 098 ÷ 2 = 46 049 + 0;
- 46 049 ÷ 2 = 23 024 + 1;
- 23 024 ÷ 2 = 11 512 + 0;
- 11 512 ÷ 2 = 5 756 + 0;
- 5 756 ÷ 2 = 2 878 + 0;
- 2 878 ÷ 2 = 1 439 + 0;
- 1 439 ÷ 2 = 719 + 1;
- 719 ÷ 2 = 359 + 1;
- 359 ÷ 2 = 179 + 1;
- 179 ÷ 2 = 89 + 1;
- 89 ÷ 2 = 44 + 1;
- 44 ÷ 2 = 22 + 0;
- 22 ÷ 2 = 11 + 0;
- 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.
47 154 667(10) = 10 1100 1111 1000 0101 1110 1011(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) 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, 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 47 154 667(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
47 154 667(10) = 0000 0010 1100 1111 1000 0101 1110 1011
Spaces were used to group digits: for binary, by 4, for decimal, by 3.