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;
- 32 768 128 ÷ 2 = 16 384 064 + 0;
- 16 384 064 ÷ 2 = 8 192 032 + 0;
- 8 192 032 ÷ 2 = 4 096 016 + 0;
- 4 096 016 ÷ 2 = 2 048 008 + 0;
- 2 048 008 ÷ 2 = 1 024 004 + 0;
- 1 024 004 ÷ 2 = 512 002 + 0;
- 512 002 ÷ 2 = 256 001 + 0;
- 256 001 ÷ 2 = 128 000 + 1;
- 128 000 ÷ 2 = 64 000 + 0;
- 64 000 ÷ 2 = 32 000 + 0;
- 32 000 ÷ 2 = 16 000 + 0;
- 16 000 ÷ 2 = 8 000 + 0;
- 8 000 ÷ 2 = 4 000 + 0;
- 4 000 ÷ 2 = 2 000 + 0;
- 2 000 ÷ 2 = 1 000 + 0;
- 1 000 ÷ 2 = 500 + 0;
- 500 ÷ 2 = 250 + 0;
- 250 ÷ 2 = 125 + 0;
- 125 ÷ 2 = 62 + 1;
- 62 ÷ 2 = 31 + 0;
- 31 ÷ 2 = 15 + 1;
- 15 ÷ 2 = 7 + 1;
- 7 ÷ 2 = 3 + 1;
- 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.
32 768 128(10) = 1 1111 0100 0000 0000 1000 0000(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 25.
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, 25,
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 32 768 128(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
32 768 128(10) = 0000 0001 1111 0100 0000 0000 1000 0000
Spaces were used to group digits: for binary, by 4, for decimal, by 3.