2. 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;
- 83 665 ÷ 2 = 41 832 + 1;
- 41 832 ÷ 2 = 20 916 + 0;
- 20 916 ÷ 2 = 10 458 + 0;
- 10 458 ÷ 2 = 5 229 + 0;
- 5 229 ÷ 2 = 2 614 + 1;
- 2 614 ÷ 2 = 1 307 + 0;
- 1 307 ÷ 2 = 653 + 1;
- 653 ÷ 2 = 326 + 1;
- 326 ÷ 2 = 163 + 0;
- 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;
3. Construct the base 2 representation of the positive number:
Take all the remainders starting from the bottom of the list constructed above.
83 665(10) = 1 0100 0110 1101 0001(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 17.
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, 17,
3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)
=== is: 32.
5. 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.
83 665(10) = 0000 0000 0000 0001 0100 0110 1101 0001
6. Get the negative integer number representation:
To write the negative integer number on 32 bits (4 Bytes),
as a signed binary in one's complement representation,
... replace all the bits on 0 with 1s and all the bits set on 1 with 0s.
Reverse the digits, flip the digits:
Replace the bits set on 0 with 1s and the bits set on 1 with 0s.
-83 665(10) = !(0000 0000 0000 0001 0100 0110 1101 0001)
Number -83 665(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:
-83 665(10) = 1111 1111 1111 1110 1011 1001 0010 1110
Spaces were used to group digits: for binary, by 4, for decimal, by 3.