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;
- 23 457 ÷ 2 = 11 728 + 1;
- 11 728 ÷ 2 = 5 864 + 0;
- 5 864 ÷ 2 = 2 932 + 0;
- 2 932 ÷ 2 = 1 466 + 0;
- 1 466 ÷ 2 = 733 + 0;
- 733 ÷ 2 = 366 + 1;
- 366 ÷ 2 = 183 + 0;
- 183 ÷ 2 = 91 + 1;
- 91 ÷ 2 = 45 + 1;
- 45 ÷ 2 = 22 + 1;
- 22 ÷ 2 = 11 + 0;
- 11 ÷ 2 = 5 + 1;
- 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.
23 457(10) = 101 1011 1010 0001(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 15.
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, 15,
3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)
=== is: 16.
5. Get the positive binary computer representation on 16 bits (2 Bytes):
If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 16.
23 457(10) = 0101 1011 1010 0001
6. Get the negative integer number representation:
To write the negative integer number on 16 bits (2 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.
-23 457(10) = !(0101 1011 1010 0001)
Number -23 457(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:
-23 457(10) = 1010 0100 0101 1110
Spaces were used to group digits: for binary, by 4, for decimal, by 3.