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;
- 1 010 012 ÷ 2 = 505 006 + 0;
- 505 006 ÷ 2 = 252 503 + 0;
- 252 503 ÷ 2 = 126 251 + 1;
- 126 251 ÷ 2 = 63 125 + 1;
- 63 125 ÷ 2 = 31 562 + 1;
- 31 562 ÷ 2 = 15 781 + 0;
- 15 781 ÷ 2 = 7 890 + 1;
- 7 890 ÷ 2 = 3 945 + 0;
- 3 945 ÷ 2 = 1 972 + 1;
- 1 972 ÷ 2 = 986 + 0;
- 986 ÷ 2 = 493 + 0;
- 493 ÷ 2 = 246 + 1;
- 246 ÷ 2 = 123 + 0;
- 123 ÷ 2 = 61 + 1;
- 61 ÷ 2 = 30 + 1;
- 30 ÷ 2 = 15 + 0;
- 15 ÷ 2 = 7 + 1;
- 7 ÷ 2 = 3 + 1;
- 3 ÷ 2 = 1 + 1;
- 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.
1 010 012(10) = 1111 0110 1001 0101 1100(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 20.
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, 20,
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.
1 010 012(10) = 0000 0000 0000 1111 0110 1001 0101 1100
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.
-1 010 012(10) = !(0000 0000 0000 1111 0110 1001 0101 1100)
Number -1 010 012(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:
-1 010 012(10) = 1111 1111 1111 0000 1001 0110 1010 0011
Spaces were used to group digits: for binary, by 4, for decimal, by 3.