1. Start with the positive version of the number:
|-654 657| = 654 657
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;
- 654 657 ÷ 2 = 327 328 + 1;
- 327 328 ÷ 2 = 163 664 + 0;
- 163 664 ÷ 2 = 81 832 + 0;
- 81 832 ÷ 2 = 40 916 + 0;
- 40 916 ÷ 2 = 20 458 + 0;
- 20 458 ÷ 2 = 10 229 + 0;
- 10 229 ÷ 2 = 5 114 + 1;
- 5 114 ÷ 2 = 2 557 + 0;
- 2 557 ÷ 2 = 1 278 + 1;
- 1 278 ÷ 2 = 639 + 0;
- 639 ÷ 2 = 319 + 1;
- 319 ÷ 2 = 159 + 1;
- 159 ÷ 2 = 79 + 1;
- 79 ÷ 2 = 39 + 1;
- 39 ÷ 2 = 19 + 1;
- 19 ÷ 2 = 9 + 1;
- 9 ÷ 2 = 4 + 1;
- 4 ÷ 2 = 2 + 0;
- 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.
654 657(10) = 1001 1111 1101 0100 0001(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.
654 657(10) = 0000 0000 0000 1001 1111 1101 0100 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.
-654 657(10) = !(0000 0000 0000 1001 1111 1101 0100 0001)
Number -654 657(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:
-654 657(10) = 1111 1111 1111 0110 0000 0010 1011 1110
Spaces were used to group digits: for binary, by 4, for decimal, by 3.