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;
- 1 512 038 331 ÷ 2 = 756 019 165 + 1;
- 756 019 165 ÷ 2 = 378 009 582 + 1;
- 378 009 582 ÷ 2 = 189 004 791 + 0;
- 189 004 791 ÷ 2 = 94 502 395 + 1;
- 94 502 395 ÷ 2 = 47 251 197 + 1;
- 47 251 197 ÷ 2 = 23 625 598 + 1;
- 23 625 598 ÷ 2 = 11 812 799 + 0;
- 11 812 799 ÷ 2 = 5 906 399 + 1;
- 5 906 399 ÷ 2 = 2 953 199 + 1;
- 2 953 199 ÷ 2 = 1 476 599 + 1;
- 1 476 599 ÷ 2 = 738 299 + 1;
- 738 299 ÷ 2 = 369 149 + 1;
- 369 149 ÷ 2 = 184 574 + 1;
- 184 574 ÷ 2 = 92 287 + 0;
- 92 287 ÷ 2 = 46 143 + 1;
- 46 143 ÷ 2 = 23 071 + 1;
- 23 071 ÷ 2 = 11 535 + 1;
- 11 535 ÷ 2 = 5 767 + 1;
- 5 767 ÷ 2 = 2 883 + 1;
- 2 883 ÷ 2 = 1 441 + 1;
- 1 441 ÷ 2 = 720 + 1;
- 720 ÷ 2 = 360 + 0;
- 360 ÷ 2 = 180 + 0;
- 180 ÷ 2 = 90 + 0;
- 90 ÷ 2 = 45 + 0;
- 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.
1 512 038 331(10) = 101 1010 0001 1111 1101 1111 1011 1011(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 31.
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, 31,
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.
1 512 038 331(10) = 0101 1010 0001 1111 1101 1111 1011 1011
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.
-1 512 038 331(10) = !(0101 1010 0001 1111 1101 1111 1011 1011)
Number -1 512 038 331(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:
-1 512 038 331(10) = 1010 0101 1110 0000 0010 0000 0100 0100
Spaces were used to group digits: for binary, by 4, for decimal, by 3.