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;
- 3 131 313 155 ÷ 2 = 1 565 656 577 + 1;
- 1 565 656 577 ÷ 2 = 782 828 288 + 1;
- 782 828 288 ÷ 2 = 391 414 144 + 0;
- 391 414 144 ÷ 2 = 195 707 072 + 0;
- 195 707 072 ÷ 2 = 97 853 536 + 0;
- 97 853 536 ÷ 2 = 48 926 768 + 0;
- 48 926 768 ÷ 2 = 24 463 384 + 0;
- 24 463 384 ÷ 2 = 12 231 692 + 0;
- 12 231 692 ÷ 2 = 6 115 846 + 0;
- 6 115 846 ÷ 2 = 3 057 923 + 0;
- 3 057 923 ÷ 2 = 1 528 961 + 1;
- 1 528 961 ÷ 2 = 764 480 + 1;
- 764 480 ÷ 2 = 382 240 + 0;
- 382 240 ÷ 2 = 191 120 + 0;
- 191 120 ÷ 2 = 95 560 + 0;
- 95 560 ÷ 2 = 47 780 + 0;
- 47 780 ÷ 2 = 23 890 + 0;
- 23 890 ÷ 2 = 11 945 + 0;
- 11 945 ÷ 2 = 5 972 + 1;
- 5 972 ÷ 2 = 2 986 + 0;
- 2 986 ÷ 2 = 1 493 + 0;
- 1 493 ÷ 2 = 746 + 1;
- 746 ÷ 2 = 373 + 0;
- 373 ÷ 2 = 186 + 1;
- 186 ÷ 2 = 93 + 0;
- 93 ÷ 2 = 46 + 1;
- 46 ÷ 2 = 23 + 0;
- 23 ÷ 2 = 11 + 1;
- 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.
3 131 313 155(10) = 1011 1010 1010 0100 0000 1100 0000 0011(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 32.
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, 32,
3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)
=== is: 64.
5. Get the positive binary computer representation on 64 bits (8 Bytes):
If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 64.
3 131 313 155(10) = 0000 0000 0000 0000 0000 0000 0000 0000 1011 1010 1010 0100 0000 1100 0000 0011
6. Get the negative integer number representation:
To write the negative integer number on 64 bits (8 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.
-3 131 313 155(10) = !(0000 0000 0000 0000 0000 0000 0000 0000 1011 1010 1010 0100 0000 1100 0000 0011)
Number -3 131 313 155(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:
-3 131 313 155(10) = 1111 1111 1111 1111 1111 1111 1111 1111 0100 0101 0101 1011 1111 0011 1111 1100
Spaces were used to group digits: for binary, by 4, for decimal, by 3.