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;
- 5 199 999 989 ÷ 2 = 2 599 999 994 + 1;
- 2 599 999 994 ÷ 2 = 1 299 999 997 + 0;
- 1 299 999 997 ÷ 2 = 649 999 998 + 1;
- 649 999 998 ÷ 2 = 324 999 999 + 0;
- 324 999 999 ÷ 2 = 162 499 999 + 1;
- 162 499 999 ÷ 2 = 81 249 999 + 1;
- 81 249 999 ÷ 2 = 40 624 999 + 1;
- 40 624 999 ÷ 2 = 20 312 499 + 1;
- 20 312 499 ÷ 2 = 10 156 249 + 1;
- 10 156 249 ÷ 2 = 5 078 124 + 1;
- 5 078 124 ÷ 2 = 2 539 062 + 0;
- 2 539 062 ÷ 2 = 1 269 531 + 0;
- 1 269 531 ÷ 2 = 634 765 + 1;
- 634 765 ÷ 2 = 317 382 + 1;
- 317 382 ÷ 2 = 158 691 + 0;
- 158 691 ÷ 2 = 79 345 + 1;
- 79 345 ÷ 2 = 39 672 + 1;
- 39 672 ÷ 2 = 19 836 + 0;
- 19 836 ÷ 2 = 9 918 + 0;
- 9 918 ÷ 2 = 4 959 + 0;
- 4 959 ÷ 2 = 2 479 + 1;
- 2 479 ÷ 2 = 1 239 + 1;
- 1 239 ÷ 2 = 619 + 1;
- 619 ÷ 2 = 309 + 1;
- 309 ÷ 2 = 154 + 1;
- 154 ÷ 2 = 77 + 0;
- 77 ÷ 2 = 38 + 1;
- 38 ÷ 2 = 19 + 0;
- 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.
5 199 999 989(10) = 1 0011 0101 1111 0001 1011 0011 1111 0101(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 33.
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, 33,
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.
5 199 999 989(10) = 0000 0000 0000 0000 0000 0000 0000 0001 0011 0101 1111 0001 1011 0011 1111 0101
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.
-5 199 999 989(10) = !(0000 0000 0000 0000 0000 0000 0000 0001 0011 0101 1111 0001 1011 0011 1111 0101)
Number -5 199 999 989(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:
-5 199 999 989(10) = 1111 1111 1111 1111 1111 1111 1111 1110 1100 1010 0000 1110 0100 1100 0000 1010
Spaces were used to group digits: for binary, by 4, for decimal, by 3.