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;
- 4 294 967 295 ÷ 2 = 2 147 483 647 + 1;
- 2 147 483 647 ÷ 2 = 1 073 741 823 + 1;
- 1 073 741 823 ÷ 2 = 536 870 911 + 1;
- 536 870 911 ÷ 2 = 268 435 455 + 1;
- 268 435 455 ÷ 2 = 134 217 727 + 1;
- 134 217 727 ÷ 2 = 67 108 863 + 1;
- 67 108 863 ÷ 2 = 33 554 431 + 1;
- 33 554 431 ÷ 2 = 16 777 215 + 1;
- 16 777 215 ÷ 2 = 8 388 607 + 1;
- 8 388 607 ÷ 2 = 4 194 303 + 1;
- 4 194 303 ÷ 2 = 2 097 151 + 1;
- 2 097 151 ÷ 2 = 1 048 575 + 1;
- 1 048 575 ÷ 2 = 524 287 + 1;
- 524 287 ÷ 2 = 262 143 + 1;
- 262 143 ÷ 2 = 131 071 + 1;
- 131 071 ÷ 2 = 65 535 + 1;
- 65 535 ÷ 2 = 32 767 + 1;
- 32 767 ÷ 2 = 16 383 + 1;
- 16 383 ÷ 2 = 8 191 + 1;
- 8 191 ÷ 2 = 4 095 + 1;
- 4 095 ÷ 2 = 2 047 + 1;
- 2 047 ÷ 2 = 1 023 + 1;
- 1 023 ÷ 2 = 511 + 1;
- 511 ÷ 2 = 255 + 1;
- 255 ÷ 2 = 127 + 1;
- 127 ÷ 2 = 63 + 1;
- 63 ÷ 2 = 31 + 1;
- 31 ÷ 2 = 15 + 1;
- 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.
4 294 967 295(10) = 1111 1111 1111 1111 1111 1111 1111 1111(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) is reserved for 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:
4 294 967 295(10) = 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111
6. Get the negative integer number representation:
To get the negative integer number representation on 64 bits (8 Bytes),
... change the first bit (the leftmost), from 0 to 1...
Number -4 294 967 295(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
-4 294 967 295(10) = 1000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.