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;
- 7 516 225 659 ÷ 2 = 3 758 112 829 + 1;
- 3 758 112 829 ÷ 2 = 1 879 056 414 + 1;
- 1 879 056 414 ÷ 2 = 939 528 207 + 0;
- 939 528 207 ÷ 2 = 469 764 103 + 1;
- 469 764 103 ÷ 2 = 234 882 051 + 1;
- 234 882 051 ÷ 2 = 117 441 025 + 1;
- 117 441 025 ÷ 2 = 58 720 512 + 1;
- 58 720 512 ÷ 2 = 29 360 256 + 0;
- 29 360 256 ÷ 2 = 14 680 128 + 0;
- 14 680 128 ÷ 2 = 7 340 064 + 0;
- 7 340 064 ÷ 2 = 3 670 032 + 0;
- 3 670 032 ÷ 2 = 1 835 016 + 0;
- 1 835 016 ÷ 2 = 917 508 + 0;
- 917 508 ÷ 2 = 458 754 + 0;
- 458 754 ÷ 2 = 229 377 + 0;
- 229 377 ÷ 2 = 114 688 + 1;
- 114 688 ÷ 2 = 57 344 + 0;
- 57 344 ÷ 2 = 28 672 + 0;
- 28 672 ÷ 2 = 14 336 + 0;
- 14 336 ÷ 2 = 7 168 + 0;
- 7 168 ÷ 2 = 3 584 + 0;
- 3 584 ÷ 2 = 1 792 + 0;
- 1 792 ÷ 2 = 896 + 0;
- 896 ÷ 2 = 448 + 0;
- 448 ÷ 2 = 224 + 0;
- 224 ÷ 2 = 112 + 0;
- 112 ÷ 2 = 56 + 0;
- 56 ÷ 2 = 28 + 0;
- 28 ÷ 2 = 14 + 0;
- 14 ÷ 2 = 7 + 0;
- 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.
7 516 225 659(10) = 1 1100 0000 0000 0000 1000 0000 0111 1011(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) 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, 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:
7 516 225 659(10) = 0000 0000 0000 0000 0000 0000 0000 0001 1100 0000 0000 0000 1000 0000 0111 1011
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 -7 516 225 659(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
-7 516 225 659(10) = 1000 0000 0000 0000 0000 0000 0000 0001 1100 0000 0000 0000 1000 0000 0111 1011
Spaces were used to group digits: for binary, by 4, for decimal, by 3.