1. 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;
- 1 100 110 000 000 ÷ 2 = 550 055 000 000 + 0;
- 550 055 000 000 ÷ 2 = 275 027 500 000 + 0;
- 275 027 500 000 ÷ 2 = 137 513 750 000 + 0;
- 137 513 750 000 ÷ 2 = 68 756 875 000 + 0;
- 68 756 875 000 ÷ 2 = 34 378 437 500 + 0;
- 34 378 437 500 ÷ 2 = 17 189 218 750 + 0;
- 17 189 218 750 ÷ 2 = 8 594 609 375 + 0;
- 8 594 609 375 ÷ 2 = 4 297 304 687 + 1;
- 4 297 304 687 ÷ 2 = 2 148 652 343 + 1;
- 2 148 652 343 ÷ 2 = 1 074 326 171 + 1;
- 1 074 326 171 ÷ 2 = 537 163 085 + 1;
- 537 163 085 ÷ 2 = 268 581 542 + 1;
- 268 581 542 ÷ 2 = 134 290 771 + 0;
- 134 290 771 ÷ 2 = 67 145 385 + 1;
- 67 145 385 ÷ 2 = 33 572 692 + 1;
- 33 572 692 ÷ 2 = 16 786 346 + 0;
- 16 786 346 ÷ 2 = 8 393 173 + 0;
- 8 393 173 ÷ 2 = 4 196 586 + 1;
- 4 196 586 ÷ 2 = 2 098 293 + 0;
- 2 098 293 ÷ 2 = 1 049 146 + 1;
- 1 049 146 ÷ 2 = 524 573 + 0;
- 524 573 ÷ 2 = 262 286 + 1;
- 262 286 ÷ 2 = 131 143 + 0;
- 131 143 ÷ 2 = 65 571 + 1;
- 65 571 ÷ 2 = 32 785 + 1;
- 32 785 ÷ 2 = 16 392 + 1;
- 16 392 ÷ 2 = 8 196 + 0;
- 8 196 ÷ 2 = 4 098 + 0;
- 4 098 ÷ 2 = 2 049 + 0;
- 2 049 ÷ 2 = 1 024 + 1;
- 1 024 ÷ 2 = 512 + 0;
- 512 ÷ 2 = 256 + 0;
- 256 ÷ 2 = 128 + 0;
- 128 ÷ 2 = 64 + 0;
- 64 ÷ 2 = 32 + 0;
- 32 ÷ 2 = 16 + 0;
- 16 ÷ 2 = 8 + 0;
- 8 ÷ 2 = 4 + 0;
- 4 ÷ 2 = 2 + 0;
- 2 ÷ 2 = 1 + 0;
- 1 ÷ 2 = 0 + 1;
2. Construct the base 2 representation of the positive number:
Take all the remainders starting from the bottom of the list constructed above.
1 100 110 000 000(10) = 1 0000 0000 0010 0011 1010 1010 0110 1111 1000 0000(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 41.
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, 41,
3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)
=== is: 64.
4. 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:
Number 1 100 110 000 000(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
1 100 110 000 000(10) = 0000 0000 0000 0000 0000 0001 0000 0000 0010 0011 1010 1010 0110 1111 1000 0000
Spaces were used to group digits: for binary, by 4, for decimal, by 3.