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;
- 1 456 777 255 814 934 ÷ 2 = 728 388 627 907 467 + 0;
- 728 388 627 907 467 ÷ 2 = 364 194 313 953 733 + 1;
- 364 194 313 953 733 ÷ 2 = 182 097 156 976 866 + 1;
- 182 097 156 976 866 ÷ 2 = 91 048 578 488 433 + 0;
- 91 048 578 488 433 ÷ 2 = 45 524 289 244 216 + 1;
- 45 524 289 244 216 ÷ 2 = 22 762 144 622 108 + 0;
- 22 762 144 622 108 ÷ 2 = 11 381 072 311 054 + 0;
- 11 381 072 311 054 ÷ 2 = 5 690 536 155 527 + 0;
- 5 690 536 155 527 ÷ 2 = 2 845 268 077 763 + 1;
- 2 845 268 077 763 ÷ 2 = 1 422 634 038 881 + 1;
- 1 422 634 038 881 ÷ 2 = 711 317 019 440 + 1;
- 711 317 019 440 ÷ 2 = 355 658 509 720 + 0;
- 355 658 509 720 ÷ 2 = 177 829 254 860 + 0;
- 177 829 254 860 ÷ 2 = 88 914 627 430 + 0;
- 88 914 627 430 ÷ 2 = 44 457 313 715 + 0;
- 44 457 313 715 ÷ 2 = 22 228 656 857 + 1;
- 22 228 656 857 ÷ 2 = 11 114 328 428 + 1;
- 11 114 328 428 ÷ 2 = 5 557 164 214 + 0;
- 5 557 164 214 ÷ 2 = 2 778 582 107 + 0;
- 2 778 582 107 ÷ 2 = 1 389 291 053 + 1;
- 1 389 291 053 ÷ 2 = 694 645 526 + 1;
- 694 645 526 ÷ 2 = 347 322 763 + 0;
- 347 322 763 ÷ 2 = 173 661 381 + 1;
- 173 661 381 ÷ 2 = 86 830 690 + 1;
- 86 830 690 ÷ 2 = 43 415 345 + 0;
- 43 415 345 ÷ 2 = 21 707 672 + 1;
- 21 707 672 ÷ 2 = 10 853 836 + 0;
- 10 853 836 ÷ 2 = 5 426 918 + 0;
- 5 426 918 ÷ 2 = 2 713 459 + 0;
- 2 713 459 ÷ 2 = 1 356 729 + 1;
- 1 356 729 ÷ 2 = 678 364 + 1;
- 678 364 ÷ 2 = 339 182 + 0;
- 339 182 ÷ 2 = 169 591 + 0;
- 169 591 ÷ 2 = 84 795 + 1;
- 84 795 ÷ 2 = 42 397 + 1;
- 42 397 ÷ 2 = 21 198 + 1;
- 21 198 ÷ 2 = 10 599 + 0;
- 10 599 ÷ 2 = 5 299 + 1;
- 5 299 ÷ 2 = 2 649 + 1;
- 2 649 ÷ 2 = 1 324 + 1;
- 1 324 ÷ 2 = 662 + 0;
- 662 ÷ 2 = 331 + 0;
- 331 ÷ 2 = 165 + 1;
- 165 ÷ 2 = 82 + 1;
- 82 ÷ 2 = 41 + 0;
- 41 ÷ 2 = 20 + 1;
- 20 ÷ 2 = 10 + 0;
- 10 ÷ 2 = 5 + 0;
- 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.
1 456 777 255 814 934(10) = 101 0010 1100 1110 1110 0110 0010 1101 1001 1000 0111 0001 0110(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 51.
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, 51,
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.
1 456 777 255 814 934(10) = 0000 0000 0000 0101 0010 1100 1110 1110 0110 0010 1101 1001 1000 0111 0001 0110
6. Get the negative integer number representation. Part 1:
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.
!(0000 0000 0000 0101 0010 1100 1110 1110 0110 0010 1101 1001 1000 0111 0001 0110)
= 1111 1111 1111 1010 1101 0011 0001 0001 1001 1101 0010 0110 0111 1000 1110 1001
7. Get the negative integer number representation. Part 2:
To write the negative integer number on 64 bits (8 Bytes),
as a signed binary in two's complement representation,
add 1 to the number calculated above
1111 1111 1111 1010 1101 0011 0001 0001 1001 1101 0010 0110 0111 1000 1110 1001
(to the signed binary in one's complement representation)
Binary addition carries on a value of 2:
0 + 0 = 0
0 + 1 = 1
1 + 1 = 10
1 + 10 = 11
1 + 11 = 100
Add 1 to the number calculated above
(to the signed binary number in one's complement representation):
-1 456 777 255 814 934 =
1111 1111 1111 1010 1101 0011 0001 0001 1001 1101 0010 0110 0111 1000 1110 1001 + 1
Number -1 456 777 255 814 934(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in two's complement representation:
-1 456 777 255 814 934(10) = 1111 1111 1111 1010 1101 0011 0001 0001 1001 1101 0010 0110 0111 1000 1110 1010
Spaces were used to group digits: for binary, by 4, for decimal, by 3.