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;
- 3 731 183 625 692 672 009 ÷ 2 = 1 865 591 812 846 336 004 + 1;
- 1 865 591 812 846 336 004 ÷ 2 = 932 795 906 423 168 002 + 0;
- 932 795 906 423 168 002 ÷ 2 = 466 397 953 211 584 001 + 0;
- 466 397 953 211 584 001 ÷ 2 = 233 198 976 605 792 000 + 1;
- 233 198 976 605 792 000 ÷ 2 = 116 599 488 302 896 000 + 0;
- 116 599 488 302 896 000 ÷ 2 = 58 299 744 151 448 000 + 0;
- 58 299 744 151 448 000 ÷ 2 = 29 149 872 075 724 000 + 0;
- 29 149 872 075 724 000 ÷ 2 = 14 574 936 037 862 000 + 0;
- 14 574 936 037 862 000 ÷ 2 = 7 287 468 018 931 000 + 0;
- 7 287 468 018 931 000 ÷ 2 = 3 643 734 009 465 500 + 0;
- 3 643 734 009 465 500 ÷ 2 = 1 821 867 004 732 750 + 0;
- 1 821 867 004 732 750 ÷ 2 = 910 933 502 366 375 + 0;
- 910 933 502 366 375 ÷ 2 = 455 466 751 183 187 + 1;
- 455 466 751 183 187 ÷ 2 = 227 733 375 591 593 + 1;
- 227 733 375 591 593 ÷ 2 = 113 866 687 795 796 + 1;
- 113 866 687 795 796 ÷ 2 = 56 933 343 897 898 + 0;
- 56 933 343 897 898 ÷ 2 = 28 466 671 948 949 + 0;
- 28 466 671 948 949 ÷ 2 = 14 233 335 974 474 + 1;
- 14 233 335 974 474 ÷ 2 = 7 116 667 987 237 + 0;
- 7 116 667 987 237 ÷ 2 = 3 558 333 993 618 + 1;
- 3 558 333 993 618 ÷ 2 = 1 779 166 996 809 + 0;
- 1 779 166 996 809 ÷ 2 = 889 583 498 404 + 1;
- 889 583 498 404 ÷ 2 = 444 791 749 202 + 0;
- 444 791 749 202 ÷ 2 = 222 395 874 601 + 0;
- 222 395 874 601 ÷ 2 = 111 197 937 300 + 1;
- 111 197 937 300 ÷ 2 = 55 598 968 650 + 0;
- 55 598 968 650 ÷ 2 = 27 799 484 325 + 0;
- 27 799 484 325 ÷ 2 = 13 899 742 162 + 1;
- 13 899 742 162 ÷ 2 = 6 949 871 081 + 0;
- 6 949 871 081 ÷ 2 = 3 474 935 540 + 1;
- 3 474 935 540 ÷ 2 = 1 737 467 770 + 0;
- 1 737 467 770 ÷ 2 = 868 733 885 + 0;
- 868 733 885 ÷ 2 = 434 366 942 + 1;
- 434 366 942 ÷ 2 = 217 183 471 + 0;
- 217 183 471 ÷ 2 = 108 591 735 + 1;
- 108 591 735 ÷ 2 = 54 295 867 + 1;
- 54 295 867 ÷ 2 = 27 147 933 + 1;
- 27 147 933 ÷ 2 = 13 573 966 + 1;
- 13 573 966 ÷ 2 = 6 786 983 + 0;
- 6 786 983 ÷ 2 = 3 393 491 + 1;
- 3 393 491 ÷ 2 = 1 696 745 + 1;
- 1 696 745 ÷ 2 = 848 372 + 1;
- 848 372 ÷ 2 = 424 186 + 0;
- 424 186 ÷ 2 = 212 093 + 0;
- 212 093 ÷ 2 = 106 046 + 1;
- 106 046 ÷ 2 = 53 023 + 0;
- 53 023 ÷ 2 = 26 511 + 1;
- 26 511 ÷ 2 = 13 255 + 1;
- 13 255 ÷ 2 = 6 627 + 1;
- 6 627 ÷ 2 = 3 313 + 1;
- 3 313 ÷ 2 = 1 656 + 1;
- 1 656 ÷ 2 = 828 + 0;
- 828 ÷ 2 = 414 + 0;
- 414 ÷ 2 = 207 + 0;
- 207 ÷ 2 = 103 + 1;
- 103 ÷ 2 = 51 + 1;
- 51 ÷ 2 = 25 + 1;
- 25 ÷ 2 = 12 + 1;
- 12 ÷ 2 = 6 + 0;
- 6 ÷ 2 = 3 + 0;
- 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.
3 731 183 625 692 672 009(10) = 11 0011 1100 0111 1101 0011 1011 1101 0010 1001 0010 1010 0111 0000 0000 1001(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 62.
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, 62,
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.
3 731 183 625 692 672 009(10) = 0011 0011 1100 0111 1101 0011 1011 1101 0010 1001 0010 1010 0111 0000 0000 1001
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.
!(0011 0011 1100 0111 1101 0011 1011 1101 0010 1001 0010 1010 0111 0000 0000 1001)
= 1100 1100 0011 1000 0010 1100 0100 0010 1101 0110 1101 0101 1000 1111 1111 0110
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
1100 1100 0011 1000 0010 1100 0100 0010 1101 0110 1101 0101 1000 1111 1111 0110
(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):
-3 731 183 625 692 672 009 =
1100 1100 0011 1000 0010 1100 0100 0010 1101 0110 1101 0101 1000 1111 1111 0110 + 1
Number -3 731 183 625 692 672 009(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:
-3 731 183 625 692 672 009(10) = 1100 1100 0011 1000 0010 1100 0100 0010 1101 0110 1101 0101 1000 1111 1111 0111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.