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;
- 487 393 651 993 223 ÷ 2 = 243 696 825 996 611 + 1;
- 243 696 825 996 611 ÷ 2 = 121 848 412 998 305 + 1;
- 121 848 412 998 305 ÷ 2 = 60 924 206 499 152 + 1;
- 60 924 206 499 152 ÷ 2 = 30 462 103 249 576 + 0;
- 30 462 103 249 576 ÷ 2 = 15 231 051 624 788 + 0;
- 15 231 051 624 788 ÷ 2 = 7 615 525 812 394 + 0;
- 7 615 525 812 394 ÷ 2 = 3 807 762 906 197 + 0;
- 3 807 762 906 197 ÷ 2 = 1 903 881 453 098 + 1;
- 1 903 881 453 098 ÷ 2 = 951 940 726 549 + 0;
- 951 940 726 549 ÷ 2 = 475 970 363 274 + 1;
- 475 970 363 274 ÷ 2 = 237 985 181 637 + 0;
- 237 985 181 637 ÷ 2 = 118 992 590 818 + 1;
- 118 992 590 818 ÷ 2 = 59 496 295 409 + 0;
- 59 496 295 409 ÷ 2 = 29 748 147 704 + 1;
- 29 748 147 704 ÷ 2 = 14 874 073 852 + 0;
- 14 874 073 852 ÷ 2 = 7 437 036 926 + 0;
- 7 437 036 926 ÷ 2 = 3 718 518 463 + 0;
- 3 718 518 463 ÷ 2 = 1 859 259 231 + 1;
- 1 859 259 231 ÷ 2 = 929 629 615 + 1;
- 929 629 615 ÷ 2 = 464 814 807 + 1;
- 464 814 807 ÷ 2 = 232 407 403 + 1;
- 232 407 403 ÷ 2 = 116 203 701 + 1;
- 116 203 701 ÷ 2 = 58 101 850 + 1;
- 58 101 850 ÷ 2 = 29 050 925 + 0;
- 29 050 925 ÷ 2 = 14 525 462 + 1;
- 14 525 462 ÷ 2 = 7 262 731 + 0;
- 7 262 731 ÷ 2 = 3 631 365 + 1;
- 3 631 365 ÷ 2 = 1 815 682 + 1;
- 1 815 682 ÷ 2 = 907 841 + 0;
- 907 841 ÷ 2 = 453 920 + 1;
- 453 920 ÷ 2 = 226 960 + 0;
- 226 960 ÷ 2 = 113 480 + 0;
- 113 480 ÷ 2 = 56 740 + 0;
- 56 740 ÷ 2 = 28 370 + 0;
- 28 370 ÷ 2 = 14 185 + 0;
- 14 185 ÷ 2 = 7 092 + 1;
- 7 092 ÷ 2 = 3 546 + 0;
- 3 546 ÷ 2 = 1 773 + 0;
- 1 773 ÷ 2 = 886 + 1;
- 886 ÷ 2 = 443 + 0;
- 443 ÷ 2 = 221 + 1;
- 221 ÷ 2 = 110 + 1;
- 110 ÷ 2 = 55 + 0;
- 55 ÷ 2 = 27 + 1;
- 27 ÷ 2 = 13 + 1;
- 13 ÷ 2 = 6 + 1;
- 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.
487 393 651 993 223(10) = 1 1011 1011 0100 1000 0010 1101 0111 1110 0010 1010 1000 0111(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 49.
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, 49,
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:
487 393 651 993 223(10) = 0000 0000 0000 0001 1011 1011 0100 1000 0010 1101 0111 1110 0010 1010 1000 0111
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 -487 393 651 993 223(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
-487 393 651 993 223(10) = 1000 0000 0000 0001 1011 1011 0100 1000 0010 1101 0111 1110 0010 1010 1000 0111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.