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;
- 9 223 371 976 725 202 943 ÷ 2 = 4 611 685 988 362 601 471 + 1;
- 4 611 685 988 362 601 471 ÷ 2 = 2 305 842 994 181 300 735 + 1;
- 2 305 842 994 181 300 735 ÷ 2 = 1 152 921 497 090 650 367 + 1;
- 1 152 921 497 090 650 367 ÷ 2 = 576 460 748 545 325 183 + 1;
- 576 460 748 545 325 183 ÷ 2 = 288 230 374 272 662 591 + 1;
- 288 230 374 272 662 591 ÷ 2 = 144 115 187 136 331 295 + 1;
- 144 115 187 136 331 295 ÷ 2 = 72 057 593 568 165 647 + 1;
- 72 057 593 568 165 647 ÷ 2 = 36 028 796 784 082 823 + 1;
- 36 028 796 784 082 823 ÷ 2 = 18 014 398 392 041 411 + 1;
- 18 014 398 392 041 411 ÷ 2 = 9 007 199 196 020 705 + 1;
- 9 007 199 196 020 705 ÷ 2 = 4 503 599 598 010 352 + 1;
- 4 503 599 598 010 352 ÷ 2 = 2 251 799 799 005 176 + 0;
- 2 251 799 799 005 176 ÷ 2 = 1 125 899 899 502 588 + 0;
- 1 125 899 899 502 588 ÷ 2 = 562 949 949 751 294 + 0;
- 562 949 949 751 294 ÷ 2 = 281 474 974 875 647 + 0;
- 281 474 974 875 647 ÷ 2 = 140 737 487 437 823 + 1;
- 140 737 487 437 823 ÷ 2 = 70 368 743 718 911 + 1;
- 70 368 743 718 911 ÷ 2 = 35 184 371 859 455 + 1;
- 35 184 371 859 455 ÷ 2 = 17 592 185 929 727 + 1;
- 17 592 185 929 727 ÷ 2 = 8 796 092 964 863 + 1;
- 8 796 092 964 863 ÷ 2 = 4 398 046 482 431 + 1;
- 4 398 046 482 431 ÷ 2 = 2 199 023 241 215 + 1;
- 2 199 023 241 215 ÷ 2 = 1 099 511 620 607 + 1;
- 1 099 511 620 607 ÷ 2 = 549 755 810 303 + 1;
- 549 755 810 303 ÷ 2 = 274 877 905 151 + 1;
- 274 877 905 151 ÷ 2 = 137 438 952 575 + 1;
- 137 438 952 575 ÷ 2 = 68 719 476 287 + 1;
- 68 719 476 287 ÷ 2 = 34 359 738 143 + 1;
- 34 359 738 143 ÷ 2 = 17 179 869 071 + 1;
- 17 179 869 071 ÷ 2 = 8 589 934 535 + 1;
- 8 589 934 535 ÷ 2 = 4 294 967 267 + 1;
- 4 294 967 267 ÷ 2 = 2 147 483 633 + 1;
- 2 147 483 633 ÷ 2 = 1 073 741 816 + 1;
- 1 073 741 816 ÷ 2 = 536 870 908 + 0;
- 536 870 908 ÷ 2 = 268 435 454 + 0;
- 268 435 454 ÷ 2 = 134 217 727 + 0;
- 134 217 727 ÷ 2 = 67 108 863 + 1;
- 67 108 863 ÷ 2 = 33 554 431 + 1;
- 33 554 431 ÷ 2 = 16 777 215 + 1;
- 16 777 215 ÷ 2 = 8 388 607 + 1;
- 8 388 607 ÷ 2 = 4 194 303 + 1;
- 4 194 303 ÷ 2 = 2 097 151 + 1;
- 2 097 151 ÷ 2 = 1 048 575 + 1;
- 1 048 575 ÷ 2 = 524 287 + 1;
- 524 287 ÷ 2 = 262 143 + 1;
- 262 143 ÷ 2 = 131 071 + 1;
- 131 071 ÷ 2 = 65 535 + 1;
- 65 535 ÷ 2 = 32 767 + 1;
- 32 767 ÷ 2 = 16 383 + 1;
- 16 383 ÷ 2 = 8 191 + 1;
- 8 191 ÷ 2 = 4 095 + 1;
- 4 095 ÷ 2 = 2 047 + 1;
- 2 047 ÷ 2 = 1 023 + 1;
- 1 023 ÷ 2 = 511 + 1;
- 511 ÷ 2 = 255 + 1;
- 255 ÷ 2 = 127 + 1;
- 127 ÷ 2 = 63 + 1;
- 63 ÷ 2 = 31 + 1;
- 31 ÷ 2 = 15 + 1;
- 15 ÷ 2 = 7 + 1;
- 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.
9 223 371 976 725 202 943(10) = 111 1111 1111 1111 1111 1111 1111 0001 1111 1111 1111 1111 1000 0111 1111 1111(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 63.
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, 63,
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:
9 223 371 976 725 202 943(10) = 0111 1111 1111 1111 1111 1111 1111 0001 1111 1111 1111 1111 1000 0111 1111 1111
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 -9 223 371 976 725 202 943(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
-9 223 371 976 725 202 943(10) = 1111 1111 1111 1111 1111 1111 1111 0001 1111 1111 1111 1111 1000 0111 1111 1111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.