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;
- 4 623 156 123 728 347 143 ÷ 2 = 2 311 578 061 864 173 571 + 1;
- 2 311 578 061 864 173 571 ÷ 2 = 1 155 789 030 932 086 785 + 1;
- 1 155 789 030 932 086 785 ÷ 2 = 577 894 515 466 043 392 + 1;
- 577 894 515 466 043 392 ÷ 2 = 288 947 257 733 021 696 + 0;
- 288 947 257 733 021 696 ÷ 2 = 144 473 628 866 510 848 + 0;
- 144 473 628 866 510 848 ÷ 2 = 72 236 814 433 255 424 + 0;
- 72 236 814 433 255 424 ÷ 2 = 36 118 407 216 627 712 + 0;
- 36 118 407 216 627 712 ÷ 2 = 18 059 203 608 313 856 + 0;
- 18 059 203 608 313 856 ÷ 2 = 9 029 601 804 156 928 + 0;
- 9 029 601 804 156 928 ÷ 2 = 4 514 800 902 078 464 + 0;
- 4 514 800 902 078 464 ÷ 2 = 2 257 400 451 039 232 + 0;
- 2 257 400 451 039 232 ÷ 2 = 1 128 700 225 519 616 + 0;
- 1 128 700 225 519 616 ÷ 2 = 564 350 112 759 808 + 0;
- 564 350 112 759 808 ÷ 2 = 282 175 056 379 904 + 0;
- 282 175 056 379 904 ÷ 2 = 141 087 528 189 952 + 0;
- 141 087 528 189 952 ÷ 2 = 70 543 764 094 976 + 0;
- 70 543 764 094 976 ÷ 2 = 35 271 882 047 488 + 0;
- 35 271 882 047 488 ÷ 2 = 17 635 941 023 744 + 0;
- 17 635 941 023 744 ÷ 2 = 8 817 970 511 872 + 0;
- 8 817 970 511 872 ÷ 2 = 4 408 985 255 936 + 0;
- 4 408 985 255 936 ÷ 2 = 2 204 492 627 968 + 0;
- 2 204 492 627 968 ÷ 2 = 1 102 246 313 984 + 0;
- 1 102 246 313 984 ÷ 2 = 551 123 156 992 + 0;
- 551 123 156 992 ÷ 2 = 275 561 578 496 + 0;
- 275 561 578 496 ÷ 2 = 137 780 789 248 + 0;
- 137 780 789 248 ÷ 2 = 68 890 394 624 + 0;
- 68 890 394 624 ÷ 2 = 34 445 197 312 + 0;
- 34 445 197 312 ÷ 2 = 17 222 598 656 + 0;
- 17 222 598 656 ÷ 2 = 8 611 299 328 + 0;
- 8 611 299 328 ÷ 2 = 4 305 649 664 + 0;
- 4 305 649 664 ÷ 2 = 2 152 824 832 + 0;
- 2 152 824 832 ÷ 2 = 1 076 412 416 + 0;
- 1 076 412 416 ÷ 2 = 538 206 208 + 0;
- 538 206 208 ÷ 2 = 269 103 104 + 0;
- 269 103 104 ÷ 2 = 134 551 552 + 0;
- 134 551 552 ÷ 2 = 67 275 776 + 0;
- 67 275 776 ÷ 2 = 33 637 888 + 0;
- 33 637 888 ÷ 2 = 16 818 944 + 0;
- 16 818 944 ÷ 2 = 8 409 472 + 0;
- 8 409 472 ÷ 2 = 4 204 736 + 0;
- 4 204 736 ÷ 2 = 2 102 368 + 0;
- 2 102 368 ÷ 2 = 1 051 184 + 0;
- 1 051 184 ÷ 2 = 525 592 + 0;
- 525 592 ÷ 2 = 262 796 + 0;
- 262 796 ÷ 2 = 131 398 + 0;
- 131 398 ÷ 2 = 65 699 + 0;
- 65 699 ÷ 2 = 32 849 + 1;
- 32 849 ÷ 2 = 16 424 + 1;
- 16 424 ÷ 2 = 8 212 + 0;
- 8 212 ÷ 2 = 4 106 + 0;
- 4 106 ÷ 2 = 2 053 + 0;
- 2 053 ÷ 2 = 1 026 + 1;
- 1 026 ÷ 2 = 513 + 0;
- 513 ÷ 2 = 256 + 1;
- 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.
4 623 156 123 728 347 143(10) = 100 0000 0010 1000 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0111(2)
3. 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) 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, 63,
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 4 623 156 123 728 347 143(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:
4 623 156 123 728 347 143(10) = 0100 0000 0010 1000 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.