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;
- 9 123 000 000 000 000 017 ÷ 2 = 4 561 500 000 000 000 008 + 1;
- 4 561 500 000 000 000 008 ÷ 2 = 2 280 750 000 000 000 004 + 0;
- 2 280 750 000 000 000 004 ÷ 2 = 1 140 375 000 000 000 002 + 0;
- 1 140 375 000 000 000 002 ÷ 2 = 570 187 500 000 000 001 + 0;
- 570 187 500 000 000 001 ÷ 2 = 285 093 750 000 000 000 + 1;
- 285 093 750 000 000 000 ÷ 2 = 142 546 875 000 000 000 + 0;
- 142 546 875 000 000 000 ÷ 2 = 71 273 437 500 000 000 + 0;
- 71 273 437 500 000 000 ÷ 2 = 35 636 718 750 000 000 + 0;
- 35 636 718 750 000 000 ÷ 2 = 17 818 359 375 000 000 + 0;
- 17 818 359 375 000 000 ÷ 2 = 8 909 179 687 500 000 + 0;
- 8 909 179 687 500 000 ÷ 2 = 4 454 589 843 750 000 + 0;
- 4 454 589 843 750 000 ÷ 2 = 2 227 294 921 875 000 + 0;
- 2 227 294 921 875 000 ÷ 2 = 1 113 647 460 937 500 + 0;
- 1 113 647 460 937 500 ÷ 2 = 556 823 730 468 750 + 0;
- 556 823 730 468 750 ÷ 2 = 278 411 865 234 375 + 0;
- 278 411 865 234 375 ÷ 2 = 139 205 932 617 187 + 1;
- 139 205 932 617 187 ÷ 2 = 69 602 966 308 593 + 1;
- 69 602 966 308 593 ÷ 2 = 34 801 483 154 296 + 1;
- 34 801 483 154 296 ÷ 2 = 17 400 741 577 148 + 0;
- 17 400 741 577 148 ÷ 2 = 8 700 370 788 574 + 0;
- 8 700 370 788 574 ÷ 2 = 4 350 185 394 287 + 0;
- 4 350 185 394 287 ÷ 2 = 2 175 092 697 143 + 1;
- 2 175 092 697 143 ÷ 2 = 1 087 546 348 571 + 1;
- 1 087 546 348 571 ÷ 2 = 543 773 174 285 + 1;
- 543 773 174 285 ÷ 2 = 271 886 587 142 + 1;
- 271 886 587 142 ÷ 2 = 135 943 293 571 + 0;
- 135 943 293 571 ÷ 2 = 67 971 646 785 + 1;
- 67 971 646 785 ÷ 2 = 33 985 823 392 + 1;
- 33 985 823 392 ÷ 2 = 16 992 911 696 + 0;
- 16 992 911 696 ÷ 2 = 8 496 455 848 + 0;
- 8 496 455 848 ÷ 2 = 4 248 227 924 + 0;
- 4 248 227 924 ÷ 2 = 2 124 113 962 + 0;
- 2 124 113 962 ÷ 2 = 1 062 056 981 + 0;
- 1 062 056 981 ÷ 2 = 531 028 490 + 1;
- 531 028 490 ÷ 2 = 265 514 245 + 0;
- 265 514 245 ÷ 2 = 132 757 122 + 1;
- 132 757 122 ÷ 2 = 66 378 561 + 0;
- 66 378 561 ÷ 2 = 33 189 280 + 1;
- 33 189 280 ÷ 2 = 16 594 640 + 0;
- 16 594 640 ÷ 2 = 8 297 320 + 0;
- 8 297 320 ÷ 2 = 4 148 660 + 0;
- 4 148 660 ÷ 2 = 2 074 330 + 0;
- 2 074 330 ÷ 2 = 1 037 165 + 0;
- 1 037 165 ÷ 2 = 518 582 + 1;
- 518 582 ÷ 2 = 259 291 + 0;
- 259 291 ÷ 2 = 129 645 + 1;
- 129 645 ÷ 2 = 64 822 + 1;
- 64 822 ÷ 2 = 32 411 + 0;
- 32 411 ÷ 2 = 16 205 + 1;
- 16 205 ÷ 2 = 8 102 + 1;
- 8 102 ÷ 2 = 4 051 + 0;
- 4 051 ÷ 2 = 2 025 + 1;
- 2 025 ÷ 2 = 1 012 + 1;
- 1 012 ÷ 2 = 506 + 0;
- 506 ÷ 2 = 253 + 0;
- 253 ÷ 2 = 126 + 1;
- 126 ÷ 2 = 63 + 0;
- 63 ÷ 2 = 31 + 1;
- 31 ÷ 2 = 15 + 1;
- 15 ÷ 2 = 7 + 1;
- 7 ÷ 2 = 3 + 1;
- 3 ÷ 2 = 1 + 1;
- 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.
9 123 000 000 000 000 017(10) = 111 1110 1001 1011 0110 1000 0010 1010 0000 1101 1110 0011 1000 0000 0001 0001(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) 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.
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 9 123 000 000 000 000 017(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
9 123 000 000 000 000 017(10) = 0111 1110 1001 1011 0110 1000 0010 1010 0000 1101 1110 0011 1000 0000 0001 0001
Spaces were used to group digits: for binary, by 4, for decimal, by 3.