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;
- 9 007 199 254 741 037 ÷ 2 = 4 503 599 627 370 518 + 1;
- 4 503 599 627 370 518 ÷ 2 = 2 251 799 813 685 259 + 0;
- 2 251 799 813 685 259 ÷ 2 = 1 125 899 906 842 629 + 1;
- 1 125 899 906 842 629 ÷ 2 = 562 949 953 421 314 + 1;
- 562 949 953 421 314 ÷ 2 = 281 474 976 710 657 + 0;
- 281 474 976 710 657 ÷ 2 = 140 737 488 355 328 + 1;
- 140 737 488 355 328 ÷ 2 = 70 368 744 177 664 + 0;
- 70 368 744 177 664 ÷ 2 = 35 184 372 088 832 + 0;
- 35 184 372 088 832 ÷ 2 = 17 592 186 044 416 + 0;
- 17 592 186 044 416 ÷ 2 = 8 796 093 022 208 + 0;
- 8 796 093 022 208 ÷ 2 = 4 398 046 511 104 + 0;
- 4 398 046 511 104 ÷ 2 = 2 199 023 255 552 + 0;
- 2 199 023 255 552 ÷ 2 = 1 099 511 627 776 + 0;
- 1 099 511 627 776 ÷ 2 = 549 755 813 888 + 0;
- 549 755 813 888 ÷ 2 = 274 877 906 944 + 0;
- 274 877 906 944 ÷ 2 = 137 438 953 472 + 0;
- 137 438 953 472 ÷ 2 = 68 719 476 736 + 0;
- 68 719 476 736 ÷ 2 = 34 359 738 368 + 0;
- 34 359 738 368 ÷ 2 = 17 179 869 184 + 0;
- 17 179 869 184 ÷ 2 = 8 589 934 592 + 0;
- 8 589 934 592 ÷ 2 = 4 294 967 296 + 0;
- 4 294 967 296 ÷ 2 = 2 147 483 648 + 0;
- 2 147 483 648 ÷ 2 = 1 073 741 824 + 0;
- 1 073 741 824 ÷ 2 = 536 870 912 + 0;
- 536 870 912 ÷ 2 = 268 435 456 + 0;
- 268 435 456 ÷ 2 = 134 217 728 + 0;
- 134 217 728 ÷ 2 = 67 108 864 + 0;
- 67 108 864 ÷ 2 = 33 554 432 + 0;
- 33 554 432 ÷ 2 = 16 777 216 + 0;
- 16 777 216 ÷ 2 = 8 388 608 + 0;
- 8 388 608 ÷ 2 = 4 194 304 + 0;
- 4 194 304 ÷ 2 = 2 097 152 + 0;
- 2 097 152 ÷ 2 = 1 048 576 + 0;
- 1 048 576 ÷ 2 = 524 288 + 0;
- 524 288 ÷ 2 = 262 144 + 0;
- 262 144 ÷ 2 = 131 072 + 0;
- 131 072 ÷ 2 = 65 536 + 0;
- 65 536 ÷ 2 = 32 768 + 0;
- 32 768 ÷ 2 = 16 384 + 0;
- 16 384 ÷ 2 = 8 192 + 0;
- 8 192 ÷ 2 = 4 096 + 0;
- 4 096 ÷ 2 = 2 048 + 0;
- 2 048 ÷ 2 = 1 024 + 0;
- 1 024 ÷ 2 = 512 + 0;
- 512 ÷ 2 = 256 + 0;
- 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.
9 007 199 254 741 037(10) = 10 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1101(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 54.
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, 54,
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 9 007 199 254 741 037(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in one's complement representation:
9 007 199 254 741 037(10) = 0000 0000 0010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 1101
Spaces were used to group digits: for binary, by 4, for decimal, by 3.