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;
- 1 010 100 109 961 ÷ 2 = 505 050 054 980 + 1;
- 505 050 054 980 ÷ 2 = 252 525 027 490 + 0;
- 252 525 027 490 ÷ 2 = 126 262 513 745 + 0;
- 126 262 513 745 ÷ 2 = 63 131 256 872 + 1;
- 63 131 256 872 ÷ 2 = 31 565 628 436 + 0;
- 31 565 628 436 ÷ 2 = 15 782 814 218 + 0;
- 15 782 814 218 ÷ 2 = 7 891 407 109 + 0;
- 7 891 407 109 ÷ 2 = 3 945 703 554 + 1;
- 3 945 703 554 ÷ 2 = 1 972 851 777 + 0;
- 1 972 851 777 ÷ 2 = 986 425 888 + 1;
- 986 425 888 ÷ 2 = 493 212 944 + 0;
- 493 212 944 ÷ 2 = 246 606 472 + 0;
- 246 606 472 ÷ 2 = 123 303 236 + 0;
- 123 303 236 ÷ 2 = 61 651 618 + 0;
- 61 651 618 ÷ 2 = 30 825 809 + 0;
- 30 825 809 ÷ 2 = 15 412 904 + 1;
- 15 412 904 ÷ 2 = 7 706 452 + 0;
- 7 706 452 ÷ 2 = 3 853 226 + 0;
- 3 853 226 ÷ 2 = 1 926 613 + 0;
- 1 926 613 ÷ 2 = 963 306 + 1;
- 963 306 ÷ 2 = 481 653 + 0;
- 481 653 ÷ 2 = 240 826 + 1;
- 240 826 ÷ 2 = 120 413 + 0;
- 120 413 ÷ 2 = 60 206 + 1;
- 60 206 ÷ 2 = 30 103 + 0;
- 30 103 ÷ 2 = 15 051 + 1;
- 15 051 ÷ 2 = 7 525 + 1;
- 7 525 ÷ 2 = 3 762 + 1;
- 3 762 ÷ 2 = 1 881 + 0;
- 1 881 ÷ 2 = 940 + 1;
- 940 ÷ 2 = 470 + 0;
- 470 ÷ 2 = 235 + 0;
- 235 ÷ 2 = 117 + 1;
- 117 ÷ 2 = 58 + 1;
- 58 ÷ 2 = 29 + 0;
- 29 ÷ 2 = 14 + 1;
- 14 ÷ 2 = 7 + 0;
- 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.
1 010 100 109 961(10) = 1110 1011 0010 1110 1010 1000 1000 0010 1000 1001(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 40.
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, 40,
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 1 010 100 109 961(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:
1 010 100 109 961(10) = 0000 0000 0000 0000 0000 0000 1110 1011 0010 1110 1010 1000 1000 0010 1000 1001
Spaces were used to group digits: for binary, by 4, for decimal, by 3.