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;
- 110 001 111 100 142 ÷ 2 = 55 000 555 550 071 + 0;
- 55 000 555 550 071 ÷ 2 = 27 500 277 775 035 + 1;
- 27 500 277 775 035 ÷ 2 = 13 750 138 887 517 + 1;
- 13 750 138 887 517 ÷ 2 = 6 875 069 443 758 + 1;
- 6 875 069 443 758 ÷ 2 = 3 437 534 721 879 + 0;
- 3 437 534 721 879 ÷ 2 = 1 718 767 360 939 + 1;
- 1 718 767 360 939 ÷ 2 = 859 383 680 469 + 1;
- 859 383 680 469 ÷ 2 = 429 691 840 234 + 1;
- 429 691 840 234 ÷ 2 = 214 845 920 117 + 0;
- 214 845 920 117 ÷ 2 = 107 422 960 058 + 1;
- 107 422 960 058 ÷ 2 = 53 711 480 029 + 0;
- 53 711 480 029 ÷ 2 = 26 855 740 014 + 1;
- 26 855 740 014 ÷ 2 = 13 427 870 007 + 0;
- 13 427 870 007 ÷ 2 = 6 713 935 003 + 1;
- 6 713 935 003 ÷ 2 = 3 356 967 501 + 1;
- 3 356 967 501 ÷ 2 = 1 678 483 750 + 1;
- 1 678 483 750 ÷ 2 = 839 241 875 + 0;
- 839 241 875 ÷ 2 = 419 620 937 + 1;
- 419 620 937 ÷ 2 = 209 810 468 + 1;
- 209 810 468 ÷ 2 = 104 905 234 + 0;
- 104 905 234 ÷ 2 = 52 452 617 + 0;
- 52 452 617 ÷ 2 = 26 226 308 + 1;
- 26 226 308 ÷ 2 = 13 113 154 + 0;
- 13 113 154 ÷ 2 = 6 556 577 + 0;
- 6 556 577 ÷ 2 = 3 278 288 + 1;
- 3 278 288 ÷ 2 = 1 639 144 + 0;
- 1 639 144 ÷ 2 = 819 572 + 0;
- 819 572 ÷ 2 = 409 786 + 0;
- 409 786 ÷ 2 = 204 893 + 0;
- 204 893 ÷ 2 = 102 446 + 1;
- 102 446 ÷ 2 = 51 223 + 0;
- 51 223 ÷ 2 = 25 611 + 1;
- 25 611 ÷ 2 = 12 805 + 1;
- 12 805 ÷ 2 = 6 402 + 1;
- 6 402 ÷ 2 = 3 201 + 0;
- 3 201 ÷ 2 = 1 600 + 1;
- 1 600 ÷ 2 = 800 + 0;
- 800 ÷ 2 = 400 + 0;
- 400 ÷ 2 = 200 + 0;
- 200 ÷ 2 = 100 + 0;
- 100 ÷ 2 = 50 + 0;
- 50 ÷ 2 = 25 + 0;
- 25 ÷ 2 = 12 + 1;
- 12 ÷ 2 = 6 + 0;
- 6 ÷ 2 = 3 + 0;
- 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.
110 001 111 100 142(10) = 110 0100 0000 1011 1010 0001 0010 0110 1110 1010 1110 1110(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 47.
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, 47,
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 110 001 111 100 142(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
110 001 111 100 142(10) = 0000 0000 0000 0000 0110 0100 0000 1011 1010 0001 0010 0110 1110 1010 1110 1110
Spaces were used to group digits: for binary, by 4, for decimal, by 3.