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;
- 105 031 943 ÷ 2 = 52 515 971 + 1;
- 52 515 971 ÷ 2 = 26 257 985 + 1;
- 26 257 985 ÷ 2 = 13 128 992 + 1;
- 13 128 992 ÷ 2 = 6 564 496 + 0;
- 6 564 496 ÷ 2 = 3 282 248 + 0;
- 3 282 248 ÷ 2 = 1 641 124 + 0;
- 1 641 124 ÷ 2 = 820 562 + 0;
- 820 562 ÷ 2 = 410 281 + 0;
- 410 281 ÷ 2 = 205 140 + 1;
- 205 140 ÷ 2 = 102 570 + 0;
- 102 570 ÷ 2 = 51 285 + 0;
- 51 285 ÷ 2 = 25 642 + 1;
- 25 642 ÷ 2 = 12 821 + 0;
- 12 821 ÷ 2 = 6 410 + 1;
- 6 410 ÷ 2 = 3 205 + 0;
- 3 205 ÷ 2 = 1 602 + 1;
- 1 602 ÷ 2 = 801 + 0;
- 801 ÷ 2 = 400 + 1;
- 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.
105 031 943(10) = 110 0100 0010 1010 1001 0000 0111(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 27.
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, 27,
3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)
=== is: 32.
4. Get the positive binary computer representation on 32 bits (4 Bytes):
If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 32:
Number 105 031 943(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
105 031 943(10) = 0000 0110 0100 0010 1010 1001 0000 0111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.