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;
- 23 042 019 ÷ 2 = 11 521 009 + 1;
- 11 521 009 ÷ 2 = 5 760 504 + 1;
- 5 760 504 ÷ 2 = 2 880 252 + 0;
- 2 880 252 ÷ 2 = 1 440 126 + 0;
- 1 440 126 ÷ 2 = 720 063 + 0;
- 720 063 ÷ 2 = 360 031 + 1;
- 360 031 ÷ 2 = 180 015 + 1;
- 180 015 ÷ 2 = 90 007 + 1;
- 90 007 ÷ 2 = 45 003 + 1;
- 45 003 ÷ 2 = 22 501 + 1;
- 22 501 ÷ 2 = 11 250 + 1;
- 11 250 ÷ 2 = 5 625 + 0;
- 5 625 ÷ 2 = 2 812 + 1;
- 2 812 ÷ 2 = 1 406 + 0;
- 1 406 ÷ 2 = 703 + 0;
- 703 ÷ 2 = 351 + 1;
- 351 ÷ 2 = 175 + 1;
- 175 ÷ 2 = 87 + 1;
- 87 ÷ 2 = 43 + 1;
- 43 ÷ 2 = 21 + 1;
- 21 ÷ 2 = 10 + 1;
- 10 ÷ 2 = 5 + 0;
- 5 ÷ 2 = 2 + 1;
- 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.
23 042 019(10) = 1 0101 1111 1001 0111 1110 0011(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 25.
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, 25,
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 23 042 019(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
23 042 019(10) = 0000 0001 0101 1111 1001 0111 1110 0011
Spaces were used to group digits: for binary, by 4, for decimal, by 3.