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;
- 2 100 421 ÷ 2 = 1 050 210 + 1;
- 1 050 210 ÷ 2 = 525 105 + 0;
- 525 105 ÷ 2 = 262 552 + 1;
- 262 552 ÷ 2 = 131 276 + 0;
- 131 276 ÷ 2 = 65 638 + 0;
- 65 638 ÷ 2 = 32 819 + 0;
- 32 819 ÷ 2 = 16 409 + 1;
- 16 409 ÷ 2 = 8 204 + 1;
- 8 204 ÷ 2 = 4 102 + 0;
- 4 102 ÷ 2 = 2 051 + 0;
- 2 051 ÷ 2 = 1 025 + 1;
- 1 025 ÷ 2 = 512 + 1;
- 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.
2 100 421(10) = 10 0000 0000 1100 1100 0101(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 22.
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, 22,
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 2 100 421(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
2 100 421(10) = 0000 0000 0010 0000 0000 1100 1100 0101
Spaces were used to group digits: for binary, by 4, for decimal, by 3.