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;
- 420 007 ÷ 2 = 210 003 + 1;
- 210 003 ÷ 2 = 105 001 + 1;
- 105 001 ÷ 2 = 52 500 + 1;
- 52 500 ÷ 2 = 26 250 + 0;
- 26 250 ÷ 2 = 13 125 + 0;
- 13 125 ÷ 2 = 6 562 + 1;
- 6 562 ÷ 2 = 3 281 + 0;
- 3 281 ÷ 2 = 1 640 + 1;
- 1 640 ÷ 2 = 820 + 0;
- 820 ÷ 2 = 410 + 0;
- 410 ÷ 2 = 205 + 0;
- 205 ÷ 2 = 102 + 1;
- 102 ÷ 2 = 51 + 0;
- 51 ÷ 2 = 25 + 1;
- 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.
420 007(10) = 110 0110 1000 1010 0111(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 19.
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, 19,
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 420 007(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:
420 007(10) = 0000 0000 0000 0110 0110 1000 1010 0111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.