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;
- 4 000 000 000 ÷ 2 = 2 000 000 000 + 0;
- 2 000 000 000 ÷ 2 = 1 000 000 000 + 0;
- 1 000 000 000 ÷ 2 = 500 000 000 + 0;
- 500 000 000 ÷ 2 = 250 000 000 + 0;
- 250 000 000 ÷ 2 = 125 000 000 + 0;
- 125 000 000 ÷ 2 = 62 500 000 + 0;
- 62 500 000 ÷ 2 = 31 250 000 + 0;
- 31 250 000 ÷ 2 = 15 625 000 + 0;
- 15 625 000 ÷ 2 = 7 812 500 + 0;
- 7 812 500 ÷ 2 = 3 906 250 + 0;
- 3 906 250 ÷ 2 = 1 953 125 + 0;
- 1 953 125 ÷ 2 = 976 562 + 1;
- 976 562 ÷ 2 = 488 281 + 0;
- 488 281 ÷ 2 = 244 140 + 1;
- 244 140 ÷ 2 = 122 070 + 0;
- 122 070 ÷ 2 = 61 035 + 0;
- 61 035 ÷ 2 = 30 517 + 1;
- 30 517 ÷ 2 = 15 258 + 1;
- 15 258 ÷ 2 = 7 629 + 0;
- 7 629 ÷ 2 = 3 814 + 1;
- 3 814 ÷ 2 = 1 907 + 0;
- 1 907 ÷ 2 = 953 + 1;
- 953 ÷ 2 = 476 + 1;
- 476 ÷ 2 = 238 + 0;
- 238 ÷ 2 = 119 + 0;
- 119 ÷ 2 = 59 + 1;
- 59 ÷ 2 = 29 + 1;
- 29 ÷ 2 = 14 + 1;
- 14 ÷ 2 = 7 + 0;
- 7 ÷ 2 = 3 + 1;
- 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.
4 000 000 000(10) = 1110 1110 0110 1011 0010 1000 0000 0000(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 32.
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, 32,
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 4 000 000 000(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:
4 000 000 000(10) = 0000 0000 0000 0000 0000 0000 0000 0000 1110 1110 0110 1011 0010 1000 0000 0000
Spaces were used to group digits: for binary, by 4, for decimal, by 3.