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;
- 823 731 420 ÷ 2 = 411 865 710 + 0;
- 411 865 710 ÷ 2 = 205 932 855 + 0;
- 205 932 855 ÷ 2 = 102 966 427 + 1;
- 102 966 427 ÷ 2 = 51 483 213 + 1;
- 51 483 213 ÷ 2 = 25 741 606 + 1;
- 25 741 606 ÷ 2 = 12 870 803 + 0;
- 12 870 803 ÷ 2 = 6 435 401 + 1;
- 6 435 401 ÷ 2 = 3 217 700 + 1;
- 3 217 700 ÷ 2 = 1 608 850 + 0;
- 1 608 850 ÷ 2 = 804 425 + 0;
- 804 425 ÷ 2 = 402 212 + 1;
- 402 212 ÷ 2 = 201 106 + 0;
- 201 106 ÷ 2 = 100 553 + 0;
- 100 553 ÷ 2 = 50 276 + 1;
- 50 276 ÷ 2 = 25 138 + 0;
- 25 138 ÷ 2 = 12 569 + 0;
- 12 569 ÷ 2 = 6 284 + 1;
- 6 284 ÷ 2 = 3 142 + 0;
- 3 142 ÷ 2 = 1 571 + 0;
- 1 571 ÷ 2 = 785 + 1;
- 785 ÷ 2 = 392 + 1;
- 392 ÷ 2 = 196 + 0;
- 196 ÷ 2 = 98 + 0;
- 98 ÷ 2 = 49 + 0;
- 49 ÷ 2 = 24 + 1;
- 24 ÷ 2 = 12 + 0;
- 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.
823 731 420(10) = 11 0001 0001 1001 0010 0100 1101 1100(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 30.
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, 30,
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 823 731 420(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
823 731 420(10) = 0011 0001 0001 1001 0010 0100 1101 1100
Spaces were used to group digits: for binary, by 4, for decimal, by 3.