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;
- 123 456 789 ÷ 2 = 61 728 394 + 1;
- 61 728 394 ÷ 2 = 30 864 197 + 0;
- 30 864 197 ÷ 2 = 15 432 098 + 1;
- 15 432 098 ÷ 2 = 7 716 049 + 0;
- 7 716 049 ÷ 2 = 3 858 024 + 1;
- 3 858 024 ÷ 2 = 1 929 012 + 0;
- 1 929 012 ÷ 2 = 964 506 + 0;
- 964 506 ÷ 2 = 482 253 + 0;
- 482 253 ÷ 2 = 241 126 + 1;
- 241 126 ÷ 2 = 120 563 + 0;
- 120 563 ÷ 2 = 60 281 + 1;
- 60 281 ÷ 2 = 30 140 + 1;
- 30 140 ÷ 2 = 15 070 + 0;
- 15 070 ÷ 2 = 7 535 + 0;
- 7 535 ÷ 2 = 3 767 + 1;
- 3 767 ÷ 2 = 1 883 + 1;
- 1 883 ÷ 2 = 941 + 1;
- 941 ÷ 2 = 470 + 1;
- 470 ÷ 2 = 235 + 0;
- 235 ÷ 2 = 117 + 1;
- 117 ÷ 2 = 58 + 1;
- 58 ÷ 2 = 29 + 0;
- 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.
123 456 789(10) = 111 0101 1011 1100 1101 0001 0101(2)
3. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 27.
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, 27,
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 123 456 789(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):
123 456 789(10) = 0000 0111 0101 1011 1100 1101 0001 0101
Spaces were used to group digits: for binary, by 4, for decimal, by 3.