2. 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;
- 987 654 297 ÷ 2 = 493 827 148 + 1;
- 493 827 148 ÷ 2 = 246 913 574 + 0;
- 246 913 574 ÷ 2 = 123 456 787 + 0;
- 123 456 787 ÷ 2 = 61 728 393 + 1;
- 61 728 393 ÷ 2 = 30 864 196 + 1;
- 30 864 196 ÷ 2 = 15 432 098 + 0;
- 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;
3. Construct the base 2 representation of the positive number:
Take all the remainders starting from the bottom of the list constructed above.
987 654 297(10) = 11 1010 1101 1110 0110 1000 1001 1001(2)
4. 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) 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, 30,
3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)
=== is: 32.
5. 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.
987 654 297(10) = 0011 1010 1101 1110 0110 1000 1001 1001
6. Get the negative integer number representation:
To write the negative integer number on 32 bits (4 Bytes),
as a signed binary in one's complement representation,
... replace all the bits on 0 with 1s and all the bits set on 1 with 0s.
Reverse the digits, flip the digits:
Replace the bits set on 0 with 1s and the bits set on 1 with 0s.
-987 654 297(10) = !(0011 1010 1101 1110 0110 1000 1001 1001)
Number -987 654 297(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in one's complement representation:
-987 654 297(10) = 1100 0101 0010 0001 1001 0111 0110 0110
Spaces were used to group digits: for binary, by 4, for decimal, by 3.