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;
- 2 147 221 290 ÷ 2 = 1 073 610 645 + 0;
- 1 073 610 645 ÷ 2 = 536 805 322 + 1;
- 536 805 322 ÷ 2 = 268 402 661 + 0;
- 268 402 661 ÷ 2 = 134 201 330 + 1;
- 134 201 330 ÷ 2 = 67 100 665 + 0;
- 67 100 665 ÷ 2 = 33 550 332 + 1;
- 33 550 332 ÷ 2 = 16 775 166 + 0;
- 16 775 166 ÷ 2 = 8 387 583 + 0;
- 8 387 583 ÷ 2 = 4 193 791 + 1;
- 4 193 791 ÷ 2 = 2 096 895 + 1;
- 2 096 895 ÷ 2 = 1 048 447 + 1;
- 1 048 447 ÷ 2 = 524 223 + 1;
- 524 223 ÷ 2 = 262 111 + 1;
- 262 111 ÷ 2 = 131 055 + 1;
- 131 055 ÷ 2 = 65 527 + 1;
- 65 527 ÷ 2 = 32 763 + 1;
- 32 763 ÷ 2 = 16 381 + 1;
- 16 381 ÷ 2 = 8 190 + 1;
- 8 190 ÷ 2 = 4 095 + 0;
- 4 095 ÷ 2 = 2 047 + 1;
- 2 047 ÷ 2 = 1 023 + 1;
- 1 023 ÷ 2 = 511 + 1;
- 511 ÷ 2 = 255 + 1;
- 255 ÷ 2 = 127 + 1;
- 127 ÷ 2 = 63 + 1;
- 63 ÷ 2 = 31 + 1;
- 31 ÷ 2 = 15 + 1;
- 15 ÷ 2 = 7 + 1;
- 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.
2 147 221 290(10) = 111 1111 1111 1011 1111 1111 0010 1010(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 31.
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, 31,
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.
2 147 221 290(10) = 0111 1111 1111 1011 1111 1111 0010 1010
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.
-2 147 221 290(10) = !(0111 1111 1111 1011 1111 1111 0010 1010)
Number -2 147 221 290(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:
-2 147 221 290(10) = 1000 0000 0000 0100 0000 0000 1101 0101
Spaces were used to group digits: for binary, by 4, for decimal, by 3.