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;
- 32 845 053 ÷ 2 = 16 422 526 + 1;
- 16 422 526 ÷ 2 = 8 211 263 + 0;
- 8 211 263 ÷ 2 = 4 105 631 + 1;
- 4 105 631 ÷ 2 = 2 052 815 + 1;
- 2 052 815 ÷ 2 = 1 026 407 + 1;
- 1 026 407 ÷ 2 = 513 203 + 1;
- 513 203 ÷ 2 = 256 601 + 1;
- 256 601 ÷ 2 = 128 300 + 1;
- 128 300 ÷ 2 = 64 150 + 0;
- 64 150 ÷ 2 = 32 075 + 0;
- 32 075 ÷ 2 = 16 037 + 1;
- 16 037 ÷ 2 = 8 018 + 1;
- 8 018 ÷ 2 = 4 009 + 0;
- 4 009 ÷ 2 = 2 004 + 1;
- 2 004 ÷ 2 = 1 002 + 0;
- 1 002 ÷ 2 = 501 + 0;
- 501 ÷ 2 = 250 + 1;
- 250 ÷ 2 = 125 + 0;
- 125 ÷ 2 = 62 + 1;
- 62 ÷ 2 = 31 + 0;
- 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.
32 845 053(10) = 1 1111 0101 0010 1100 1111 1101(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 25.
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, 25,
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.
32 845 053(10) = 0000 0001 1111 0101 0010 1100 1111 1101
6. Get the negative integer number representation. Part 1:
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.
!(0000 0001 1111 0101 0010 1100 1111 1101)
= 1111 1110 0000 1010 1101 0011 0000 0010
7. Get the negative integer number representation. Part 2:
To write the negative integer number on 32 bits (4 Bytes),
as a signed binary in two's complement representation,
add 1 to the number calculated above
1111 1110 0000 1010 1101 0011 0000 0010
(to the signed binary in one's complement representation)
Binary addition carries on a value of 2:
0 + 0 = 0
0 + 1 = 1
1 + 1 = 10
1 + 10 = 11
1 + 11 = 100
Add 1 to the number calculated above
(to the signed binary number in one's complement representation):
-32 845 053 =
1111 1110 0000 1010 1101 0011 0000 0010 + 1
Number -32 845 053(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:
-32 845 053(10) = 1111 1110 0000 1010 1101 0011 0000 0011
Spaces were used to group digits: for binary, by 4, for decimal, by 3.