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;
- 11 011 046 ÷ 2 = 5 505 523 + 0;
- 5 505 523 ÷ 2 = 2 752 761 + 1;
- 2 752 761 ÷ 2 = 1 376 380 + 1;
- 1 376 380 ÷ 2 = 688 190 + 0;
- 688 190 ÷ 2 = 344 095 + 0;
- 344 095 ÷ 2 = 172 047 + 1;
- 172 047 ÷ 2 = 86 023 + 1;
- 86 023 ÷ 2 = 43 011 + 1;
- 43 011 ÷ 2 = 21 505 + 1;
- 21 505 ÷ 2 = 10 752 + 1;
- 10 752 ÷ 2 = 5 376 + 0;
- 5 376 ÷ 2 = 2 688 + 0;
- 2 688 ÷ 2 = 1 344 + 0;
- 1 344 ÷ 2 = 672 + 0;
- 672 ÷ 2 = 336 + 0;
- 336 ÷ 2 = 168 + 0;
- 168 ÷ 2 = 84 + 0;
- 84 ÷ 2 = 42 + 0;
- 42 ÷ 2 = 21 + 0;
- 21 ÷ 2 = 10 + 1;
- 10 ÷ 2 = 5 + 0;
- 5 ÷ 2 = 2 + 1;
- 2 ÷ 2 = 1 + 0;
- 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.
11 011 046(10) = 1010 1000 0000 0011 1110 0110(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 24.
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, 24,
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.
11 011 046(10) = 0000 0000 1010 1000 0000 0011 1110 0110
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 0000 1010 1000 0000 0011 1110 0110)
= 1111 1111 0101 0111 1111 1100 0001 1001
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 1111 0101 0111 1111 1100 0001 1001
(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):
-11 011 046 =
1111 1111 0101 0111 1111 1100 0001 1001 + 1
Number -11 011 046(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:
-11 011 046(10) = 1111 1111 0101 0111 1111 1100 0001 1010
Spaces were used to group digits: for binary, by 4, for decimal, by 3.