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;
- 682 732 848 ÷ 2 = 341 366 424 + 0;
- 341 366 424 ÷ 2 = 170 683 212 + 0;
- 170 683 212 ÷ 2 = 85 341 606 + 0;
- 85 341 606 ÷ 2 = 42 670 803 + 0;
- 42 670 803 ÷ 2 = 21 335 401 + 1;
- 21 335 401 ÷ 2 = 10 667 700 + 1;
- 10 667 700 ÷ 2 = 5 333 850 + 0;
- 5 333 850 ÷ 2 = 2 666 925 + 0;
- 2 666 925 ÷ 2 = 1 333 462 + 1;
- 1 333 462 ÷ 2 = 666 731 + 0;
- 666 731 ÷ 2 = 333 365 + 1;
- 333 365 ÷ 2 = 166 682 + 1;
- 166 682 ÷ 2 = 83 341 + 0;
- 83 341 ÷ 2 = 41 670 + 1;
- 41 670 ÷ 2 = 20 835 + 0;
- 20 835 ÷ 2 = 10 417 + 1;
- 10 417 ÷ 2 = 5 208 + 1;
- 5 208 ÷ 2 = 2 604 + 0;
- 2 604 ÷ 2 = 1 302 + 0;
- 1 302 ÷ 2 = 651 + 0;
- 651 ÷ 2 = 325 + 1;
- 325 ÷ 2 = 162 + 1;
- 162 ÷ 2 = 81 + 0;
- 81 ÷ 2 = 40 + 1;
- 40 ÷ 2 = 20 + 0;
- 20 ÷ 2 = 10 + 0;
- 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.
682 732 848(10) = 10 1000 1011 0001 1010 1101 0011 0000(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.
682 732 848(10) = 0010 1000 1011 0001 1010 1101 0011 0000
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.
!(0010 1000 1011 0001 1010 1101 0011 0000)
= 1101 0111 0100 1110 0101 0010 1100 1111
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
1101 0111 0100 1110 0101 0010 1100 1111
(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):
-682 732 848 =
1101 0111 0100 1110 0101 0010 1100 1111 + 1
Number -682 732 848(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:
-682 732 848(10) = 1101 0111 0100 1110 0101 0010 1101 0000
Spaces were used to group digits: for binary, by 4, for decimal, by 3.