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;
- 982 274 421 ÷ 2 = 491 137 210 + 1;
- 491 137 210 ÷ 2 = 245 568 605 + 0;
- 245 568 605 ÷ 2 = 122 784 302 + 1;
- 122 784 302 ÷ 2 = 61 392 151 + 0;
- 61 392 151 ÷ 2 = 30 696 075 + 1;
- 30 696 075 ÷ 2 = 15 348 037 + 1;
- 15 348 037 ÷ 2 = 7 674 018 + 1;
- 7 674 018 ÷ 2 = 3 837 009 + 0;
- 3 837 009 ÷ 2 = 1 918 504 + 1;
- 1 918 504 ÷ 2 = 959 252 + 0;
- 959 252 ÷ 2 = 479 626 + 0;
- 479 626 ÷ 2 = 239 813 + 0;
- 239 813 ÷ 2 = 119 906 + 1;
- 119 906 ÷ 2 = 59 953 + 0;
- 59 953 ÷ 2 = 29 976 + 1;
- 29 976 ÷ 2 = 14 988 + 0;
- 14 988 ÷ 2 = 7 494 + 0;
- 7 494 ÷ 2 = 3 747 + 0;
- 3 747 ÷ 2 = 1 873 + 1;
- 1 873 ÷ 2 = 936 + 1;
- 936 ÷ 2 = 468 + 0;
- 468 ÷ 2 = 234 + 0;
- 234 ÷ 2 = 117 + 0;
- 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.
982 274 421(10) = 11 1010 1000 1100 0101 0001 0111 0101(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.
982 274 421(10) = 0011 1010 1000 1100 0101 0001 0111 0101
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.
!(0011 1010 1000 1100 0101 0001 0111 0101)
= 1100 0101 0111 0011 1010 1110 1000 1010
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
1100 0101 0111 0011 1010 1110 1000 1010
(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):
-982 274 421 =
1100 0101 0111 0011 1010 1110 1000 1010 + 1
Number -982 274 421(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:
-982 274 421(10) = 1100 0101 0111 0011 1010 1110 1000 1011
Spaces were used to group digits: for binary, by 4, for decimal, by 3.