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;
- 1 591 903 041 ÷ 2 = 795 951 520 + 1;
- 795 951 520 ÷ 2 = 397 975 760 + 0;
- 397 975 760 ÷ 2 = 198 987 880 + 0;
- 198 987 880 ÷ 2 = 99 493 940 + 0;
- 99 493 940 ÷ 2 = 49 746 970 + 0;
- 49 746 970 ÷ 2 = 24 873 485 + 0;
- 24 873 485 ÷ 2 = 12 436 742 + 1;
- 12 436 742 ÷ 2 = 6 218 371 + 0;
- 6 218 371 ÷ 2 = 3 109 185 + 1;
- 3 109 185 ÷ 2 = 1 554 592 + 1;
- 1 554 592 ÷ 2 = 777 296 + 0;
- 777 296 ÷ 2 = 388 648 + 0;
- 388 648 ÷ 2 = 194 324 + 0;
- 194 324 ÷ 2 = 97 162 + 0;
- 97 162 ÷ 2 = 48 581 + 0;
- 48 581 ÷ 2 = 24 290 + 1;
- 24 290 ÷ 2 = 12 145 + 0;
- 12 145 ÷ 2 = 6 072 + 1;
- 6 072 ÷ 2 = 3 036 + 0;
- 3 036 ÷ 2 = 1 518 + 0;
- 1 518 ÷ 2 = 759 + 0;
- 759 ÷ 2 = 379 + 1;
- 379 ÷ 2 = 189 + 1;
- 189 ÷ 2 = 94 + 1;
- 94 ÷ 2 = 47 + 0;
- 47 ÷ 2 = 23 + 1;
- 23 ÷ 2 = 11 + 1;
- 11 ÷ 2 = 5 + 1;
- 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.
1 591 903 041(10) = 101 1110 1110 0010 1000 0011 0100 0001(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.
1 591 903 041(10) = 0101 1110 1110 0010 1000 0011 0100 0001
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.
!(0101 1110 1110 0010 1000 0011 0100 0001)
= 1010 0001 0001 1101 0111 1100 1011 1110
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
1010 0001 0001 1101 0111 1100 1011 1110
(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):
-1 591 903 041 =
1010 0001 0001 1101 0111 1100 1011 1110 + 1
Number -1 591 903 041(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:
-1 591 903 041(10) = 1010 0001 0001 1101 0111 1100 1011 1111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.