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;
- 1 598 898 269 ÷ 2 = 799 449 134 + 1;
- 799 449 134 ÷ 2 = 399 724 567 + 0;
- 399 724 567 ÷ 2 = 199 862 283 + 1;
- 199 862 283 ÷ 2 = 99 931 141 + 1;
- 99 931 141 ÷ 2 = 49 965 570 + 1;
- 49 965 570 ÷ 2 = 24 982 785 + 0;
- 24 982 785 ÷ 2 = 12 491 392 + 1;
- 12 491 392 ÷ 2 = 6 245 696 + 0;
- 6 245 696 ÷ 2 = 3 122 848 + 0;
- 3 122 848 ÷ 2 = 1 561 424 + 0;
- 1 561 424 ÷ 2 = 780 712 + 0;
- 780 712 ÷ 2 = 390 356 + 0;
- 390 356 ÷ 2 = 195 178 + 0;
- 195 178 ÷ 2 = 97 589 + 0;
- 97 589 ÷ 2 = 48 794 + 1;
- 48 794 ÷ 2 = 24 397 + 0;
- 24 397 ÷ 2 = 12 198 + 1;
- 12 198 ÷ 2 = 6 099 + 0;
- 6 099 ÷ 2 = 3 049 + 1;
- 3 049 ÷ 2 = 1 524 + 1;
- 1 524 ÷ 2 = 762 + 0;
- 762 ÷ 2 = 381 + 0;
- 381 ÷ 2 = 190 + 1;
- 190 ÷ 2 = 95 + 0;
- 95 ÷ 2 = 47 + 1;
- 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.
1 598 898 269(10) = 101 1111 0100 1101 0100 0000 0101 1101(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.
1 598 898 269(10) = 0101 1111 0100 1101 0100 0000 0101 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.
!(0101 1111 0100 1101 0100 0000 0101 1101)
= 1010 0000 1011 0010 1011 1111 1010 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
1010 0000 1011 0010 1011 1111 1010 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):
-1 598 898 269 =
1010 0000 1011 0010 1011 1111 1010 0010 + 1
Number -1 598 898 269(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:
-1 598 898 269(10) = 1010 0000 1011 0010 1011 1111 1010 0011
Spaces were used to group digits: for binary, by 4, for decimal, by 3.