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;
- 834 407 174 ÷ 2 = 417 203 587 + 0;
- 417 203 587 ÷ 2 = 208 601 793 + 1;
- 208 601 793 ÷ 2 = 104 300 896 + 1;
- 104 300 896 ÷ 2 = 52 150 448 + 0;
- 52 150 448 ÷ 2 = 26 075 224 + 0;
- 26 075 224 ÷ 2 = 13 037 612 + 0;
- 13 037 612 ÷ 2 = 6 518 806 + 0;
- 6 518 806 ÷ 2 = 3 259 403 + 0;
- 3 259 403 ÷ 2 = 1 629 701 + 1;
- 1 629 701 ÷ 2 = 814 850 + 1;
- 814 850 ÷ 2 = 407 425 + 0;
- 407 425 ÷ 2 = 203 712 + 1;
- 203 712 ÷ 2 = 101 856 + 0;
- 101 856 ÷ 2 = 50 928 + 0;
- 50 928 ÷ 2 = 25 464 + 0;
- 25 464 ÷ 2 = 12 732 + 0;
- 12 732 ÷ 2 = 6 366 + 0;
- 6 366 ÷ 2 = 3 183 + 0;
- 3 183 ÷ 2 = 1 591 + 1;
- 1 591 ÷ 2 = 795 + 1;
- 795 ÷ 2 = 397 + 1;
- 397 ÷ 2 = 198 + 1;
- 198 ÷ 2 = 99 + 0;
- 99 ÷ 2 = 49 + 1;
- 49 ÷ 2 = 24 + 1;
- 24 ÷ 2 = 12 + 0;
- 12 ÷ 2 = 6 + 0;
- 6 ÷ 2 = 3 + 0;
- 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.
834 407 174(10) = 11 0001 1011 1100 0000 1011 0000 0110(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.
834 407 174(10) = 0011 0001 1011 1100 0000 1011 0000 0110
6. Get the negative integer number representation:
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.
-834 407 174(10) = !(0011 0001 1011 1100 0000 1011 0000 0110)
Number -834 407 174(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in one's complement representation:
-834 407 174(10) = 1100 1110 0100 0011 1111 0100 1111 1001
Spaces were used to group digits: for binary, by 4, for decimal, by 3.