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;
- 11 111 011 005 ÷ 2 = 5 555 505 502 + 1;
- 5 555 505 502 ÷ 2 = 2 777 752 751 + 0;
- 2 777 752 751 ÷ 2 = 1 388 876 375 + 1;
- 1 388 876 375 ÷ 2 = 694 438 187 + 1;
- 694 438 187 ÷ 2 = 347 219 093 + 1;
- 347 219 093 ÷ 2 = 173 609 546 + 1;
- 173 609 546 ÷ 2 = 86 804 773 + 0;
- 86 804 773 ÷ 2 = 43 402 386 + 1;
- 43 402 386 ÷ 2 = 21 701 193 + 0;
- 21 701 193 ÷ 2 = 10 850 596 + 1;
- 10 850 596 ÷ 2 = 5 425 298 + 0;
- 5 425 298 ÷ 2 = 2 712 649 + 0;
- 2 712 649 ÷ 2 = 1 356 324 + 1;
- 1 356 324 ÷ 2 = 678 162 + 0;
- 678 162 ÷ 2 = 339 081 + 0;
- 339 081 ÷ 2 = 169 540 + 1;
- 169 540 ÷ 2 = 84 770 + 0;
- 84 770 ÷ 2 = 42 385 + 0;
- 42 385 ÷ 2 = 21 192 + 1;
- 21 192 ÷ 2 = 10 596 + 0;
- 10 596 ÷ 2 = 5 298 + 0;
- 5 298 ÷ 2 = 2 649 + 0;
- 2 649 ÷ 2 = 1 324 + 1;
- 1 324 ÷ 2 = 662 + 0;
- 662 ÷ 2 = 331 + 0;
- 331 ÷ 2 = 165 + 1;
- 165 ÷ 2 = 82 + 1;
- 82 ÷ 2 = 41 + 0;
- 41 ÷ 2 = 20 + 1;
- 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.
11 111 011 005(10) = 10 1001 0110 0100 0100 1001 0010 1011 1101(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 34.
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, 34,
3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)
=== is: 64.
5. Get the positive binary computer representation on 64 bits (8 Bytes):
If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 64.
11 111 011 005(10) = 0000 0000 0000 0000 0000 0000 0000 0010 1001 0110 0100 0100 1001 0010 1011 1101
6. Get the negative integer number representation. Part 1:
To write the negative integer number on 64 bits (8 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.
!(0000 0000 0000 0000 0000 0000 0000 0010 1001 0110 0100 0100 1001 0010 1011 1101)
= 1111 1111 1111 1111 1111 1111 1111 1101 0110 1001 1011 1011 0110 1101 0100 0010
7. Get the negative integer number representation. Part 2:
To write the negative integer number on 64 bits (8 Bytes),
as a signed binary in two's complement representation,
add 1 to the number calculated above
1111 1111 1111 1111 1111 1111 1111 1101 0110 1001 1011 1011 0110 1101 0100 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):
-11 111 011 005 =
1111 1111 1111 1111 1111 1111 1111 1101 0110 1001 1011 1011 0110 1101 0100 0010 + 1
Number -11 111 011 005(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:
-11 111 011 005(10) = 1111 1111 1111 1111 1111 1111 1111 1101 0110 1001 1011 1011 0110 1101 0100 0011
Spaces were used to group digits: for binary, by 4, for decimal, by 3.