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;
- 1 006 632 916 ÷ 2 = 503 316 458 + 0;
- 503 316 458 ÷ 2 = 251 658 229 + 0;
- 251 658 229 ÷ 2 = 125 829 114 + 1;
- 125 829 114 ÷ 2 = 62 914 557 + 0;
- 62 914 557 ÷ 2 = 31 457 278 + 1;
- 31 457 278 ÷ 2 = 15 728 639 + 0;
- 15 728 639 ÷ 2 = 7 864 319 + 1;
- 7 864 319 ÷ 2 = 3 932 159 + 1;
- 3 932 159 ÷ 2 = 1 966 079 + 1;
- 1 966 079 ÷ 2 = 983 039 + 1;
- 983 039 ÷ 2 = 491 519 + 1;
- 491 519 ÷ 2 = 245 759 + 1;
- 245 759 ÷ 2 = 122 879 + 1;
- 122 879 ÷ 2 = 61 439 + 1;
- 61 439 ÷ 2 = 30 719 + 1;
- 30 719 ÷ 2 = 15 359 + 1;
- 15 359 ÷ 2 = 7 679 + 1;
- 7 679 ÷ 2 = 3 839 + 1;
- 3 839 ÷ 2 = 1 919 + 1;
- 1 919 ÷ 2 = 959 + 1;
- 959 ÷ 2 = 479 + 1;
- 479 ÷ 2 = 239 + 1;
- 239 ÷ 2 = 119 + 1;
- 119 ÷ 2 = 59 + 1;
- 59 ÷ 2 = 29 + 1;
- 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.
1 006 632 916(10) = 11 1011 1111 1111 1111 1111 1101 0100(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.
1 006 632 916(10) = 0011 1011 1111 1111 1111 1111 1101 0100
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 1011 1111 1111 1111 1111 1101 0100)
= 1100 0100 0000 0000 0000 0000 0010 1011
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 0100 0000 0000 0000 0000 0010 1011
(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):
-1 006 632 916 =
1100 0100 0000 0000 0000 0000 0010 1011 + 1
Number -1 006 632 916(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:
-1 006 632 916(10) = 1100 0100 0000 0000 0000 0000 0010 1100
Spaces were used to group digits: for binary, by 4, for decimal, by 3.