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;
- 65 101 111 067 ÷ 2 = 32 550 555 533 + 1;
- 32 550 555 533 ÷ 2 = 16 275 277 766 + 1;
- 16 275 277 766 ÷ 2 = 8 137 638 883 + 0;
- 8 137 638 883 ÷ 2 = 4 068 819 441 + 1;
- 4 068 819 441 ÷ 2 = 2 034 409 720 + 1;
- 2 034 409 720 ÷ 2 = 1 017 204 860 + 0;
- 1 017 204 860 ÷ 2 = 508 602 430 + 0;
- 508 602 430 ÷ 2 = 254 301 215 + 0;
- 254 301 215 ÷ 2 = 127 150 607 + 1;
- 127 150 607 ÷ 2 = 63 575 303 + 1;
- 63 575 303 ÷ 2 = 31 787 651 + 1;
- 31 787 651 ÷ 2 = 15 893 825 + 1;
- 15 893 825 ÷ 2 = 7 946 912 + 1;
- 7 946 912 ÷ 2 = 3 973 456 + 0;
- 3 973 456 ÷ 2 = 1 986 728 + 0;
- 1 986 728 ÷ 2 = 993 364 + 0;
- 993 364 ÷ 2 = 496 682 + 0;
- 496 682 ÷ 2 = 248 341 + 0;
- 248 341 ÷ 2 = 124 170 + 1;
- 124 170 ÷ 2 = 62 085 + 0;
- 62 085 ÷ 2 = 31 042 + 1;
- 31 042 ÷ 2 = 15 521 + 0;
- 15 521 ÷ 2 = 7 760 + 1;
- 7 760 ÷ 2 = 3 880 + 0;
- 3 880 ÷ 2 = 1 940 + 0;
- 1 940 ÷ 2 = 970 + 0;
- 970 ÷ 2 = 485 + 0;
- 485 ÷ 2 = 242 + 1;
- 242 ÷ 2 = 121 + 0;
- 121 ÷ 2 = 60 + 1;
- 60 ÷ 2 = 30 + 0;
- 30 ÷ 2 = 15 + 0;
- 15 ÷ 2 = 7 + 1;
- 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.
65 101 111 067(10) = 1111 0010 1000 0101 0100 0001 1111 0001 1011(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 36.
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, 36,
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.
65 101 111 067(10) = 0000 0000 0000 0000 0000 0000 0000 1111 0010 1000 0101 0100 0001 1111 0001 1011
6. Get the negative integer number representation:
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.
-65 101 111 067(10) = !(0000 0000 0000 0000 0000 0000 0000 1111 0010 1000 0101 0100 0001 1111 0001 1011)
Number -65 101 111 067(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:
-65 101 111 067(10) = 1111 1111 1111 1111 1111 1111 1111 0000 1101 0111 1010 1011 1110 0000 1110 0100
Spaces were used to group digits: for binary, by 4, for decimal, by 3.