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;
- 3 134 249 702 ÷ 2 = 1 567 124 851 + 0;
- 1 567 124 851 ÷ 2 = 783 562 425 + 1;
- 783 562 425 ÷ 2 = 391 781 212 + 1;
- 391 781 212 ÷ 2 = 195 890 606 + 0;
- 195 890 606 ÷ 2 = 97 945 303 + 0;
- 97 945 303 ÷ 2 = 48 972 651 + 1;
- 48 972 651 ÷ 2 = 24 486 325 + 1;
- 24 486 325 ÷ 2 = 12 243 162 + 1;
- 12 243 162 ÷ 2 = 6 121 581 + 0;
- 6 121 581 ÷ 2 = 3 060 790 + 1;
- 3 060 790 ÷ 2 = 1 530 395 + 0;
- 1 530 395 ÷ 2 = 765 197 + 1;
- 765 197 ÷ 2 = 382 598 + 1;
- 382 598 ÷ 2 = 191 299 + 0;
- 191 299 ÷ 2 = 95 649 + 1;
- 95 649 ÷ 2 = 47 824 + 1;
- 47 824 ÷ 2 = 23 912 + 0;
- 23 912 ÷ 2 = 11 956 + 0;
- 11 956 ÷ 2 = 5 978 + 0;
- 5 978 ÷ 2 = 2 989 + 0;
- 2 989 ÷ 2 = 1 494 + 1;
- 1 494 ÷ 2 = 747 + 0;
- 747 ÷ 2 = 373 + 1;
- 373 ÷ 2 = 186 + 1;
- 186 ÷ 2 = 93 + 0;
- 93 ÷ 2 = 46 + 1;
- 46 ÷ 2 = 23 + 0;
- 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.
3 134 249 702(10) = 1011 1010 1101 0000 1101 1010 1110 0110(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 32.
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, 32,
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.
3 134 249 702(10) = 0000 0000 0000 0000 0000 0000 0000 0000 1011 1010 1101 0000 1101 1010 1110 0110
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.
-3 134 249 702(10) = !(0000 0000 0000 0000 0000 0000 0000 0000 1011 1010 1101 0000 1101 1010 1110 0110)
Number -3 134 249 702(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:
-3 134 249 702(10) = 1111 1111 1111 1111 1111 1111 1111 1111 0100 0101 0010 1111 0010 0101 0001 1001
Spaces were used to group digits: for binary, by 4, for decimal, by 3.