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;
- 3 414 649 298 ÷ 2 = 1 707 324 649 + 0;
- 1 707 324 649 ÷ 2 = 853 662 324 + 1;
- 853 662 324 ÷ 2 = 426 831 162 + 0;
- 426 831 162 ÷ 2 = 213 415 581 + 0;
- 213 415 581 ÷ 2 = 106 707 790 + 1;
- 106 707 790 ÷ 2 = 53 353 895 + 0;
- 53 353 895 ÷ 2 = 26 676 947 + 1;
- 26 676 947 ÷ 2 = 13 338 473 + 1;
- 13 338 473 ÷ 2 = 6 669 236 + 1;
- 6 669 236 ÷ 2 = 3 334 618 + 0;
- 3 334 618 ÷ 2 = 1 667 309 + 0;
- 1 667 309 ÷ 2 = 833 654 + 1;
- 833 654 ÷ 2 = 416 827 + 0;
- 416 827 ÷ 2 = 208 413 + 1;
- 208 413 ÷ 2 = 104 206 + 1;
- 104 206 ÷ 2 = 52 103 + 0;
- 52 103 ÷ 2 = 26 051 + 1;
- 26 051 ÷ 2 = 13 025 + 1;
- 13 025 ÷ 2 = 6 512 + 1;
- 6 512 ÷ 2 = 3 256 + 0;
- 3 256 ÷ 2 = 1 628 + 0;
- 1 628 ÷ 2 = 814 + 0;
- 814 ÷ 2 = 407 + 0;
- 407 ÷ 2 = 203 + 1;
- 203 ÷ 2 = 101 + 1;
- 101 ÷ 2 = 50 + 1;
- 50 ÷ 2 = 25 + 0;
- 25 ÷ 2 = 12 + 1;
- 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.
3 414 649 298(10) = 1100 1011 1000 0111 0110 1001 1101 0010(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.
3 414 649 298(10) = 0000 0000 0000 0000 0000 0000 0000 0000 1100 1011 1000 0111 0110 1001 1101 0010
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.
-3 414 649 298(10) = !(0000 0000 0000 0000 0000 0000 0000 0000 1100 1011 1000 0111 0110 1001 1101 0010)
Number -3 414 649 298(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:
-3 414 649 298(10) = 1111 1111 1111 1111 1111 1111 1111 1111 0011 0100 0111 1000 1001 0110 0010 1101
Spaces were used to group digits: for binary, by 4, for decimal, by 3.