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;
- 2 147 418 113 ÷ 2 = 1 073 709 056 + 1;
- 1 073 709 056 ÷ 2 = 536 854 528 + 0;
- 536 854 528 ÷ 2 = 268 427 264 + 0;
- 268 427 264 ÷ 2 = 134 213 632 + 0;
- 134 213 632 ÷ 2 = 67 106 816 + 0;
- 67 106 816 ÷ 2 = 33 553 408 + 0;
- 33 553 408 ÷ 2 = 16 776 704 + 0;
- 16 776 704 ÷ 2 = 8 388 352 + 0;
- 8 388 352 ÷ 2 = 4 194 176 + 0;
- 4 194 176 ÷ 2 = 2 097 088 + 0;
- 2 097 088 ÷ 2 = 1 048 544 + 0;
- 1 048 544 ÷ 2 = 524 272 + 0;
- 524 272 ÷ 2 = 262 136 + 0;
- 262 136 ÷ 2 = 131 068 + 0;
- 131 068 ÷ 2 = 65 534 + 0;
- 65 534 ÷ 2 = 32 767 + 0;
- 32 767 ÷ 2 = 16 383 + 1;
- 16 383 ÷ 2 = 8 191 + 1;
- 8 191 ÷ 2 = 4 095 + 1;
- 4 095 ÷ 2 = 2 047 + 1;
- 2 047 ÷ 2 = 1 023 + 1;
- 1 023 ÷ 2 = 511 + 1;
- 511 ÷ 2 = 255 + 1;
- 255 ÷ 2 = 127 + 1;
- 127 ÷ 2 = 63 + 1;
- 63 ÷ 2 = 31 + 1;
- 31 ÷ 2 = 15 + 1;
- 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.
2 147 418 113(10) = 111 1111 1111 1111 0000 0000 0000 0001(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 31.
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, 31,
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.
2 147 418 113(10) = 0111 1111 1111 1111 0000 0000 0000 0001
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.
!(0111 1111 1111 1111 0000 0000 0000 0001)
= 1000 0000 0000 0000 1111 1111 1111 1110
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
1000 0000 0000 0000 1111 1111 1111 1110
(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):
-2 147 418 113 =
1000 0000 0000 0000 1111 1111 1111 1110 + 1
Number -2 147 418 113(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:
-2 147 418 113(10) = 1000 0000 0000 0000 1111 1111 1111 1111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.