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;
- 2 146 232 058 ÷ 2 = 1 073 116 029 + 0;
- 1 073 116 029 ÷ 2 = 536 558 014 + 1;
- 536 558 014 ÷ 2 = 268 279 007 + 0;
- 268 279 007 ÷ 2 = 134 139 503 + 1;
- 134 139 503 ÷ 2 = 67 069 751 + 1;
- 67 069 751 ÷ 2 = 33 534 875 + 1;
- 33 534 875 ÷ 2 = 16 767 437 + 1;
- 16 767 437 ÷ 2 = 8 383 718 + 1;
- 8 383 718 ÷ 2 = 4 191 859 + 0;
- 4 191 859 ÷ 2 = 2 095 929 + 1;
- 2 095 929 ÷ 2 = 1 047 964 + 1;
- 1 047 964 ÷ 2 = 523 982 + 0;
- 523 982 ÷ 2 = 261 991 + 0;
- 261 991 ÷ 2 = 130 995 + 1;
- 130 995 ÷ 2 = 65 497 + 1;
- 65 497 ÷ 2 = 32 748 + 1;
- 32 748 ÷ 2 = 16 374 + 0;
- 16 374 ÷ 2 = 8 187 + 0;
- 8 187 ÷ 2 = 4 093 + 1;
- 4 093 ÷ 2 = 2 046 + 1;
- 2 046 ÷ 2 = 1 023 + 0;
- 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.
2 146 232 058(10) = 111 1111 1110 1100 1110 0110 1111 1010(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.
2 146 232 058(10) = 0111 1111 1110 1100 1110 0110 1111 1010
6. Get the negative integer number representation:
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.
-2 146 232 058(10) = !(0111 1111 1110 1100 1110 0110 1111 1010)
Number -2 146 232 058(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:
-2 146 232 058(10) = 1000 0000 0001 0011 0001 1001 0000 0101
Spaces were used to group digits: for binary, by 4, for decimal, by 3.