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;
- 2 013 252 436 ÷ 2 = 1 006 626 218 + 0;
- 1 006 626 218 ÷ 2 = 503 313 109 + 0;
- 503 313 109 ÷ 2 = 251 656 554 + 1;
- 251 656 554 ÷ 2 = 125 828 277 + 0;
- 125 828 277 ÷ 2 = 62 914 138 + 1;
- 62 914 138 ÷ 2 = 31 457 069 + 0;
- 31 457 069 ÷ 2 = 15 728 534 + 1;
- 15 728 534 ÷ 2 = 7 864 267 + 0;
- 7 864 267 ÷ 2 = 3 932 133 + 1;
- 3 932 133 ÷ 2 = 1 966 066 + 1;
- 1 966 066 ÷ 2 = 983 033 + 0;
- 983 033 ÷ 2 = 491 516 + 1;
- 491 516 ÷ 2 = 245 758 + 0;
- 245 758 ÷ 2 = 122 879 + 0;
- 122 879 ÷ 2 = 61 439 + 1;
- 61 439 ÷ 2 = 30 719 + 1;
- 30 719 ÷ 2 = 15 359 + 1;
- 15 359 ÷ 2 = 7 679 + 1;
- 7 679 ÷ 2 = 3 839 + 1;
- 3 839 ÷ 2 = 1 919 + 1;
- 1 919 ÷ 2 = 959 + 1;
- 959 ÷ 2 = 479 + 1;
- 479 ÷ 2 = 239 + 1;
- 239 ÷ 2 = 119 + 1;
- 119 ÷ 2 = 59 + 1;
- 59 ÷ 2 = 29 + 1;
- 29 ÷ 2 = 14 + 1;
- 14 ÷ 2 = 7 + 0;
- 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.
2 013 252 436(10) = 111 0111 1111 1111 1100 1011 0101 0100(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.
2 013 252 436(10) = 0111 0111 1111 1111 1100 1011 0101 0100
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 0111 1111 1111 1100 1011 0101 0100)
= 1000 1000 0000 0000 0011 0100 1010 1011
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 1000 0000 0000 0011 0100 1010 1011
(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):
-2 013 252 436 =
1000 1000 0000 0000 0011 0100 1010 1011 + 1
Number -2 013 252 436(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:
-2 013 252 436(10) = 1000 1000 0000 0000 0011 0100 1010 1100
Spaces were used to group digits: for binary, by 4, for decimal, by 3.