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;
- 89 703 850 ÷ 2 = 44 851 925 + 0;
- 44 851 925 ÷ 2 = 22 425 962 + 1;
- 22 425 962 ÷ 2 = 11 212 981 + 0;
- 11 212 981 ÷ 2 = 5 606 490 + 1;
- 5 606 490 ÷ 2 = 2 803 245 + 0;
- 2 803 245 ÷ 2 = 1 401 622 + 1;
- 1 401 622 ÷ 2 = 700 811 + 0;
- 700 811 ÷ 2 = 350 405 + 1;
- 350 405 ÷ 2 = 175 202 + 1;
- 175 202 ÷ 2 = 87 601 + 0;
- 87 601 ÷ 2 = 43 800 + 1;
- 43 800 ÷ 2 = 21 900 + 0;
- 21 900 ÷ 2 = 10 950 + 0;
- 10 950 ÷ 2 = 5 475 + 0;
- 5 475 ÷ 2 = 2 737 + 1;
- 2 737 ÷ 2 = 1 368 + 1;
- 1 368 ÷ 2 = 684 + 0;
- 684 ÷ 2 = 342 + 0;
- 342 ÷ 2 = 171 + 0;
- 171 ÷ 2 = 85 + 1;
- 85 ÷ 2 = 42 + 1;
- 42 ÷ 2 = 21 + 0;
- 21 ÷ 2 = 10 + 1;
- 10 ÷ 2 = 5 + 0;
- 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.
89 703 850(10) = 101 0101 1000 1100 0101 1010 1010(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 27.
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, 27,
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.
89 703 850(10) = 0000 0101 0101 1000 1100 0101 1010 1010
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.
!(0000 0101 0101 1000 1100 0101 1010 1010)
= 1111 1010 1010 0111 0011 1010 0101 0101
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
1111 1010 1010 0111 0011 1010 0101 0101
(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):
-89 703 850 =
1111 1010 1010 0111 0011 1010 0101 0101 + 1
Number -89 703 850(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:
-89 703 850(10) = 1111 1010 1010 0111 0011 1010 0101 0110
Spaces were used to group digits: for binary, by 4, for decimal, by 3.