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;
- 5 643 703 ÷ 2 = 2 821 851 + 1;
- 2 821 851 ÷ 2 = 1 410 925 + 1;
- 1 410 925 ÷ 2 = 705 462 + 1;
- 705 462 ÷ 2 = 352 731 + 0;
- 352 731 ÷ 2 = 176 365 + 1;
- 176 365 ÷ 2 = 88 182 + 1;
- 88 182 ÷ 2 = 44 091 + 0;
- 44 091 ÷ 2 = 22 045 + 1;
- 22 045 ÷ 2 = 11 022 + 1;
- 11 022 ÷ 2 = 5 511 + 0;
- 5 511 ÷ 2 = 2 755 + 1;
- 2 755 ÷ 2 = 1 377 + 1;
- 1 377 ÷ 2 = 688 + 1;
- 688 ÷ 2 = 344 + 0;
- 344 ÷ 2 = 172 + 0;
- 172 ÷ 2 = 86 + 0;
- 86 ÷ 2 = 43 + 0;
- 43 ÷ 2 = 21 + 1;
- 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.
5 643 703(10) = 101 0110 0001 1101 1011 0111(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 23.
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, 23,
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.
5 643 703(10) = 0000 0000 0101 0110 0001 1101 1011 0111
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 0000 0101 0110 0001 1101 1011 0111)
= 1111 1111 1010 1001 1110 0010 0100 1000
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 1111 1010 1001 1110 0010 0100 1000
(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):
-5 643 703 =
1111 1111 1010 1001 1110 0010 0100 1000 + 1
Number -5 643 703(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:
-5 643 703(10) = 1111 1111 1010 1001 1110 0010 0100 1001
Spaces were used to group digits: for binary, by 4, for decimal, by 3.