## How to convert signed integers from decimal system to binary code

### Follow the steps below to convert a signed base ten integer number to signed binary:

- 1. In a signed binary, first bit (leftmost) is reserved for sign: 0 = positive integer number, 1 = positive integer number. If the number to be converted is negative, start with the positive version of it.
- 2. Divide repeatedly by 2 the integer number that is to be converted to binary, keeping track of each remainder. STOP when the last quotient of the operations is ZERO.
- 3. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
- 4. Binary numbers represented in computer language have to have 4, 8, 16, 32, 64, ... bit length (power of 2) - if needed, fill in '0' bits in front of the base 2 number (leftmost), up to the right length; this way the first bit (leftmost) will always be '0', as for the positive representation
- 5. To get the negative version of the binary, simply switch the first bit (leftmost), from '0' to '1'.

### Example: convert negative number -63 from decimal system (base ten) to signed binary code:

- 1. Start with the positive version of the number: |-63| = 63
- 2. Divide repeatedly 63 by 2, keeping track of each remainder:
iteration division quotient remainder 1 63 : 2 = 31 **1**2 31 : 2 = 15 **1**3 15 : 2 = 7 **1**4 7 : 2 = 3 **1**5 3 : 2 = 1 **1**6 1 : 2 = 0 **1**Last quotient is ZERO => FULL STOP - 3. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above:

63_{(10)}= 11 1111_{(2)} - 4. Length of base 2 representation is 6, so the positive binary computer representation of a signed binary will take 8 bits (the least power of 2 higher than 6)- add '0' in front of the base 2 number, up to the required length, this way the first bit (leftmost) will always be '0', as for a positive:

63_{(10)}= 0011 1111_{(2)} - 5. To get the negative integer number representation simply change the first bit (leftmost), from '0' to '1':

-63_{(10)}= 1011 1111