# Boolean questions

Create the following variables.

```
a = Bits("11110000")
b = Bits("10101010")
```

For each of the following bytes, give an equivalent 
expression which uses only `a`, `b`, and bit operators.
The answers to the first two questions are given.

1. 01010101

~b

2. 00000101

~a & ~b

3. 00000001

a >> 7

4. 10000000

a << 3

5. 01010000

~b << 4

6. 00001010

b >> 4

7. 01010000

~b << 4

8. 10101011

a ^ b >> 2

## Integer questions

These questions are difficult! Try exploring ideas with `Bits` 
in Terminal, a paper and pencil, and a whiteboard. And definitely
talk with others.

9. If `a` represents a positive integer, and `one = Bits(1, length=len(a))`, 
   give an expression equivalent to `-a`, but which does not use negation.

   (~a) + one

10. It is extremely easy to double a binary number: just shift all the bits 
    to the left. (`a << 1` is twice `a`.) Explain why this trick works. 

shifting left is the same as multiplying by 2 becouse each shift moves every bit to a position worth twice as much

11. Consider the following: 
   ```
   >>> hundred = Bits(100, 8)
   >>> hundred
   01100100
   >>> (hundred + hundred)
   11001000
   >>> (hundred + hundred).int
   -56
   ```
   Apparently 100 + 100 = -56. What's going on here?

   100 + 100 results in -56 because the sum (200) exceeds the maximum value that can be stored in 8 bits, causing an overflow and making the binary result interpreted as a negative number using twos complement.

12. What is the bit representation of negative zero? Explain your answer.

    negative zero has the same bit representation as zero (00000000) because flipping all bits of zero and adding one results in zero again, so there is no distinct negative zero.

13. What's the largest integer that can be represented in a single byte? 
    Explain your reasoning.

    The largest integer in a single byte is 127 because, the leftmost bit is reserved for the sign, leaving 7 bits for the value, which gives a maximum of 01111111 = 127.

14. What's the smallest integer that can be represented in a single byte? 
    Explain your reasoning.

    The smallest integer in a single byte is -128 because in 8-bit, the bit pattern 10000000 represents -128, which is the lowest possible value when the leftmost bit is used as the sign.

15. What's the largest integer that can be represented in `n` bits? 
    Explain your reasoning.



## Text questions

16. Look at the bits for a few different characters using the `utf8` encoding. 
    You will notice they have different bit lengths:

    ```
    >>> Bits('a', encoding='utf8')
    01100001
    >>> Bits('ñ', encoding='utf8')
    1100001110110001
    >>> Bits('♣', encoding='utf8')
    111000101001100110100011
    >>> Bits('😍', encoding='utf8')
    11110000100111111001100010001101
    ```
    
    When it's time to decode a sequence of utf8-encoded bits, the decoder
    somehow needs to decide when it has read enough bits to decode a character, 
    and when it needs to keep reading. For example, the decoder will produce 
    'a' after reading 8 bits but after reading the first 8 bits of 'ñ', the 
    decoder realizes it needs to read 8 more bits. 

    Make a hypothesis about how this could work. 

    The decoder looks at the first 8 bits of a byte to determine how many total bytes the character uses.