# 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 >> 3

4. 10000000

a << 3

5. 01010000

~b & a

6. 00001010

~a & b

7. 01010000

~b << 4

8. 10101011

~a >> 3 | b

## 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=91
>>>a
01011011
>>>one=Bits(1, 8)
>>>one
00000001
>>>~a
10100100
**>>>~a + one**
10100101
>>>-a
10100101
>>>-a.int
-91

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.
    Every binary digit place, from right to left, each left shift of one, 
    raises the digit to the power of 2.

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? 
   The range of Bits in this case is from -128 to 128, in base 10. If you change the first line to: **hundred = Bits(100, 9)**, that ninth bit (on the left) increases the Bits range to from -256 to 256, in base 10. So, then you get the right answer: 200.

12. What is the bit representation of negative zero? Explain your answer.
"What is negative 0 in binary?
In a 1+7-bit sign-and-magnitude representation for integers, negative zero is represented by the bit string 1000 0000 . In an 8-bit ones' complement representation, negative zero is represented by the bit string 1111 1111 . In all these three encodings, positive or unsigned zero is represented by 0000 0000 ." - Google Search

13. What's the largest integer that can be represented in a single byte? (8 bits in a byte).
    Explain your reasoning. 127
   "For a signed integer (the most common representation in modern computing, using two's complement), the range is -128 to 127."
   - Google Search

14. What's the smallest integer that can be represented in a single byte? (8 bits in a byte).
    Explain your reasoning. -128
    "For a signed integer (the most common representation in modern computing, using two's complement), the range is -128 to 127."
   - Google Search

15. What's the largest integer that can be represented in `n` bits? ((2**n)/2) - 1.
    Explain your reasoning.
    Again, power of 2 for each bit, as you go to the left, one bit at a time.
    Also, the range of integers must include negative numbers, so you divide the total by 2. 
    Then you subtract 1 because 0 is included in the range.

## 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.