lab_encoding/questions.md

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 & b

4. 10000000 a & (b ^ b)

5. 01010000 a & ~b

6. 00001010 ~a & b

7. 01010000 a & ~b

8. 10101011

b | (~a & ~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 + 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.

Every bit position represents a power of 2. Shifting all bits one place to the left multiplies every bit's value by 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?

100 + 100 = 200 in bit it is 11001000. Because the first is 1 the bit represents a negative number -56.

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

There is no negative zero zero is only represented by 00000000.

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

127 is the largest because the first 1 represents a negative hence 01111111 = 127

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

-128 is the smallest where the smallest is 10000000.

15. What's the largest integer that can be represented in n bits? Explain your reasoning. 2^(n-1) -1 : one bit is used for the sign example if n = 3 bits then it would be 2^2 - 1 = 3 leading to 011.

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.

    Use the leading bits of the first byte to indicate the total length of the characters where 0--------- = 1 byte "a" then 110------- = 2 byte "n" characters, 1110------ = 3 byte, shade characters then each byte has specific meaning as used above.