lab_encoding/questions.md

Nelson Mason

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.