Extension: Bitwise Operators, Numeral Systems, and Byte Strings

Way back in What is a computer? you were introduced to the idea that all information in a computer is stored as a sequence of bits. Python is a high-level language that thankfully keeps us from having to know such low-level details in order do most of the useful work we want done with it, but it does have features that allow us to work directly with bits if we want to. Since bitwise operators are part of the language tested on the PCEP, we will explore them in this extension.

1. Enter each of the following expressions into the Python shell and record what you get as a response:

   • >>> 5 | 3

   • >>> 9 & 3

   • >>> 12 ^ 5

   • >>> 1 << 3

   • >>> ~42

In order to understand the results you got in this exercise, you need to understand binary numbers and two’s complement. Take time to read through these Wikipedia entries carefully, until you can convince yourself you understand both binary numbers and two’s complement.

Then read the entry on bitwise operations. See if you can determine which symbol Python uses for each of the operations described in the article.

Finally, revisit exercise 1, this time making sure you understand the results.

2. Now enter the following into the Python shell and record what you get as a response:

   >>> for n in [0b1001, 0b1100, 0o10, 0xA, 0x12]:
   ...     print(n, end=' ')
   ... else:
   ...     print()
   ...
   

Binary, Octal, and Hexidecimal Numbers

Octal and hexadecimal (or hex) are the two other numeral systems commonly used in computer science to represent bits.

Now look at the following table:

Decimal, Binary, Octal and Hexadecimal Numbers

Decimal	Binary	Octal	Hexadecimal
0	0b0000	0o0	0x0
1	0b0001	0o1	0x1
2	0b0010	0o2	0x2
3	0b0011	0o3	0x3
4	0b0100	0o4	0x4
5	0b0101	0o5	0x5
6	0b0110	0o6	0x6
7	0b0111	0o7	0x7
8	0b1000	0o10	0x8
9	0b1001	0o11	0x9
10	0b1010	0o12	0xA
11	0b1011	0o13	0xB
12	0b1100	0o14	0xC
13	0b1101	0o15	0xD
14	0b1110	0o16	0xE
15	0b1111	0o17	0xF
16	0b10000	0o20	0x10
17	0b10001	0o21	0x11
18	0b10010	0o22	0x12

Bytes Objects

A byte is sequence of 8 bits. Python has a data type for working with a sequence of bytes.

Take a look at the following Python shell session:

>>> message = b'A message from GHC'
>>> type(message)
<class 'bytes'>
>>> for byte in message:
...     print(byte)
...
65
32
109
101
115
115
97
103
101
32
102
114
111
109
32
71
72
67

Run this yourself and confirm that you get the same result. Then take time to understand it.

PCEP Practice

3. What is the output of the following code?

   x = 1
   y = 0
   z = x & y
   w = x | y
   u = x ^ y
   print(z, w,  u)
   

4. What is the output of the following code?

a, b = 0, 1
print(a ^ a, a ^ b, b ^ a, b ^ b)

5. What is the output of the following code?

var = 17
var_right = var >> 1
var_left = var << 2
print(var, var_left, var_right)

6. What is the output of the following code?

x = 1
y = 0
z = ((x == y) and (x == y)) or not(x == y)
print(not(z))

7. What is the output of the following code?

x = 8
y = 1

a = x & y
b = x | y
c = ~x
d = x ^ 5
e = x >> 2
f = x << 2

print(a, b, c, d, e, f)