Lecture 5 numbers and built in functions

1 of 45Module 5 : Numbers and Built‐in Functions
Introduction to
Computational Thinking
Module 5 :
Numbers and Built‐in Functions
Asst Prof Chi‐Wing FU, Philip
Office: N4‐02c‐104
email: cwfu[at]ntu.edu.sg

Topics
• More on Numbers
• Built-in functions
• Comparing floating point numbers
• Case Study: Finding Square Root
• Self Study: Bitwise operators (optional)

More on Numbers
• Integers and Floating point numbers
• Their representations in computers
• Data range
• Optional: Bitwise operators on integers

Memory: Bits and Bytes
• 1 Byte = 8 bits (each bit either 0 or 1)
• 1 Byte has 28 different variations, i.e., 256
• How about 4 bytes?
0
0
0
0
0
0
0
0 0
0
00
1
11
1
1
1
1
1
1
1111000
1001
1002
Contents of
memory
Memory
This is also covered in
Introduction to Computing Systems

Integers (others)
• In most programming languages, an
integer takes up 4 bytes. The first bit for
sign: +ve or -ve, i.e., 31 bits left
• Since there is a zero needed to be
represented (as all zeros in the 32 bits),
the available data range is [-231, +231 – 1 ]

Integers (Python)
• Integers in Python has unlimited precision
• This is a useful feature in Python; otherwise,
we (programmers) have to handle them
ourselves through programming effort

Float
• For real numbers with decimals
• Limited precision because we use limited
number of bits to store the data in a special
format (you will learn later in other course):
• Max. representable finite float
• Min. positive normalized float
Want to know more? http://docs.python.org/tutorial/floatingpoint.html
http://en.wikipedia.org/wiki/Double_precision_floating-point_format

Float
• and it has limited precision…
• Hence, computation results with floats
may not be exact

Integer VS Float
• When writing programs, variables that are
numeric can be integers or floats
• So… how to choose?
Think about the context!!!
Need decimal values? Want precision?

Integer VS Float
• For the followings, which type is
more appropriate?
• Age
• Height
• Volume of a container
• Exchange rate
• Voltage and current
• Number of students in a class
• Bank account balance

Topics
• More on Numbers

What are Built-in functions
• Built-in functions are subroutines, procedures,
or methods that are built into Python itself; each
provides a specific functionality
• See the Python standard library:
http://docs.python.org/library/functions.html
• In fact, you knew some of them already:
• print (for displaying data)
• input (for reading user input)
• float, int, str (for converting data)
• type (for checking data type)
• exec (for executing a string in Python), …, etc.

• abs(x) – Return absolute value of x
• min(x,y) and max(x,y) – Return minimum
and maximum of x and y, respectively
They allow two or
more arguments
Let’s learn more: calling & arg(s)
Takes one argument

More Built-in Functions
• math.ceil(x) and math.floor(x) –
Return the ceiling and floor of x as “int”
Remember to
import math

• math.pow(x,y) – Return x raised to the power y
• math.sqrt(x) – Return the square root of x
• math.log(x) – Return natural logarithm of x
• math.log10(x) – Return base-10 logarithm of x
Math functions
usually return
“float”

• math.sin(x), math.cos(x), math.tan(x)
– Return the sine, cosine, and tangent of x (radian)
• math.asin(x),math.acos(x),math.atan(x)
– Return arc sin, cos, and tan of x in radian
(Note: they all use radian!)
• math.degrees(x) and math.radians(x)
– Return x in degrees and radians, respectively

Help Function
• help(“...”) – display help information on a
Python function or module

Topics
• More on Numbers

No Equality!
• If we want to know if two floating point numbers
are equal or not, can we use "==" ??
• It is tricky!!!
They are not equal! Why?
So… how to fix it?

So… how to check?
• First, we can define a relatively small floating
point value, usually called epsilon or tolerance
• IDEA: if the absolute difference between the
two floating point numbers is smaller than
epsilon, we say that their values are the same
Key idea!!!!
(computationally but not mathematically)

Topics
• More on Numbers

Case Study
• Here we want to implement an algorithm to
compute the square root of a value without
using math.sqrt
You should implement, run, and try the case
study yourself in the course

Algorithm: compute square root
• We can compute the square root of a value,
say x, iteratively using the following idea:
Given x,
first compute an initial guess: guess = x/2
then diff = guess* guess - x
if abs(diff) < epsilon, we are done
if diff > 0, guess is too large
if diff < 0, guess is too small
update guess so that it gets closer, then
Repeat

• But… how to update guess so that it gets
closer in every iteration…
• Let’s check x/guess…
In case x=100 and guess=50,
x/guess = 2
the targeting square root value should always
lie between guess and x/guess!
• So… we may update guess as mean value of
guess and x/guess and make it closer!

Let’s do this test…
• Iteration #1
x=100 and guess=50
x/guess=2 -> new guess = mean = 26
• Iteration #2
x=100 and guess=26
x/guess=3.846 -> new guess = mean = 14.923
• Iteration #3
x=100 and guess=14.923
x/guess=6.701 -> new guess = mean = 10.812
…………
Really getting closer and closer!!!

Implementation
• Here is the Python implementation
We will learn “while” in module 6.2

Implementation
• Add variable "iter" to count number of iterations
Add iteration counter

Verification… by math.sqrt
• Lastly… we must verify our program!!!
Report the difference
This is a verification

Testing #1
• We try with x = 100
Our result
Difference is very small

Testing #2
• We try with x = 100000
Our result
But…
need more
iterations for
larger num.
Difference
still small

Testing #3
• Lastly… try with x = 2
Difference is still very small
Fewer
iterations
this time
Our result

Topics
• More on Numbers

What are Bitwise operators
• Recall that integers are stored in binary, e.g.,

NOT~
SHIFT (right)>>
SHIFT (left)<<
Bitwise XOR^
Bitwise OR|
Bitwise AND&
meaningoperator
What are Bitwise operators
• Python provides bitwise operators that process
input integers bit by bit correspondingly
• Treat 1 as true and 0 as false
Input:
15 -> 0000 1111
17 -> 0001 0001
What is 15 | 17 ?
All are binary
operations except ~,
which is unary

00 & 0
00 & 1
01 & 0
11 & 1
ResultAND
Truth tables of &, |, ^, ~
00 | 0
10 | 1
11 | 0
11 | 1
ResultOR
00 ^ 0
10 ^ 1
11 ^ 0
01 ^ 1
ResultXOR
• XOR is called exclusive or; it is true ONLY if one
of the two bits is true (but not both)
• ~x is a special operation called 2’s complement,
which is just -x-1, i.e., ~2 gives -3

Examples: &, |, ^
&
|
^

Examples: shift <<, >>
<<
>>

3 << 1 gives 6SHIFT (left)<<
~3 gives -4NOT~
3 >> 1 gives 1SHIFT (right)>>
3 ^ 2 gives 1Bitwise XOR^
3 | 2 gives 3Bitwise OR|
3 & 2 gives 2Bitwise AND&
examplesmeaningoperator
More examples
Try to work them out yourself

Any Application?
1. Simple Data Encryption and Decryption
using the xor operator
2. Pseudo Random Number Generator
(PRNG)

Application #1
• xor can be used for doing simple encryption
and decryption
• Given
D – our data (any integer, i.e., 32 bits)
K – our key in encryption (which is an integer)
• We can perform encryption by:
E = D xor K where E is the encrypted integer
• and decryption by
E xor K gives D
We can recover D from E using K
XOR has this interesting property
Why? Study the truth table…

Application #2
• Pseudo Random Number Generator (PRNG):
• Seed – A number to initialize the generator
• Random number sequence – Use seed as
input, we apply the generator to obtain the first
random number; then use it as input to
generate the next random number, and so on…
See http://docs.python.org/library/random.html
• In designing a PRNG, bitwise operations are
often used, see next slide for a simple example
Interesting article:
http://spectrum.ieee.org/semiconductors/processors/behind-intels-new-randomnumber-generator

Application #2 (cont.)
• Example: Define a function “parityOf” to
compute the parity of an integer:
A bit to add to an integer to make the
number of 1s in its binary form to be even
Define a simple 12-bit PRNG
A sequence of random number
(binary numbers) can be
generated starting from the seed
note: 0x indicates
hexidecimal format

Application #2 (cont.)
• Note:
• You will learn how to use Python keyword “def” to
define functions later in this course
• Using bitwise operators, we can efficiently compute the
parity and develop pseudorandom number generators.
There are many PRNGs in the world; some are more
complicated, and they could have different quality…
• Since the numbers generated in the example code
form a binary number sequence, we also call it a
Pseudo Random Binary Sequence Generator (PRBSG)
• Related courses in the future:
“Computer Organization and Architecture” and
“Microprocessor-based System Design”

Take Home Messages
• Integers, Long, and Floating point numbers
– Representation and property: data range, minimum and
maximum values, and precision
• Built-in functions are useful resources built into
Python itself
• Comparing floating point numbers can be tricky
(and error-prone)
– Use absolute difference against a small epsilon to test
equality for floating point numbers

Reading Assignment
• Textbook
Chapter 1: Beginnings
1.8 to 1.10
Note: Though some material (1.10) in textbook is
not directly related to the lecture material, you can
learn more from them.

Lecture 5 numbers and built in functions

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