Python Programming: An Introduction to Computer Science Chapter 3 Computing with Numbers Python Programming, 1/e 1 Slideshow and powerpoint viewer: Objectives To understand the con

Objectives To understand the concept of data types. To be familiar with the basic numeric data types in Python. To understand the fundamental principles of how numbers are represented on a computer. Python Programming, 1/e 2

Objectives (cont.) To be able to use the Python math library. To understand the accumulator program pattern. To be able to read and write programs that process numerical data. Python Programming, 1/e 3

Numeric Data Types The information that is stored and manipulated bu computers programs is referred to as data. There are two different kinds of numbers! (5, 4, 3, 6) are whole numbers – they don’t have a fractional part (.25, .10, .05, .01) are decimal fractions Python Programming, 1/e 4

Numeric Data Types Inside the computer, whole numbers and decimal fractions are represented quite differently! We say that decimal fractions and whole numbers are two different data types. The data type of an object determines what values it can have and what operations can be performed on it. Python Programming, 1/e 5

Numeric Data Types Whole numbers are represented using the integer (int for short) data type. These values can be positive or negative whole numbers. Python Programming, 1/e 6

Numeric Data Types Numbers that can have fractional parts are represented as floating point (or float) values. How can we tell which is which? A numeric literal without a decimal point produces an int value A literal that has a decimal point is represented by a float (even if the fractional part is 0) Python Programming, 1/e 7

Numeric Data Types Python has a special function to tell us the data type of any value. >>> type (3) >>> type (3.1) >>> type(3.0) >>> myint = -32 >>> type(myint) >>> myfloat = 32.0 >>> type(myfloat) >>> mystery = myint * myfloat >>> type(mystery) Python Programming, 1/e 8

Numeric Data Types Why do we need two number types? Values that represent counts can’t be fractional (you can’t have 3 ½ quarters) Most mathematical algorithms are very efficient with integers The float type stores only an approximation to the real number being represented! Since floats aren’t exact, use an int whenever possible! Python Programming, 1/e 9

Numeric Data Types Integer division always produces an integer, discarding any fractional result. That’s why 10/3 = 3! Think of it as ‘gozinta’, where 10/3 = 3 since 3 gozinta (goes into) 10 3 times (with a remainder of 1) 10%3 = 1 is the remainder of the integer division of 10 by 3. Python Programming, 1/e 11

Numeric Data Types Now you know why we had to use 9.0/5.0 rather than 9/5 in our Celsius to Fahrenheit conversion program! a = (a/b)(b) + (a%b) Python Programming, 1/e 12

Using the Math Library Besides (+, -, *, /, **, %, abs), we have lots of other math functions available in a math library. A library is a module with some useful definitions/functions. Python Programming, 1/e 13

Using the Math Library Let’s write a program to compute the roots of a quadratic equation! b b 2 4ac x 2a The only part of this we don’t know how to do is find a square root… but it’s in the math library! Python Programming, 1/e 14

Using the Math Library To use a library, we need to make sure this line is in our program: import math Importing a library makes whatever functions are defined within it available to the program. Python Programming, 1/e 15

Using the Math Library To access the sqrt library routine, we need to access it as math.sqrt(x). Using this dot notation tells Python to use the sqrt function found in the math library module. To calculate the root, you can do discRoot = math.sqrt(b*b – 4*a*c) Python Programming, 1/e 16

Using the Math Library # quadratic.py # A program that computes the real roots of a quadratic equation. # Illustrates use of the math library. # Note: This program crashes if the equation has no real roots. import math # Makes the math library available. def main(): print "This program finds the real solutions to a quadratic" print a, b, c = input("Please enter the coefficients (a, b, c): ") discRoot = math.sqrt(b * b - 4 * a * c) root1 = (-b + discRoot) / (2 * a) root2 = (-b - discRoot) / (2 * a) print print "The solutions are:", root1, root2 main() Python Programming, 1/e 17

Using the Math Library This program finds the real solutions to a quadratic Please enter the coefficients (a, b, c): 3, 4, -1 The solutions are: 0.215250437022 -1.54858377035 What do you suppose this means? This program finds the real solutions to a quadratic Please enter the coefficients (a, b, c): 1, 2, 3 Traceback (most recent call last): File "", line 1, in -toplevelmain() File "C:\Documents and Settings\Terry\My Documents\Teaching\W04\CS 120\Textbook\code\chapter3\quadratic.py", line 14, in main discRoot = math.sqrt(b * b - 4 * a * c) ValueError: math domain error >>> Python Programming, 1/e 18

Math Library If a = 1, b = 2, c = 3, then we are trying to take the square root of a negative number! Using the sqrt function is more efficient than using **. How could you use ** to calculate a square root? Python Programming, 1/e 19

Accumulating Results: Factorial Say you are waiting in a line with five other people. How many ways are there to arrange the six people? 720 -- 720 is the factorial of 6 (abbreviated 6!) Factorial is defined as: n! = n(n-1)(n-2)…(1) So, 6! = 6*5*4*3*2*1 = 720 Python Programming, 1/e 20

Accumulating Results: Factorial How we could we write a program to do this? Input number to take factorial of, n Compute factorial of n, fact Output fact Python Programming, 1/e 21

Accumulating Results: Factorial How did we calculate 6!? 6*5 = 30 Take that 30, and 30 * 4 = 120 Take that 120, and 120 * 3 = 360 Take that 360, and 360 * 2 = 720 Take that 720, and 720 * 1 = 720 Python Programming, 1/e 22

Accumulating Results: Factorial What’s really going on? We’re doing repeated multiplications, and we’re keeping track of the running product. This algorithm is known as an accumulator, because we’re building up or accumulating the answer in a variable, known as the accumulator variable. Python Programming, 1/e 23

Accumulating Results: Factorial The general form of an accumulator algorithm looks like this: Initialize the accumulator variable Loop until final result is reached update the value of accumulator variable Python Programming, 1/e 24

Accumulating Results: Factorial It looks like we’ll need a loop! fact = 1 for factor in [6, 5, 4, 3, 2, 1]: fact = fact * factor Let’s trace through it to verify that this works! Python Programming, 1/e 25

Accumulating Results: Factorial Why did we need to initialize fact to 1? There are a couple reasons… Each time through the loop, the previous value of fact is used to calculate the next value of fact. By doing the initialization, you know fact will have a value the first time through. If you use fact without assigning it a value, what does Python do? Python Programming, 1/e 26

Accumulating Results: Factorial Since multiplication is associative and commutative, we can rewrite our program as: fact = 1 for factor in [2, 3, 4, 5, 6]: fact = fact * factor Great! But what if we want to find the factorial of some other number?? Python Programming, 1/e 27

Accumulating Results: Factorial Using this souped-up range statement, we can do the range for our loop a couple different ways. We can count up from 2 to n: range(2, n+1) (Why did we have to use n+1?) We can count down from n to 1: range(n, 1, -1) Python Programming, 1/e 30

Accumulating Results: Factorial Our completed factorial program: # factorial.py # Program to compute the factorial of a number # Illustrates for loop with an accumulator def main(): n = input("Please enter a whole number: ") fact = 1 for factor in range(n,1,-1): fact = fact * factor print "The factorial of", n, "is", fact main() Python Programming, 1/e 31

The Limits of Int What is 100!? >>> main() Please enter a whole number: 100 The factorial of 100 is 93326215443944152681699238856266700490715968 26438162146859296389521759999322991560894146 39761565182862536979208272237582511852109168 64000000000000000000000000 Wow! That’s a pretty big number! Python Programming, 1/e 32

The Limits of Int What’s going on? While there are an infinite number of integers, there is a finite range of ints that can be represented. This range depends on the number of bits a particular CPU uses to represent an integer value. Typical PCs use 32 bits. Python Programming, 1/e 34

The Limits of Int Typical PCs use 32 bits That means there are 232 possible values, centered at 0. This range then is –231 to 231-1. We need to subtract one from the top end to account for 0. We can test this with an old version of Python. Python Programming, 1/e 35

The Limits of Int It blows up between 230 and 231 as we expected. Can we calculate 231-1? >>> 2**31-1 Traceback (innermost last): File "", line 1, in ? 2**31-1 OverflowError: integer pow() What happened? It tried to evaluate 231 first! Python Programming, 1/e 37

The Limits of Int We need to be more clever! 231 = 230+230 231-1 = 230-1+230 We’re subtracting one from each side! >>> 2**30-1+2**30 2147483647 >>> 2147483647+1 Traceback (innermost last): File "", line 1, in ? 2147483647+1 OverflowError: integer addition >>> Python Programming, 1/e 38

The Limits of Int What have we learned? The largest int value we can represent is 2147483647 How do modern versions of Python handle this? Python 2.3.3 (#51, Dec 18 2003, 20:22:39) [MSC v.1200 32 bit (Intel)] on win32 Type "copyright", "credits" or "license()" for more information. >>> 2**40 1099511627776L Python Programming, 1/e 39

Handling Large Numbers: Long Ints Does switching to float data types get us around the limitations of ints? If we initialize the accumulator to 1.0, we get >>> main() Please enter a whole number: 15 The factorial of 15 is 1.307674368e+012 We no longer get an exact answer! Python Programming, 1/e 40

Handling Large Numbers: Long Ints Very large and very small numbers are expressed in scientific or exponential notation. 1.307674368e+012 means 1.307674368 * 1012 Here the decimal needs to be moved right 12 decimal places to get the original number, but there are only 9 digits, so 3 digits of precision have been lost. Python Programming, 1/e 41

Handling Large Numbers: Long Ints Floats are approximations Floats allow us to represent a larger range of values, but with lower precision. Python has a solution, the long int! Long Ints are not a fixed size and expand to handle whatever value it holds. Python Programming, 1/e 42

Handling Large Numbers: Long Ints To get a long int, put “L” on the end of a numeric literal. 5 is an int representation of five 5L is a long int representation of five >>> 2L 2L >>> 2L**31 2147483648L >>> type(2L) >>> 100000000000000000000000000000000000L + 25 100000000000000000000000000000000025L Python Programming, 1/e 43

Handling Large Numbers: Long Ints Calculations involving long int produce long int results. Newer versions of Python automatically convert your ints to long ints when they grow so large as to overflow. >>> x = 2147483647 >>> x = x + 1 >>> x 2147483648L >>> type (x) >>> print x 2147483648 Python Programming, 1/e 44

Handling Large Numbers: Long Ints We started out with x assigned the largest integer value, and then added 1. x was automatically changed to type long int. When we print long ints, the ‘L’ is dropped Why not use long ints all the time? – Less efficient, slow computations Python Programming, 1/e 45

Type Conversions We know that combining an int with an int produces an int, and combining a float with a float produces a float. What happens when you mix an int and float in an expression? x = 5.0 / 2 What do you think should happen? Python Programming, 1/e 46

Type Conversions For Python to evaluate this expression, it must either convert 5.0 to 5 and do an integer division, or convert 2 to 2.0 and do a floating point division. Converting a float to an int will lose information Ints can be converted to floats by adding “.0” Python Programming, 1/e 47

Type Conversion In mixed-typed expressions Python will convert ints to floats. Sometimes we want to control the type conversion. This is called explicit typing. average = sum / n If the numbers to be averaged are 4, 5, 6, 7, then sum is 22 and n is 4, so sum/n is 5, not 5.5! Python Programming, 1/e 48

Type Conversions To fix this problem, tell Python to change one of the values to floating point: average = float(sum)/n We only need to convert the numerator because now Python will automatically convert the denominator. Python Programming, 1/e 49

Type Conversions Why doesn’t this work? average = float(sum/n) sum = 22, n = 5, sum/n = 4, float(sum/n) = 4.0! Python also provides int(), and long() functions to convert numbers into ints and longs. Python Programming, 1/e 50

Type Conversions The round function returns a float, rounded to the nearest whole number. >>> round(3.9) 4.0 >>> round(3) 3.0 >>> int(round(3.9)) 4 Python Programming, 1/e 52