Overview

1. Finishing and Testing our Counter Object
2. Strange Arithmetic

Last Time

We will defined a Counter object!

• Internal state:
  • store count as an int
• Instance methods:
  • get count
  • increment counter
  • reset
• Constructor

Today

Finish and test the implementation!

Finished Implementation

public class Counter {
    private int count;
    
	// constructor defines how to initialize instance
    public Counter () {
	    count = 0;
	}
	
	// getter method for count
	public int getCount () {
	    return count;
	}
	
	// increment the counter
	public void increment () {
	    count++;
	}
	
	// reset the counter
	public void reset () {
	    count = 0;
	}
}

How can we use the Counter?

Create an instance of the Counter class:

// create a Counter instance
Counter myCounter = new Counter();
		
// increment the counter: call instance method		
myCounter.increment();

// print the current count
System.out.println("Current count: " + myCounter.getCount());

Why Make a Counter Class?

Couldn’t we have just used an int?

Reasons Counter is Preferable to int

1. Semantics. a Counter signals object is being used to count something
   • an int could signify anything!
2. Safety. a Counter restricts the operations
   • cannot do something accidentally that would mess up count
3. Separate implementation from interface.
   • can change implementation without affecting code that uses counter
   • can use Counter objects without knowledge of internal workings
4. Extensibility. can add functionality without affecting the code using Counter

Strange Arithmetic

Middle School Arithmetic

Given any positive integer $a$, we have

• \[1 = \frac{a}{a} = a \cdot \frac 1 a.\]

Writing the multiplication as repeated addition, we get

• \[a \cdot \frac 1 a = \underset{a \text{ terms}}{\underbrace{\frac 1 a + \frac 1 a + \cdots + \frac 1 a}} = 1\]

What does Java think of this?

Testing the Identity

Consider the following piece of code

    int a = 10;
    double recip = 1.0 / a;
    double product = 0;
    
    // add 1/a to itself a times
    for (int i = 0; i < a; i++) {
      product += recip;
    }

Questions

1. What do you expect the value of product to be?
2. What is the value of product?
3. What should it be?

Test Code

What happened?

Why?

Does it matter?

Worse arithmetic

Now consider an execution of the following code:

    double one = product;
	int iter = 50;
	for (int i = 0; i < iter; i++) {
	    one = one * one;
	}

Same Questions

1. What do you expect the value of power to be?
2. What is the value of power?
3. What should it be?

What is Going on Here?

• Our calculations that should all give a value of 1

• But can give values arbitrarily far from 1

  • Why?
  • How can we fix this?

Representation of floats/doubles

• Java represents floating point numbers (float and double) in binary
• For many fractional values, this behavior leads to round-off errors
  • the number 1 / 10 does not have a finite binary expression, so it is rounded off
• when we perform more arithmetic operations, round-off error can add up to be (unacceptably) large!

How can we fix this problem?

How can we ensure that we have, for example,

• \(\frac 1 6 + \frac 1 6 + \frac 1 6 + \frac 1 6 + \frac 1 6 + \frac 1 6 = 1\)?

Increasing the precision of float and double won’t fix the problem!

(Maybe) better design?

Design a Fraction object to represent fractional values!

• Follow the link from class to work on a group activity

State of a Fraction Object

• What instance variables to store?

How to Implement Arithmetic (+ and *)?

What Subtleties/problems/inconveniences of Using Fraction?

Next Time

We’ll write and test our Fraction class!