Arithmetic on a Circle

Arithmetic on a Circle
A Programming Pattern

Prof. David Bernstein
James Madison University

Computer Science Department

bernstdh@jmu.edu

Motivation

Frequently:
- Values can be increased without bound (e.g., if someone keeps giving you dollar bills)
In Some Cases:
- Values don't keep increasing, they "repeat" (e.g., the hour of the day)

Review

Addition on the Line

Subtraction on the Line

Thinking About the Problem

Addition on the Circle

Subtraction on the Circle

The Pattern

Notation:
- current denotes the current value
- change denotes the change (either positive or negative)
- cardinality denotes the number of elements in the set of values
The Process:
- Use a 0-based set of consecutive integer values
- Use integer arithmetic
- If needed, calculate the number of times 0 is passed using:
  
  javaexamples/programmingpatterns/ArithmeticOnTheCircle.java (Fragment: pattern1)
```
        passes = (current + change) / cardinality;        
        
```
- Calculate the new value as:
  
  javaexamples/programmingpatterns/ArithmeticOnTheCircle.java (Fragment: pattern2)
```
        remainder = (current + change) % cardinality;
        
```

The Example of Military Time

The Values:
- 0 to 23
Start at 17:00:
- In 12 hours it will be 29 % 24 or 5:00
- In 93 hours it will be 110 % 24 or 14:00

Other Time/Date Examples

Other Examples: Weights and Distances

Weights:
- Ounces go from 0 to 15 (then use pounds)
- Pounds go from 0 to 1999 (then use tons)
Distances:
- Inches go from 0 to 11 (then use feet)
- Feet go from 0 to 2 (then use yards)
- Yards go from 0 to 1759 (then use miles)

Other Examples: Even/Odd Numbers

Definition:
- A number is said to be even if it can be divided by 2 with no remainder
An Observation:
- If we think of all numbers as being either even or odd we can think about this as counting on a circle
Using %:
- Does x % 2 equal 0?

Other Examples: Cycling through a Set

Examples:
- Turn-taking by different players
- Cycling through a set of colors
- Repeating a set of instructions
An Observation:
- An element in a set can be identified by its index (starting at 0)

Humor

(Courtesy of xkcd)