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Circular Nature of the Integer Data Type Family

Module by: Kenneth Leroy Busbee. E-mail the author

Summary: An explanation of how modular arithmetic concepts apply to the integer data type family and counting loops.

General Discussion

There are times when character and integer data types are lumped together because they both act the same (often called the integer family). Maybe we should say they act differently than the floating-point data types. The integer family values jump from one value to another. There is nothing between 6 and 7 nor between 'A' and 'B'. It could be asked why not make all your numbers floating-point data types. The reason is twofold. First, some things in the real world are not fractional. A dog, even with only 3 legs, is still one dog not three fourths of a dog. Second, the integer data type is often used to control program flow by counting (counting loops). The integer family has a circular wrap around feature. Using a two byte integer, the next number bigger than 32767 is negative 32768 (character acts the same way going from 255 to 0. We could also reverse that to be the next smaller number than negative 32768 is positive 32767. This can be shown by using a normal math line, limiting the domain and then connecting the two ends to form a circle.

Figure 1
Figure 1 (graphics1.jpg)

This circular nature of the integer family works for both integer and character data types. In theory, it should work for the Boolean data type as well; but in most programming languages it does not for various technical reasons.

"In mathematics, modular arithmetic (sometimes called clock arithmetic) is a system of arithmetic for integers where numbers "wrap around" after they reach a certain value — the modulus. …

A familiar use of modular arithmetic is its use in the 12 hour clock the arithmetic of time-keeping in which the day is divided into two 12 hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Usual addition would suggest that the later time should be 7 + 8 = 15, but this is not the answer because clock time "wraps around" every 12 hours; there is no "15 o'clock". Likewise, if the clock starts at 12:00 (noon) and 21 hours elapse, then the time will be 9:00 the next day, rather than 33:00. Since the hour number starts over when it reaches 12, this is arithmetic modulo 12.

Figure 2
Figure 2 (graphics2.png)

Time-keeping on a clock gives an example of modular arithmetic." (Modular arithmetic from Wikipedia)

The use of the modulus operator in integer division is tied to the concepts used in modular arithmetic.

Implications When Executing Loops

If a programmer sets up a counting loop incorrectly, usually one of three things happen:

  • Infinite loop – usually caused by missing update attribute.
  • Loop never executes – usually the text expression is wrong with the direction of the less than or greater than relationship needing to be switched.
  • Loop executes more times than desired – update not properly handled. Usually the direction of counting (increment or decrement) need to be switched.

Let’s give an example of the loop executing for what appears to be for infinity (the third item on our list).

Example 1: C++ source code


for (int x = 0; x < 10; x--)
  {
  cout << x << endl;
  }

The above code accidently decrements and the value of x goes in a negative way towards -2147483648 (the largest negative value in a normal four byte signed integer data type). It might take a while (thus it might appear to be in an infinite loop) for it to reach the negative 2 billion plus value, before finally decrementing to positive 2147483647 which would, incidentally, stop the loop execution.

Demonstration Program in C++

Creating a Folder or Sub-Folder for Source Code Files

Depending on your compiler/IDE, you should decide where to download and store source code files for processing. Prudence dictates that you create these folders as needed prior to downloading source code files. A suggested sub-folder for the Bloodshed Dev-C++ 5 compiler/IDE might be named:

  • Demo_Programs

If you have not done so, please create the folder(s) and/or sub-folder(s) as appropriate.

Download the Demo Program

Download and store the following file(s) to your storage device in the appropriate folder(s). Following the methods of your compiler/IDE, compile and run the program(s). Study the source code file(s) in conjunction with other learning materials.

Download from Connexions: Demo_Circular_Nature_Integer.cpp

Definitions

Definition 1: circular nature
Connecting the negative and positive ends of the domain of an integer family data type.
Definition 2: loop control
Making sure the attributes of a loop are properly handled.
Definition 3: modular arithmetic
A system of arithmetic for integers where numbers "wrap around".

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