The Law of Trichotomy

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Law of Trichotomy: If a and b are real numbers, then one and only one of the following is true:

  1. a < b
  2. a = b
  3. a > b

The Law of Trichotomy is a simple idea that every schoolchild realizes at an early age. I suspect a teacher’s interference is not required to establish this property of the Real numbers.

I am firmly convinced that the obvious nature of The Law of Trichotomy is precisely why students find it difficult to use it in applications.  It is so obvious that a newcomer doesn’t even think of it when the opportunity for use presents itself.

Teachers from the earliest grades to college level courses must emphasize the power of The Law of Trichotomy by constantly pointing it out.

To fully understand and use The Law of Trichotomy it is necessary to know a variety of different, but equivalent, statements and interpretations.  In the following I present six equivalent statements.  Strictly for reference I have given each of them distinct names.

These seven statements are EQUIVALENT so it is completely permissible to use whichever form matches the application at hand.

Whether you are teacher or learner you should always relate these statements to The Number Line.

Law of Trichotomy (English)If two real numbers are compared, exactly one of the following is true:

  1. The first is less than the second
  2. The first is equal to the second
  3. The first is greater than the second

Law of Trichotomy (Graphical): Two points A and B on the number line represent two real numbers, then exactly one of the following is true:

  1. A is to the left of B
  2. A and B are the same point
  3. A is to the right of B
  4. Law of Trichotomy (Difference): If a and b are real numbers, then one and only one of the following is true:

    1. a – b < 0
    2. a – b = 0
    3. a – b > 0

    Law of Trichotomy (one number is zero): If a is a real number, then one and only one of the following is true:

    1. a < 0
    2. a = 0
    3. a > 0

    Law of Trichotomy (one number is zero – English): If a is a real number, then one and only one of the following is true:

    1. a is negative
    2. a is zero
    3. a is positive

    Law of Trichotomy (one number is zero – English): If a is a real number, then one and only one of the following is true:

    1. a is left of zero
    2. a is zero
    3. a is right of zero

    This topic requires quite a bit of discussion in the classroom and here in this blog.  Consequently, I will post a few more essays related to The Law of Trichotomy before moving on to other topics.

    Suggested Flash Cards

    Click on the FlashCards link to see some suggested flashcards to assist in the study of The The Law of Trichotomy.  At the top left of the Flash Card Presenter is a drop-down list of available flash card decks.  Select “The Law of Trichotomy” to gain access to a deck containing a number of cards.  Then use the buttons around the perimeter to navigate through the deck for low cost interleaved retrieval practice.  Note some cards are not directly related to The Law of Trichotomy.  This is a simple attempt to do some interleaved retrieval practice.

    FlashCards

    http://www.drdelmath.com/study_aids/flash_cards/flash_card_presenter.html

    Another good way to practice retrieval is to take low-cost or no-cost quizzes.  It will be a few weeks before I will have working online versions of such quizzes.  When they are ready, I will present samples for each of The Big Four.

    Some of you may have noticed I ran into a problem with the flash card program so I fell behind in my schedule.  I expect to get caught up soon. I also hope to get the quiz programs working in the near future.  Keep watching.

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