Classroom Aid - Rational Number Line
Channel: David Butler
Date: 2018-09-19

Text http://howfarawayisit.com/wp-content/uploads/2018/09/How-Far-Mathmatical-Foundations-Direct-Measurement.pdf

A significant amount of mathematics is used in the How Far Away Is It channel video books. Although mathematical equations are identified, they were not the focus. They served to deepen understanding of the physical observations. In this video book, we will begin from first principles and develop the foundation for the math used. But the focus will not be on proofs and notation, but rather on the principles and postulates and key theorems. 

In order to better understand ‘direct measurement’, we’ll develop the real number system from counting numbers. We’ll add zero and then negative numbers to get the integer number line. At that point, we introduce the basic mathematical operations of addition, subtraction, multiplication and division. With these we use Peano’s Postulates to identify the associative, commutative and distributive properties of whole numbers. 

We then extend the integer number line to the rational number line and illustrate the Trichotomy property. We then cover irrational numbers – going back to the ancient Greek philosopher Hippassus – including his proof published by Euclid. Combining sets, we’ll construct the dense and continuous real number line, and identify the problems with irrational numbers that persisted until the late 1800s when Richard Dedekind developed his real number line cuts. 
And finally, the real number line is then used as the bases for direct measurement where we will identify a key difference between Math and Physics. 

Along the way, we’ll see: the Bakhshali manuscript where we see the first use of zero; the earliest written reference to negative numbers in the Chinese book “The Nine Chapters on the Mathematical Art”; an algebraic exercise that purports to show that the number one equals the number two; some key issues with the number zero, division and exponents; and a way to multiply by doubling and halving.

This real number line will be the bases for all subsequent mathematical analysis. I trust you’ll find it informative and entertaining.