Tuesday, May 27, 2014

Nature of Mathematics

Understanding Our Number System

In order to be confident in our use of numbers, we must understand how our number system works and how numbers are related to each other.  We use a single number structure.  All numbers in our single number structure can be represented using combinations of a finite set of digits--0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
An important understanding of how our number system works comes from understanding "place value." This means that a digit can carry a different value depending on where in a number it is located.  We use a base 10 system.  Depending on where a digit is placed, that digit represents a power of 10.  

When learning about number systems, it is important to remember 3 things.
1.)  Use of numbers in the world around us can take on many different forms.
2.)  The same digit takes on different values depending on where it is placed .
3.)  The number system is finite.  
Below is an interactive chart that helps explain the different ideas behind the number system.

Ideally, what our numeral system does is it
  • Represents a useful set of numbers
  • Give every number represented a unique representation 
  • Reflect the algebraic and arithmetic structure of the numbers. 
Our numeral system can represent a useful set of numbers such as integers, rational numbers, or irrational numbers, for example. Our numeral system gives every number represented a unique representation.  For example, the usual decimal representation of whole numbers gives every non zero whole number a unique representation as a finite sequence of digit, beginning by a non zero digit.  However, when decimal representation is used with rational or real numbers, such numbers have an infinite number of representations (2.31, 2.310, 2.309999999).  The following link gives great insight into how our numeral system reflects the algebraic and arithmetic structure of numbers.  




  1. This is a great way to show our number system. I really liked the graph that linked the different aspects of our number system together. That is a great visual aid that could be shown to students. Nice job on this post!

  2. To be an exemplar, you'd want to have more of your own thinking. How would you draw the diagram? Is there some way to distinguish between ideas (proportion) and models (number line)? What other models could go on this? Or you could expand on that last paragraph. How does our hindu-arabic number system do these things?