[HN Gopher] Intermediate Algebra
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       Intermediate Algebra
        
       Author : parsecs
       Score  : 22 points
       Date   : 2021-09-06 22:02 UTC (57 minutes ago)
        
 (HTM) web link (saylordotorg.github.io)
 (TXT) w3m dump (saylordotorg.github.io)
        
       | [deleted]
        
       | Gehinnn wrote:
       | > Irrational numbers are defined as any numbers that cannot be
       | written as a ratio of two integers.
       | 
       | > Finally, the set of real numbers, denoted R, is defined as the
       | set of all rational numbers combined with the set of all
       | irrational numbers.
       | 
       | I'm sorry, this is not how math works. What is a number?
       | 
       | Also the definition of Q is missing the quotient construction (or
       | any motivation of how to deal with ka/kb = a/b).
        
         | syops wrote:
         | I think it's clear you've never taught low level mathematics
         | courses. There is a lot of hand waving and brain washing that
         | happens. The vast majority of people don't know what a number
         | is in a precise, mathematical sense. At the level of the
         | intended audience it would be wholly inappropriate talk about
         | the definition of a number.
         | 
         | My background on this topic is that I've taught intermediate
         | algebra for over 20 years.
        
           | Gehinnn wrote:
           | You are right, I didn't teach low level math courses, but
           | this brain washing is also precisely why I didn't understand
           | math in high school. You cannot argue with this kind of
           | definitions. Everything feels as if it was randomly defined
           | by the teacher. This "intuition" simplifies teaching, but
           | makes understanding harder. It is like a game where you
           | invent rules as you play. No student can win this game.
        
             | JeremyBanks wrote:
             | A problem is that lots of lower-education math instructors
             | don't understand these concepts deeply themselves. I think
             | it could be okay if these things were clearly framed as
             | "true for the problems we're looking at, but not
             | universal", but they were typically presented as universal
             | by teachers who themselves don't know any better, and that
             | really caught me up too.
        
             | syops wrote:
             | Here's the definition of 2 using the standard construction
             | with the Peano axioms. It's the set containing 0 and 1. The
             | number 1 is the set containing 0 and 0 exists by one of the
             | axioms. It's not something a person in intermediate algebra
             | can understand. For one, the natural question then is,
             | "what is a set?". Whatever one does there has to be some
             | brain washing in order to get started. This is unavoidable
             | unless one thinks _Principia Mathematica_ should be the
             | starting point.
        
         | threatofrain wrote:
         | A number as a general concept is an informal term within
         | mathematics.
         | 
         | Defining a rational number as a form consisting of pairs of
         | integers without a 0 denominator is pretty typical. Defining
         | the reals as Q completed by the irrational numbers is also
         | pretty typical.
         | 
         | I see this as a volume meant for an educator who is teaching
         | what is known as Algebra 1/2 in the US, and as such, it is very
         | high quality and thorough. It is very typical in pedagogy to
         | first teach the objects and their behavior for intuition, and
         | only them do you discuss formalisms.
        
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