[HN Gopher] Intermediate Algebra ___________________________________________________________________ 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. ___________________________________________________________________ (page generated 2021-09-06 23:00 UTC)