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Intermediate Algebra

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Re: Intermediate Algebra

#11

Post by Gehinnn »

syops wrote: post #8
Gehinnn wrote: post #6

Earlier quoted context omitted.

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.

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…

Well, the peano arithmetic can be described directly as first order logic without set theory ;)

I'm fine with having an intuition for sets, but I think reals really should be defined properly. At least, R should not be confused with the algebraic closure of Q.

Re: Intermediate Algebra

#12

Post by threatofrain »

Gehinnn wrote: post #11
syops wrote: post #8

Earlier quoted context omitted.

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…

Well, the peano arithmetic can be described directly as first order logic without set theory ;) I'm fine with having an intuition for sets, but I think reals really should be defined properly. At least, R should not be confused with the algebraic closure of Q.

This volume does not confuse R as the algebraic completion of Q. It is completely reasonable for an Algebra 1/2 teacher to wait for a Calculus or Analysis teacher to discuss the metric completion of Q. Describing R as rational + irrational numbers is a completely solid description.

Re: Intermediate Algebra

#13

Post by Aicy »

raegis wrote: post #9
Gehinnn wrote: post #3

> 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. Reals are basically defined as number. But what is a number? Also the definition of Q is missing the quotient construction (or any motivatio…

I'm completely serious here. Could you give us your definition of "number" which you would teach at this level? Please, please, I'm not trying to be an ass or anything. We could start a great conversation here.

From Wikipedia:

> In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line

That's the intuitive definition that makes the most sense to me.

Re: Intermediate Algebra

#14

Post by sunneversets »

Gehinnn wrote: post #3

> 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. Reals are basically defined as number. But what is a number? Also the definition of Q is missing the quotient construction (or any motivatio…

From the TOC, it looks like an elementary algebra book.

Students usually don't see rigorous construction of integers and rationals till classes titled something to the effect of "intro to proofs 101", "intro to discrete analysis", "abstract algebra 101", "elementary general algebra", "elementary set theory", "basics of elementary number theory"... And the construction of reals would have to wait till something like "elementary real analysis 101".

Usually, after elementary algebra come 3 semesters of calculus. Subject as vague and, uhh, as un-mathematical as it contains neither concrete definitions(other than the one for derivative), nor any theorems. Nothing substantial to grasp at when drowning.

Traditionally, actual learnable definition-lemma-theorem-corollary-examples style math starts after the calculus sequence. So the audience for the linked book is at least 2 classes removed from the time they get to see number construction.

While elementary algebra is both useful and unavoidable, I don't think the same about the calculus. IMHO, the latter is just a waste of time.

The sequence elementary algebra -> intro to math proofs -> elementary real analysis -> Lebesgue Integral by way of Daniell-Riesz -> complex analysis -> non-Euclidean/abstract topology -> measure theory -> probability theory -> differential geometry would give anyone world- class education into the nature of functions which the calculus sequence is kind-sorta supposed to give you a tiny and very distant taste of.

Re: Intermediate Algebra

#15

Post by Aicy »

threatofrain wrote: post #10
raegis wrote: post #9

Earlier quoted context omitted.

I'm completely serious here. Could you give us your definition of "number" which you would teach at this level? Please, please, I'm not trying to be an ass or anything. We could start a great conversation here.

A number is an informal term within mathematics.

Not exactly true. You can formally construct the set of real numbers with set theory and then say a number is an element of that set Wikipedia page on different approaches: https://en.m.wikipedia.org/wiki/Construction_of_the_real_num...

Re: Intermediate Algebra

#16

Post by young_unixer »

syops wrote: post #4
Gehinnn wrote: post #3

> 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. Reals are basically defined as number. But what is a number? Also the definition of Q is missing the quotient construction (or any motivatio…

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.

What should I read if I want to learn what a number is?

Re: Intermediate Algebra

#17

Post by Gehinnn »

raegis wrote: post #9
Gehinnn wrote: post #3

> 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. Reals are basically defined as number. But what is a number? Also the definition of Q is missing the quotient construction (or any motivatio…

I'm completely serious here. Could you give us your definition of "number" which you would teach at this level? Please, please, I'm not trying to be an ass or anything. We could start a great conversation here.

I'm sure there are more elegant approaches, but I would start like this:

A number is an imaginary (as in made up by humans, not as in complex numbers) object that solves certain equations. It is not well defined and should not be used as basis for other definitions.

"Natural numbers" can be constructed directly and have a well known notation. Integer numbers are used to make the equation "a + x = b" solvable by x for every natural number a and b.

Rational numbers (Q) make the equation "ax = b" solvable by x for any non-zero integer a and b.

The algebraic closure of Q (=: Q*) makes the equation P(x)=0 solvable by x for any polynomial P.

The definition of R is more complex. I guess I would simply say it contains numbers to solve equations like sin(x)=0 by x.

This makes it very clear that -3 or 0 apples don't exist. These numbers just have the purpose to solve equations which makes them handy if you want to compute stuff.

Re: Intermediate Algebra

#18

Post by threatofrain »

Aicy wrote: post #15
threatofrain wrote: post #10

Earlier quoted context omitted.

A number is an informal term within mathematics.

Not exactly true. You can formally construct the set of real numbers with set theory and then say a number is an element of that set Wikipedia page on different approaches: https://en.m.wikipedia.org/wiki/Construction_of_the_real_num...

If we're talking about "number" in general, yes, it is true that this is an informal term among mathematicians.

Re: Intermediate Algebra

#19

Post by Gehinnn »

threatofrain wrote: post #12
Gehinnn wrote: post #11

Earlier quoted context omitted.

Well, the peano arithmetic can be described directly as first order logic without set theory ;) I'm fine with having an intuition for sets, but I think reals really should be defined properly. At least, R should not be confused with the algebraic closure of Q.

This volume does not confuse R as the algebraic completion of Q. It is completely reasonable for an Algebra 1/2 teacher to wait for a Calculus or Analysis teacher to discuss the metric completion of Q. Describing R as rational + irrational numbers is a completely solid description.

It is only solid as long as you don't define irrational numbers literally as everything that is not rational.

Re: Intermediate Algebra

#20

Post by Aicy »

threatofrain wrote: post #18
Aicy wrote: post #15

Earlier quoted context omitted.

Not exactly true. You can formally construct the set of real numbers with set theory and then say a number is an element of that set Wikipedia page on different approaches: https://en.m.wikipedia.org/wiki/Construction_of_the_real_num...

If we're talking about "number" in general, yes, it is true that this is an informal term among mathematicians.

What mathematicians are you talking about? Set theorists and number theoriests have very precise definitions of numbers.

Maybe by "number" in general do you mean something that encapsulates both "real numbers" as in lengths, and "integers" as in the discrete counting numbers? In which you can quite easily do so by defining real numbers, either formally or saying that something like the Wikipedia definition that they are a distance along a line, and then saying integers are a subset of those numbers.

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