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Intermediate Algebra

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Re: Intermediate Algebra

#31

Post by alisonkisk »

CamperBob2 wrote: post #25
Gehinnn wrote: post #19

Earlier quoted context omitted.

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

What would be an example of a real number that's neither rational nor irrational? (I'm not a math guy, in case it's not obvious)

There aren't any, but it's circular reasoning.

It's exactly the same as saying "real numbers are the union of rational numbers and gargoyle numbers". What's a gargoyle number? "It's any number that's not a rational numbers." What's a number? "It's anything that's a rational numbers or a gargoyle number"

Re: Intermediate Algebra

#32

Post by alisonkisk »

cubano wrote: post #21
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.

You "win at the game" by learning what is taught, getting the "A", and then doing you own in-depth research about what interests you on your own time. K-12 was, of course, invented by the Germans in order to create good little factory workers that would get up early and work all day and not complain too much. The fact we still use the word Kindergarten is a nod to this origin story. They weren't at all interested in…

Why on earth would a factory worker go to high school? This argument makes no sense.

Re: Intermediate Algebra

#33

Post by Koshkin »

antegamisou wrote: post #23
Gehinnn wrote: post #17

Earlier quoted context omitted.

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…

>A number is an imaginary object and now the time has come for Intro to Complex Analysis, good luck not leaving everyone completely dumbfounded.

Also, a natural number is not “imaginary” in any sense. (E.g. you do not imagine the number of fingers on your hand - you discover it, by counting.)

Re: Intermediate Algebra

#34

Post by jtimdwyer »

Aicy wrote: post #13
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.

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.

Now define distance :)

Re: Intermediate Algebra

#35

Post by zsmi »

young_unixer wrote: post #16
syops wrote: post #4

Earlier quoted context omitted.

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?

I once tried to go down this rabbit hole but gave up quickly as it's a really deep subject. Maybe start here:

https://en.wikipedia.org/wiki/Set-theoretic_definition_of_na...

Re: Intermediate Algebra

#36

Post by syops »

young_unixer wrote: post #16
syops wrote: post #4

Earlier quoted context omitted.

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?

It depends on your mathematical background. I don’t know of any books about this at the level below senior undergraduate mathematics. If you are familiar with sets I can outline the idea behind how to define non negative integers.

We assume the empty set exists and call this 0. We define 1 to be the set containing 0. So 1 = {0}. We define 2 to be the set containing 0 and 1. So 2 = {0, 1}.

Let’s look at this set: {a, b}. I know this set has size 2 and not 1 because I can map {a, b} to {0, 1} in a one-to-fashion. I can’t map {a, b} to {0} in a one-to-one fashion. We say any set has size 2 if it can be mapped to {0, 1} in a one-to-one fashion.

Re: Intermediate Algebra

#37

Post by Koshkin »

CamperBob2 wrote: post #25
Gehinnn wrote: post #19

Earlier quoted context omitted.

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

What would be an example of a real number that's neither rational nor irrational? (I'm not a math guy, in case it's not obvious)

∞

Re: Intermediate Algebra

#38

Post by syops »

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.

The algebraic closure of the rationals is not the reals. The reals are the completion of the rationals using the standard metric.

The intention of my post was to point out the complexity of not brain washing students at a low level. Your comments have enhanced my point by bring up considerations I didn’t want to get into!

Re: Intermediate Algebra

#39

Post by syops »

CamperBob2 wrote: post #25
Gehinnn wrote: post #19

Earlier quoted context omitted.

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

What would be an example of a real number that's neither rational nor irrational? (I'm not a math guy, in case it's not obvious)

There is no such example. All real numbers are either rational or irrational.

Re: Intermediate Algebra

#40

Post by syops »

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 don’t give a definition. The students know real numbers are the “decimal numbers”. They don’t need to know a definition at this point and giving one more precise than “decimal numbers” will just confuse them. They gain familiarity by practice in the same way a Fahrenheit person like myself can only get intuition with Celsius temperatures is by using them.
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