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Intermediate Algebra

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Re: Intermediate Algebra

#41

Post by zsmi »

Gehinnn wrote: post #6
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.

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.

> Everything feels as if it was randomly defined by the teacher.

I suppose you prefer things randomly defined by Euclid? Just kidding... kinda. Seriously though, randomly defining things and then working through the consequences of that definition is a totally valid way to do math. Those random definitions are called postulates.

Re: Intermediate Algebra

#43

Post by zsmi »

jtimdwyer wrote: post #34
Aicy wrote: post #13

Earlier quoted context omitted.

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 :)

Don't forget "continuous" and "line". :)

Re: Intermediate Algebra

#44

Post by kenjackson »

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)

I think by the definition given at the beginning of the thread irrational numbers are numbers that can’t be expressed as a ratio of two natural numbers. i is irrational by that definition, but not a real.

But I must admit I haven’t read the whole post.

Re: Intermediate Algebra

#45

Post by penteract »

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.

At this level, associating numbers with potentially infinite decimals would not be too bad. The examples of π and √2 are shown as decimals, suggesting that students are already expected to think something like this.

I think that this pair of definitions for real and irrational numbers are unacceptable - the examples of irrationals mean that they don't mislead anyone (unless you can find someone who thinks of complex numbers as numbers but hasn't encountered the term "real numbers"), but it's wrong to pretend that this is a mathematical definition. Even just removing the words "defined as" would be fine.

Re: Intermediate Algebra

#46

Post by [deleted] »

Gehinnn wrote: post #27
Aicy wrote: post #22

Earlier quoted context omitted.

Personally I find these definitions a bit circular. Equations like "a + x = b", "ax = b" and "sin(x) = 0" have no meaning unless you have an understanding of what the terms mean, or could mean.

You can easily define + on natural numbers and * on integer numbers without running into problems! Thus there is no cirularity here.

[deleted]

Re: Intermediate Algebra

#47

Post by Aicy »

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 don't think this is a fair critisim - they've just said it reverse order.

They should start with the set of real numbers, denoted R, and then this can be split into two disjoint subsets - the rational numbers and the irrational numbers. From there the natural numbers are a subet of the rational numbers.

Re: Intermediate Algebra

#48

Post by Aicy »

jtimdwyer wrote: post #34
Aicy wrote: post #13

Earlier quoted context omitted.

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 :)

You can make this argument on literally any definition - you have to start with something.

So dictionary.com says the defintion of distance is:

> the extent or amount of space between two things, points, lines, etc.

That works for me.

I think it makes sense to start with something physical, in the real world that people - and especially kids, can intuatively understand. Our brains by default understand what distance is. Maybe that's true for numbers as well- and that's ok too.

Re: Intermediate Algebra

#50

Post by yodsanklai »

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.

At that level, I'd use a geometric definition. I would postulate the existence of a set R of numbers which represent the distance along a line based on a given unit. From there, you "show" that natural and rational numbers belong to R. You show that some numbers are irrational (easy to prove that sqrt(2) is).

There are some postulate there but I think it provides the right intuition and it's not a obviously circular definition.

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