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Intermediate Algebra

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Re: Intermediate Algebra

#51

Post by im3w1l »

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?

A common construction is this:

First define natural numbers as sysops did (or you can use peanos axioms).

Then add the negative numbers (I actuallly don't remember how this is done, ig it's usually hand waved as trivial). The negative and natural numbers together make up the integers.

The rationals are introduced as a pair of numbers (a, b) where a is an integer and b is a positive integer. (a, b) is considered the same rational as (c, d) if a * d = b * c.

The reals are finally introduced as sets of rationals with a certain property, namely that if p is in S, then all smaller rationals must also be in S. Edit: there are a few more properties, see https://en.wikipedia.org/wiki/Dedekind_cut

Re: Intermediate Algebra

#53

Post by Aicy »

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.

I understand very well that the peano axioms for example are logical and not circular.

My point was that I think your explanation relies on the experience on what a number is, having used those equations before with the terms actually representing real numbers (such as 3 + 2 = 5 instead of a + x = b and sin(180) = 0 instead of sin(x) = 0).

So as a definition of what a number is it's circular as you presume experience with numbers and what they mean.

Re: Intermediate Algebra

#54

Post by glitchc »

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…

Any given irrational number may not be possible to write down.

a/b is a rational number. The set of real numbers is the union of rational and irrational number sets i.e. all possible rational and irrational numbers combined form the real number set.

Re: Intermediate Algebra

#55

Post by Koshkin »

kenjackson wrote: post #44
CamperBob2 wrote: post #25

Earlier quoted context omitted.

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.

Looks like the set of reals is simply taken as the number universe (i.e. all numbers are already assumed to be real).

Re: Intermediate Algebra

#56

Post by Aicy »

glitchc wrote: post #54
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…

Any given irrational number may not be possible to write down. a/b is a rational number. The set of real numbers is the union of rational and irrational number sets i.e. all possible rational and irrational numbers combined form the real number set.

> Any given irrational number may not be possible to write down.

There's no "may" about it!

By definition an irrational number has an infinitely long decimal expantion that does not repeat, so none of them can be written down (other than in symbolic representations like pi)

Re: Intermediate Algebra

#57

Post by yodsanklai »

penteract wrote: post #45
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.

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…

> At this level, associating numbers with potentially infinite decimals would not be too bad.

I find number with an infinite number of decimals quite an abstract concept. Another issue with this is that it confuses numbers with their representation.

Re: Intermediate Algebra

#58

Post by javajosh »

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?

Numbers start with counting. You can build up more kinds of numbers, but tearing down counting into smaller pieces is surprisingly hard. Personally, I think it's good enough to start with any two distinguishable states. You can then keep combining those states in various ways to form the counting numbers. Note that more complex symbols like "1 2 3" are themselves effectively a form of tally, where you are counting the number of angles. You can combine your method with geometry or a clock and call it "measurement" of space and time, respectively.

(Real numbers are my favorite construction because I was enamored with Cantor's diagonal argument when I first learned it. It's quite clever and hints at the magic mathematicians are capable of. Although most mathematicians (algebra peeps) seem to like the classic "root 2 is irrational" proof more.)

Re: Intermediate Algebra

#59

Post by Aicy »

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?

Learn set theory! Specifically ordinals (just one formal defintion of a number)

This vsauce video is actually a really accurate (and entertaining!) introduction to set theory: https://youtu.be/s86-Z-CbaHA

Re: Intermediate Algebra

#60

Post by SatvikBeri »

Aicy wrote: post #20
threatofrain wrote: post #18

Earlier quoted context omitted.

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…

Mathematicians have precise definitions of "real numbers", "complex numbers", etc. but not of "number." For example, are the hyperreals numbers? Nonstandard integers? Quaternions?
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