Mathematical symbols question (Myrtle?)

[Plaid Dad]

I'm taking myself through the Teaching Company's Algebra I course, and last night I was watching the second lesson. The teacher was going through the various sets of numbers: natural, whole, integers, rational and irrational, real. She noted that some of these sets have internationally recognized symbols, such as N for the natural numbers. For two sets, she created symbols for use in the course, but she said they were not standard mathematical symbols. Are there no internationally recognized symbols for whole and for irrational numbers? I searched around on the web, but couldn't find a clear answer. TIA!

[Jane in NC]

While the set of whole numbers is used in elementary school, it is not a common set for mathematicians. The natural numbers are used often in series or iterative processes--if one wants the count to start at 0, an adjustment is made--but there is usually no reason to specify the set of Whole Numbers in a larger mathematical context.

Regarding irrationals: The properties of the reals are true for irrationals. It is rare to want to work only with the set of irrationals--more common to work over the set of rationals or all reals. For example, in basic algebra, exponents are first defined for natural numbers, then integers, then rationals, finally reals. There is no reason that I can apparently see for defining an irrational exponent.

Of course, people can make up their own symbolic notation and do. For example, someone first used boldface Z for integers (Bourbaki, I believe) or "e" for the number (base of the natural log), named after Euler. The creation of notation has contributed to students' frustrations. For example, the derivative notion used by Newton was different than that used by Leibniz. Modern students thus learn several ways to denote the derivative (prime, dot, d/dx).

Best,
Jane

[Plaid Dad]

Thanks so much, Jane! That's really helpful. :)

[Myrtle]

The natural numbers are used often in series or iterative processes
Best,
Jane

Unless you are Keith Devlin.

HIJACK: Have you seen his latest? "Multiplication Ain't no repeated addition"

And if you could read it and shed some light on what he could possible mean I'd love to know. We're scratching our heads at our house and I just did a blog entry on it. It turned into an entertaining and amusing flame war on Denis's "Let's Play Math" blog (she closed the comments but it's all still there)

It would be perhaps elucidating for me to see this discussed by cooler heads. I asked my FIL who dismissed it as crackpottery and in fact, I'm having problems getting anyone with a degree in math to take what he's saying seriously and explain it to me.

The only person who is taking Devlin's side on this recommended an analysis text which supposedly develops the real numbers from four axioms. It's by a Polish mathematican called Mikusinksi and I went ahead and ordered the book (it was only $12)

And the person that originally recommended this book also kindly emailed me the first 10 or so pages. After reviewing them it would seem that perhaps Mikusinski is able to use only four axioms because he assumes the rest informally. In fact, in the first paragraph of the book he dismisses the need to define inequality or introduce ordering axioms because "any school child understands this"

I will suspend judgment until I see the rest of the book though.

Plaid Dad:

I think set theory was invented by Cantor and it was Richard Dedekind who then showed how the different sets are related. Prior to Dedekind everyone was using Euclid to explain rational numbers and Dedekind was the first to say, "It's not a good idea (proof wise it didn't work)to use geometry to explain arithmetic, arithmetic and algebra need their own set of axioms" He was disatisfied with the gaps in logic that the explanations involved and tightened up the mathematical arguments used. In other words, there were mathematical problems that came up and Euclidean axioms and ideas couldn't be used to solve these by 19th c mathematicians. These problems were solved with Cantor's set theory, Dedekind, and Peano.


Cantor and Dedekind are very important in the history of ideas, not just math, but because they contributed a lot to how we view "infinity" and the origin of these musings of infinity was with the Greeks. If one is watching how the infinity concept baton is handed off in Western Civ, Cantor and Dedekind can be seen running with it.

I guess I should add that infinity was explored in the idea of "infinite" sets, and infinite numbers (continuity) between any two given numbers. At the time in math they were some glaring mistakes made by Euler, (for one,) on how to handle this. (It comes up as a division by zero problem, for example, back in the day people didn't say that this was undefined) and also dealing with the "infintesimals" ...an idea used in 18th century calculus. The ancient Greeks, preferring to avoid assumptions, avoided the idea of infinity as much as they could in their math and there is some neat stuff in Euclid where you can see him contorting his proofs to avoid bringing it up directly.

I'm having a bit of notational issues with set theory myself. The Germans seemed to have used some interesting German letters for this and I can't make out what's up with it.

[nmoira]

Wow Myrtle, I learn something almost every time you post. I couldn't rep you though... too soon.

:sad:

[Caroline]

Myrtle, I ordered the book. And I, too, learn something every time you post.