Friday, May 19, 2006

'Rithmetic

Make sure to read my other posts on The Three R's:
The Three R's
Reading
Reading Comprehension Test
C.S. Lewis on "Literary" and "Unliterary" Reading
'Riting
A Few Samples of Past Writings
Prologue to the Story I'm Writing

'Rithmetic - the last of the Three R's, and probably my favorite. As I mentioned in my first post on The Three R's, I really think it would be more parallel to list mathematics with reading and writing, but I'll try not to be too picky :).

I'm guessing that most if not all people reading this post know by now that math is a major part of my life. It may have something to do with the fact that I was raised in a heavily math environment - just a hunch ;). My dad has taught high school (and some college) math for close to 20 years, my mom and I both teach math to homeschoolers, and I also tutor math to a number of public school students. Brother Dear and Sister Dear are also very gifted in math (in fact, both scored higher than me on the math portion of the SAT), though they use it less on a daily basis than Mother Dear, Father Dear, and I do. Especially given the growing problem of innumeracy in our nation, I'm very thankful for the mathematical background that I have :).

I've already blogged quite a bit about math, so I'll try not to muse too long in this post, instead just give a brief overview of some of the antics of my math family.

You see, when it came to liking math, I really didn't have a choice in the matter ;). My parents gave me bedtime math problems growing up. We also played a lot of math games like Muggins, or games that encouraged quick mental arithmetic, like Yahtzee, and we watched math-oriented TV shows like Square One TV. It has been rumored that we sometimes debate the coolest number, but there is little evidence for that ;). We do make sure to celebrate an important mathematical holiday every year, and note the mathematical significance of ages and license plates. I admittedly enjoy doing problem-solving exercises just for fun, and I'm not above occasionally participating in a mathematical duel. And yes, I occasionally have random mathematical thoughts, I admit, but if someone tells you that I purposely bordered a quilt with golden rectangles, don't believe them, and if someone refers to my family as those weird math people, it's a gross exaggeration! Okay, okay, on occasion my family does enjoy giving mathematically-themed gifts, and completing mathematical color-by-numbers. And even Brother Dear has written some clever mathematical sweet-nothings that are sure to win any girl's heart.

Hmm, maybe we are a bit abnormal. . . or just special :-D.

Oh, oh, in related news, we recently acquired a slide rule, so I'm going to play around with that this summer :). Yay!

To tie this post into my previous post on reading, I really must recommend an excellent book that relates to mathematics. It's not a textbook, but a juvenile biography on Nathaniel Bowditch. If you haven't already, you really need to read Carry On, Mr. Bowditch, by Jean Lee Latham! It's an excellent true story of a child with a hunger for knowledge and a strong self-determination to excel, even though that often meant teaching himself. Forced to leave school at a young age, he continued his education on his own, and managed to teach himself French and Latin, in addition to excelling in the sciences of navigation and mathematics. He was instrumental in the improvement of many aspects of navigation, through his knowledge of mathematics. Nat Bowditch's advancements in navigation were instrumental in the popularization of "book sailing," or sailing by mathematical charts and tables. If you want to know why precision in mathematics is so important, you need to read this book!

Okay, I'm almost done, but I cannot close this post on math without sharing with all of you my absolute favorite mathematical proof:

Theorem: All positive integers are interesting.

Proof: Assume the contrary. Then by the well-ordering principle, there is a lowest non-interesting positive integer. But, hey, that's pretty interesting! A contradiction. QED

:-D

17 comments:

Adrian C. Keister said...

A nice summary of mathematical wierdness. Naturally, who am I to talk?

Ah, yes, Carry On, Mr. Bowditch. Fantastic book; more people should read it. I like how he always learned a new language starting with John 1:1. Perhaps a not-so-well known fact: Bowditch's book is still used in the US Navy.

In Christ.

Ben Garrison said...

I had forgotten about that book.

OH MY GOSH IT WAS SO FRUSTRATING WHEN THE CABLE KEPT SNAPPING!!!

Mrs Blythe said...

Maths is my Achilles heel. I failed GCSE 3 times before finally passing (I needed at least grade C along with my A levels to get into University). I hated my maths teacher, she couldn't control the class and made it as boring as possible. I have jokingly said that I have number dyslexia, but I think it was a mix of bad teaching and anxiety! I did well with statistics though, funnily enough, which were necessary for some of the experiments and studies for my course. I just hope that my kiddies fare better but I'm just not sure how to encourage them given my hopeless background in the subject. :o.

helen said...

LOL and your ending! Thanks for the laugh =D

Susan said...

Um, wrong book, Ben. You're thinking of Young Man in a Hurry about Cyrus McCormick. It was also written by Jean Lee Latham, but not the same. AND YES, IT WAS FRUSTRATING WHEN THE CABLE KEPT SNAPPING!!!

Susan said...

Okay, I'm slow again, Adrian. I just realized you meant Bowditch's book on navigation. I was trying to figure out why Carry On, Mr. Bowditch was used in the US Navy. It's sad, isn't it? Anyway *smoothing back blond hairs again* I had no idea his book was still used. I thought it would be obsolete with all the new technology. That's pretty neat.

It's interesting, Mrs. Blythe, because statistics really is a whole other discipline from mathematics, though it's related and often categorized as the same. I've known people who were horrible at math but loved statistics. It's a very practical subject :).

une_fille_d'Ève said...

'Rithmetic - the last of the Three R's, and probably my favorite.

I'm actually shocked that you would say that, Sister Dear! Sure, you're a math nerd, but you value math over reading? That's extreme...

And I was with you, Su, interpreting Andrian's comment to mean that they used Carry On, Mr. Bowditch in the Navy. *laughs* And people think we're intelligent. Boy, do we have them fooled! *also smooths back blond hairs*

une_fille_d'Ève said...

er, make that "Adrian's comment"

Susan said...

Well, note, Hannah that I said probably. It's really hard to choose between math and reading. It's like picking a favorite star in the sky :-D.

Yay! Clueless blondes of the world unite! I'm reminded of Mrs. Z's comment after I took my own piece in chess :-D. It's hard to believe she's as smart as she is. . . . or something like that ;). Hehe.

Ashley said...

Hi!

I have nothing to say about math.

John Dekker said...

Great post! I haven't followed all the links yet, but I know I'm going to enjoy doing so.

My favourite number is, of course, the lowest number that cannot be described in less than thirteen words.

Susan said...

Hehe. In case anyone cares, I meant to say Cyrus Fields, not Cyrus McCormick :). I was rereading my comment and thinking, Hmmmm, I thought Cyrus McCormick invented the reaper. Google agrees.

John,

If you actually do read through all the links, I'll be impressed. The ensuing comments on a few of the links are actually better than the posts themselves :).

I don't have time now to figure out what your favorite number is, but I'm going to have to figure it out, or I won't be able to sleep tonight :). I refuse to use google as an aid.

Anyway, thanks for stopping by.

John Dekker said...

I'm glad I've set you a challenge! ;) I look forward to hearing your answer.

But feel free to use google...

Susan said...

Okay, I know it took me a while to respond, but I've been at a conference the last 3 days, and barely have had time to breathe :). In reference to your favorite number, I've come to two conclusions:
(a) It is a trick question, and there is no such number, or
(b) You inadequately worded your description.
Here is why:

Your description is the lowest number that cannot be described in less than thirteen words, which allows for the use of negative numbers. With negative numbers, there is (in general - note: I am currently only addressing the integer case) a pretty strong inverse relationship between the "smallness" of the number and the words needed to describe that number. I realize there is the occasional exception, such as "negative one-billion" or the like; I am speaking of a general correlation. So, once I reach a negative number that must be described in thirteen words, I can easily find an infinite set of smaller numbers by continually subtracting from it to get smaller numbers that will (in general) have 13+ word descriptions as well. In this way I can find an infinite set of negative numbers that cannot be described in less than thirteen words. Note that the corresponding positive set is irrelevant here, since any negative number is smaller than any positive number, and we are concerned with the smallest number.

Now, in that (hypothetical) infinite set of negative numbers that fit the number-of-words requirement, there is no least number. This does not contradict the Well-Ordering principle (which I cited in the proof at the end of my post), since the principle does not apply here. In the proof that "every positive integer is interesting," the principle only worked because the problem dealt with the natural numbers. The Well-Ordering principle states that each subset of the natural numbers has a least element, but infinite sets of numbers in general (or even a set of only the integers) do not necessarily have a least element. I mentioned the corresponding positive set of numbers that cannot be described in less than thirteen words. This set, though infinite, does have a "smallest number," but your description is not restricted to only positive numbers, so this positive set is irrelevant, since I can easily construct a smaller number by tacking "negative" onto any number in that set.

So, your favorite number is non-existant unless "non-negative" is specified. One could find a positive integer that would fit your description. Even then, though, is "45" one word or two, since some people write it as "forty five" and others as "forty-five." In addition, must "and" be included in a written description of 105? - "one-hundred and five" or "one-hundred five", for example. So, even finding a positive integer that fits the description is difficult. Okay, let's set the negative numbers aside and assume we have agreed on a way to describe the numbers (using "and", hyphens, etc.). As I said, we could find such an integer, but if we look at numbers besides the integers, we run into problems.

First of all, complex numbers are out, since they are not an ordered set, but even if we look at the rationals or the irrationals, we have tons of ambiguities. For instance, the irrational number phi (my personal favorite number), one could argue, cannot be described in fewer words than nine, since it is "one plus the square root of five over two," but actually it requires more words to be clear. I would argue that it would have to be described in no fewer than eleven words: "quantity, one plus the square root of five, quantity, over two." But someone else may argue that my own description is not adequate or conversely, superfluous, in describing phi. Irrational numbers are much more difficult to succinctly describe, and often have geometric (non-numeric) descriptions.

The rational and irrational numbers, unlike the integers, are not organized to a list as nicely (the irrationals are uncountable, or un-listable, in fact) so we can't simply make a list and notice that phi is smaller than e which is smaller than pi (which is true), etc., and decide we've covered everything. What about the number "phi over pi"? We can always find a way to construct a smaller rational number, whose description may or may not necessitate 13+ words. The number of words need to describe positive integers increases somewhat steadily with the value of the number, but this is far from true with the irrational numbers. For example the number "quantity, phi plus pi plus e plus three, quantity, over, quantity, phi plus pi plus e plus four, quantity" is approximately .913, but I could make a much smaller number with many fewer words by making the numerator simply "one", or the number "one over, quantity, phi plus pi plus e plus four, quantity" - or approximately .087. This illustrates the fact that there is nothing close to a direct relationship between the number of words needed to describe a rational or irrational number and the "smallness" of such a number.

So, in summary, there would be no way to find such a number, even if the number was restricted to the positive numbers. The exception would be if your description was re-worded as the lowest positive integer that cannot be described in fewer than thirteen words, but even then, you would have to clarify what constitutes superfluous wording.

John Dekker said...

Wow.

You're right, of course - I should have used "positive integers". But I don't think a negative integer of large magnitude is "small" in any meaningful sense.

{I should also have used "fewer" - I can't believe I'm still making that mistake. Why is it that only homeschoolers know grammar?}

You were right to pick up on the slippery nature of that word "describe". (Though perhaps not quite as slippery as "interesting"). Yet it can't have been a trick question, because it wasn't even a question. ;)

Having said all that, I think you missed the obvious. :)

Oh, and I've just coined a new word - "sproof". I'll be publishing a sproof on my blog tomorrow.

Looking forward to your post about the conference.

Susan said...

Okay, trick description then ;), not question. I was so careful to say "description" all through my comment, but slipped up in that one instance. Oops.

Yes, certainly "interesting" is even more ambiguous, which is the only way a proof like that works. Too true :).

So what is the obvious that I missed? I'm an analytical blonde, so I have the habit of thinking too deeply about things (as evidenced by my previous comment) but missing the really obvious.

In my case, I remember to distinguish "fewer" and "less" by conditioning. My mom twitches when those terms are used incorrectly. And yes, I was homeschooled :).

Sproof? Is that a pseudo-proof, perhaps? I guess I'll wait and see.

John Dekker said...

OK, I think I nailed it. But I had to change "thirteen" to "fourteen", of course. :)