"By replacing post and beam construction with cantileverage supports, walls were no longer viewed as enclosing walls, but rather as independent and unattached. Any one of these walls could be modified by either shortening, extending, or redividing."
This is the last page of the section on Frank Lloyd Wright. Pappas is in the middle of telling us how the architect "destroyed the box" in order to make the room look more like open space than a box. So she continues:
"Wright did not stop with freeing the horizontal plan, but also the vertical."
As she explains, Wright opened up the top to the sky by using columns to hold up the ceiling. He referred to this as organic architecture. Pappas ends this section with a quote from the architect:
"So organic architecture is architecture in which you feel and see all this happen as a third dimension ... space alive by way of the third dimension."
There is another picture on this page, but of course you can't see it. Then again, even I who can see the picture am looking at only a 2D representation of the house -- as Wright points out here, we need to see the actual 3D house in order to feel the true effect of organic architecture.
Chapter 5 of Stanley Ogilvy's Excursions in Number Theory is called "Irrationals and Iterations." He begins like this:
"The infinite sequence of the perfect squares begins like this: 1, 4, 9, 16, 25, 36, 49, 64, .... It is perhaps not immediately apparent that nowhere in this sequence is there one number that is twice another."
Ogilvy is, of course, writing about irrational numbers -- in particular, the irrationality of sqrt(2). You may recall that irrationals are part of the first Common Core standard for eighth grade -- 8.NS1. A little over a year ago, I wrote about how I found out the proof that sqrt(2) is irrational:
Give an indirect proof to show that sqrt(2) is not a/b for any whole numbers a and b...that is, show that sqrt(2) is irrational! I remember when I first saw the famous indirect proof of the irrationality of the sqrt(2) -- it was given in a sidebar in my old Algebra I text. I had heard for a while that sqrt(2) is irrational, but I didn't realize that it was something that could be proved.
I tried to tell my eighth graders about the proof, and I wrote about it last year:
I might give my students a proof of the irrationality of sqrt(2). In fact, I'd love to tell the story such as the math YouTube star Vi Hart does:
In her video, Hart discusses how Pythagoras and his disciples eschewed beans. I definitely like Hart's pun about how one of the followers "spilled the beans" about the irrationality proof!
Today I start to tell the story about Pythagoras, but my eighth graders are a little confused as to why I'm telling them this story. I ask them what sqrt(2) is, and one student tries 1, until I tell them that 1 times 1 is 1. And of course they shouldn't even bother with 2, since 2 times 2 is 4. Someone suggests 1 1/2, or 3/2, which is a good guess, but its square is 9/4, which is a little more than 2.
But then the story falls apart when I try to suggest 7/5, or 1.4, as the next guess, since the students are confused where these numbers come from. And besides, the story works better when sticking to fractions, rather than decimals -- otherwise, students may wonder why sqrt(2) is "impossible" when 1.414213562... is apparently sqrt(2). (Though in the end, one student does figure out what my story is leading up to.)
Actually, the story might have gone better if I'd begun just as Ogilvy does, showed the students the sequence 1, 4, 9, 16, 25, 36, 49, 64, and asked them to find one number that's double another. Then again, we recall that students talked loudly a lot in that class, and so they might not have been able to appreciate any explanation that I gave them.
Ogilvy gives a straightforward proof of the irrationality of sqrt(2) -- if it were a rational a/b, then its square a^2/b^2 must simplify to 2. But then the factors of b^2, which appear in pairs, somehow cancel out the factors of a^2, which appear in pairs, to leave only a 2, which is impossible. Therefore sqrt(2) is irrational -- and he goes on to explain that all square roots of non-perfect squares and cube roots of non-perfect cubes are irrational.
This is actually all Ogilvy has to say on irrational numbers. Instead, he suddenly switches to a pattern with multiplication:
-- 1 * 142857 = 142857
-- 2 * 142857 = 285714
-- 3 * 142857 = 428571
-- 4 * 142857 = 571428
-- 5 * 142857 = 714285
-- 6 * 142857 = 857142
-- 7 * 142857 = 999999.
The author explains that the reason for this cyclic permutations is the decimal expansion of 1/7. He writes out the division:
1/7 = 0.142857,14...
He writes out the remainders and show that they repeat after six digits (at the comma). The multiples of 142857 are the periods of the fractions 2/7 to 6/7, while 7/7 is just 0.999999,999999... = 1.
Ogilvy tells us that 1/7 is a "full repetend" prime -- one whose period (or repetend) is of the maximum length, which is one less than the prime itself. In long division, all the remainders appear except zero. He informs us that the next full repetend prime is 17:
1/17 = 0.588235294117647,0588...
But unfortunately, no one knows how to determine whether a prime is full repetend. The author does connect the length of the period to Fermat's Little Theorem:
10^6 == 1 (mod 7)
This tells us that 7 divides a string of six 9's (with the quotient equal to the period), but it doesn't tell us whether 6 is the smallest exponent such that:
10^e - 1 == 0 (mod 7)
With 7 it happens to be. But for instance,
10^10 - 1 == 0 (mod 11),
and we find that
1/11 = .090909...
Thus 11 is not a full repetend prime, since its repetend has only two digits, 09.
At this point Ogilvy suddenly switches to primes consisting of only 1's -- repunit primes. The first repunit prime is 11, and the next two have 19 and 23 1's. At the time Ogilvy wrote his book, this was the largest known repunit prime, but since then primes with 317 and 1031 1's have been found.
The author tells us that repunits and repetends are related through the following equation:
p_1 * p_2 * ... * p_n = (10^k - 1)/9
where the right-hand side is the repunit of length k. Then the left-hand side contains all the possible primes whose period is of length k. His example is 239 * 4649 = 1,111,111, so 239 and 4649 are the only primes with period 7.
Ogilvy provides a complete list of all the primes of period less than 21. I don't wish to give the full list, so let's cut it off at period 7 (the example given above):
Period Length Primes
1 3
2 11
3 37
4 101
5 41, 271
6 7, 13
7 239, 4649
Repeating decimals is actually another part of Common Core Standard 8.NS1. (Wow, so we cover all of 8.NS1 by reading Chapter 5 in Ogilvy!) I wrote last year that perhaps it's better to teach repeating decimals before irrationals, so that students can appreciate irrationals as decimals that don't repeat.
Everything I wrote about repeating decimals applies only to base 10, decimal. In other bases, periods are of different lengths. Periods of length 2 are possible only for prime divisors of the alpha (i.e., the repunit of length 2), while periods of length 1 are for prime divisors of the omega. Thus bases with composites as their alpha and omega have more primes with brief periods, while the other bases are have more small primes with longer periods (possibly full repetend).
All odd bases have 2 as a full repetend prime, while 3 is full repetend in bases congruent to 2 mod 3 (as these have 3 in the alpha). Perfect square bases have no full repetend primes (except for 2 in odd square bases). This is because each digit of base b^2 corresponds to two digits of base b. If there were a prime of maximal length in base b^2 it would be above maximal length in base b, a contradiction.
The remaining sections in this chapter have nothing to do with Common Core 8. First, Ogilvy writes about iterated radicals (which admittedly look better in the book than in ASCII):
sqrt(n + sqrt(n + sqrt(n + sqrt (n + ...
This question often shows up on the Pappas calendar. Usually this would be on the left-hand side and the right side would be a given integer x, and she asks us to solve for n. Now Ogilvy solves all such problems at once:
x = sqrt(n + sqrt(n + sqrt(n + sqrt (n + ... = sqrt(n + x)
Squaring both sides,
x^2 = n + x
n = x(x - 1)
For x = 2, we see that n is also 2. Ogilvy calculates a few values just to be sure:
sqrt(2) = 1.414...
sqrt(2 + sqrt(2)) = 1.848...
sqrt(2 + sqrt(2 + sqrt(2))) = 1.962...
sqrt(2 + sqrt(2 + sqrt(2 + sqrt(2)))) = 1.990...
The final example in this chapter asks about raising a number to an irrational power:
x = 2^sqrt(2)
Ogilvy tells us that in his day, one can take the logarithm of both sides:
log x = sqrt(2)log(2)
and then use log tables to find the answer. Nowadays we can just use a calculator. But that still does not tell us what kind of number the answer is.
We know what it means to raise a number to a rational power p/q (that is, take the q'th root and then raise it to the p'th power), and Ogilvy does so:
2^(3/2) = sqrt(2^3) = sqrt(8)
But Ogilvy tells us that we can find take many fractions p/q that approach sqrt(2) in the limit, and then the powers 2^(p/q) approach 2^sqrt(2) in the limit. Ironically, I believe that modern AP Calculus books actually take the first approach, using natural logs to define irrational exponents (where the natural log itself is defined as a certain integral).
The author wraps up the chapter with a classic question -- prove that there exists an irrational power of an irrational number that is rational. The usual answer is to consider sqrt(2)^sqrt(2) -- if this is rational, then the proof is trivial. If it's irrational, then we let it be the base and put another sqrt(2) in as the exponent:
x = (sqrt(2)^sqrt(2))^sqrt(2)
So the base and exponent are both irrational. But
x = sqrt(2)^(sqrt(2)sqrt(2)) = sqrt(2)^2 = 2
is not only rational but integral. QED
As it turns out, 2^sqrt(2) is actually transcendental (like pi) rather than algebraic. Therefore
2^sqrt(2) = x^sqrt(2) = (((sqrt(2)^sqrt(2))^sqrt(2))^sqrt(2) = (sqrt(2)^sqrt(2))^2
is transcendental. I'll let Ogilvy take us home:
"Therefore neither is sqrt(2)^sqrt(2), and since x = 2, we have an integer expressed as an irrational power of a transcendental number (see first definition of x above)."
Ogilvy's chapter today deals with so many different definitions -- such as the definitions of irrational, cyclic permutation, full repetend prime, and so on. This leads us into the next lesson:
Lesson 2-5 of the U of Chicago text is called "Good Definitions." (It appears as Lesson 2-4 in the modern edition of the text.)
This is what I wrote two years ago about today's lesson:
Lesson 2-5 of the U of Chicago text deals with definitions -- the backbone of mathematical logic. Many problems in geometry -- both proof and otherwise -- are simplified when students know the definition.
Consider the following non-mathematical example:
Given: My friend is Canadian.
Prove: My friend comes from Canada.
The proof, of course, is obvious. The friend comes from Canada because that's the definition of Canadian -- that's what Canadian means. But many English speakers don't think about this -- if I were to say to someone, "My friend is Canadian. Prove that my friend comes from Canada," the thought process would be, "Didn't you just tell me that?" Most people would think that "My friend is Canadian" and "My friend is from Canada" as being two identical statements -- rather than two nonidentical statements that are related in that the second follows from the first from the definition of Canadian. Yet this is precisely how a mathematician thinks -- and how a student must think if he or she wants to be successful in mathematics.
And so, let's take the first definition given in this section -- that of midpoint -- and consider:
Given: M is the midpoint of
Prove: AM = MB
The proof is once again trivial -- AM = MB comes directly from the definition of midpoint.
The text proceeds with the definition of a few other terms -- equidistant, circle, and a few terms closely related to circles. Then the text emphasizes biconditional statements -- that is, statements containing the phrase "if and only if." Some mathematicians abbreviate this phrase as "iff" -- but very few textbooks actually use this abbreviation.
Notice that Dr. Franklin Mason does give the "iff" abbreviation in his text. Last year, I noticed that his Lesson 2.4 on biconditionals had an (H) symbol -- which stands for honors. It's interesting how Dr. M once considered this to be an honors-only topic. Now, he wants to emphasize the importance of definitions and biconditionals to all his students, not just his honors students.
Every definition, according to the U of Chicago text, is a biconditional statement, with one direction being called the "meaning" and the other the "sufficient" condition. Mathematicians often use the terms "necessary" and "sufficient." Many texts use the word "if" in definitions when "if and only if" would be proper -- but our U of Chicago text is careful to use "if and only if" always with definitions.
Before I leave, I point out Fawn Nguyen's latest announcement -- after reading her blog for so long, I have the chance to meet her in person when she gives a workshop in December. But alas, I'm no longer a math teacher, and so I won't be able to attend.
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