"By 1800 mathematicians and physicists had developed calculus into an indispensable tool for the study of the natural world."
This chapter is all about proving the theorems of Calculus -- something that is often taught nowadays in upper division college-level classes called "Real Analysis." The plan is to satisfy Bishop Berkeley and his objection that Calculus is all about "the ghosts of departed quantities."
We began with the late 18th century mathematicians, who began to grow concerned:
"In particular, they were uncertain how to handle discontinuous functions, such as f (x) = 0 if x < 0, f (x) = 1 if x > 0. This function suddenly jumps from 0 to 1 as x passes through 0."
And the problems were exposed by Joseph Fourier:
"And it was Fourier who really got up their noses, with his amazing ideas about writing any function as an infinite series of sines and cosines, developed in his study of heat flow."
The functions expressible as a Fourier series include, believe it or not, discontinuous functions:
"Slowly the mathematicians of the 19th century started separating the difficult conceptual issues in this difficult area."
And the first important concept that needed to be defined was "continuous function":
"The person who made the first serious start on sorting out this mess was a Bohemian priest, philosopher, and mathematician."
That mathematician was Bernhard Bolzano. He came up with a rigorous definition of continuity and proved, for example, that polynomial functions are continuous. Stewart also writes about Niels Abel (another mathematician I discussed in side-along reading a few years ago), who complained about the lack of rigor up until that point:
"His criticisms struck home, and gradually order emerged from the chaos. Bolzano's ideas set these improvements in motion. He made it possible to define the limit of an infinite sequence of numbers, and from that the series, which is the sum of an infinite sequence."
(I'm sorry, but considering that today is Game 7 of the World Series, I can't avoid the joke -- Bolzano proved that the World Series converges. OK, replace "World Series" with 1 + 1/2 + 1/4 + 1/8 + ....)
So Bolzano rigorously defined the limits that are still taught today in Calculus AB. But his definition was more formal that anything that appears on the AP: the sequence a_n tends to limit a as n tends to infinity if for every epsilon > 0, there exists N > 0 such that for any n > N, |a - a_n| < epsilon. (A few years ago I wrote about Erdos, who called little children "epsilons." This is where that name comes from -- "epsilon" is supposed to be a small number.)
Stewart tells us that this idea extends to Fourier series:
"There are several variations on the basic theme, and for Fourier series you have to pick the right ones. Weierstrauss realized that the same ideas work for complex numbers as well as real numbers."
It was this mathematician, Weierstrauss, who first came up with the idea of power series (as in Taylor or Maclaurin series), as well as some other interesting functions:
"One of his most surprising theorems proves that there exists a function f (x) of a real variable x, which is continuous at every point, but differentiable at no point."
Recall the math teacher blogger Sam Shah, who actually named his blog after this function -- "Continuous Everywhere but Differentiable Nowhere." Here's a link to his latest post:
https://samjshah.com/2019/10/08/a-short-whiteboard-activity-to-check-understanding/
(In this October 8th post, Shah has his Algebra II class draw some functions on whiteboards. Notice that these functions are definitely not continuous everywhere.)
But writing down power series without worrying about convergence leads to problems:
"Even Euler stated some pretty stupid things. For instance, he started from the power series 1 + x + x^2 + x^3 + x^4 + ... which sums to 1/(1 - x), set x = -1 and deduced that 1 - 1 + 1 - 1 + 1 - 1 + ... = 1/2 which is nonsense."
There was a recent debate involving series that don't converge -- recall that Numberphile claimed in a video that 1 + 2 + 3 + 4 + ... = -1/12, and Mathologer corrected him.
On this note, Stewart concludes the chapter by discussing infinitesimals -- a nonstandard system in which Berkeley's "ghost" numbers actually exist:
"In spirit, it is close to the way Cauchy thought. It remains a minority speciality -- but watch this space."
The sidebars in this chapter are all about the famous Riemann hypothesis (named after the Riemann who also formalized spherical geometry), what analysis did for them, and what analysis does for us.
Lesson 5-4 of the U of Chicago text is called "Properties of Kites." In the modern Third Edition of the text, kites appear in Lesson 6-5.
This is what I wrote last year about today's topic:
This is what I wrote last year about today's topic:
Section 5-4 of the U of Chicago text covers kites. The kite is a relatively new quadrilateral classification. Not only did Euclid never define kite, but many texts made no mention of kites -- including my class geometry textbook from 20 years ago. Nowadays most texts define kite, but some include kites only in bonus questions, not in the main text.
Here's what John Conway wrote about the kite -- over 20 years ago, right around the time that I was taking my geometry class:
In fact it's not quite true, either, because "kite" is not
a very traditional name - it was obviously inserted because
this was a type of quadrilateral that SHOULD have received a
traditional name, but didn't, until recently.
Why do we include the kite - plainly because it represents
the one type of symmetry not otherwise mentioned. But this
reason suggests we should also EXCLUDE the non-isosceles
trapezoid.
a very traditional name - it was obviously inserted because
this was a type of quadrilateral that SHOULD have received a
traditional name, but didn't, until recently.
Why do we include the kite - plainly because it represents
the one type of symmetry not otherwise mentioned. But this
reason suggests we should also EXCLUDE the non-isosceles
trapezoid.
David Joyce, meanwhile, doesn't find kites to be necessary at all:
Too much is included in this chapter. The sections on rhombuses, trapezoids, and kites are not important and should be omitted.
Notice that both Conway and Joyce want to exclude trapezoids -- but Joyce is the only writer I know who wants to omit rhombuses. Of course, for Joyce, the emphasis should be on triangles and parallel lines, not quadrilaterals like rhombuses. Also, notice that kites are not specifically mentioned in the Common Core Geometry Standards. But I find that students can identify a kite more readily than a rhombus or trapezoid.
Now every kite contains two special vertices, known as its "ends." As defined by the U of Chicago:
"The common vertices of the equal sides of a kite are the ends of the kite."
Notice that Michael Serra doesn't define ends, but does give a name to the two angles whose vertices are the ends of the kite -- the vertex angles, in analogy with the vertex angle of an isosceles triangle. I see that from a proof standpoint, this makes sense, since the first thing that we do in the proof of our main theorem is divide the kite into two isosceles triangles.
And now here is our main theorem: the Kite Symmetry Theorem. As I mentioned back in the lesson on isosceles triangles, we use symmetry to determine the properties of kites. (A pre-Common Core proof might divide the kite into two triangles to be proved congruent by SSS.) I will post a proof here of the Kite Symmetry Theorem. It is taken directly from the U of Chicago, except that I, as always, add a Given step to the beginning of the proof. Since the U of Chicago's proof has eight steps, mine has nine:
Kite Symmetry Theorem:
The line containing the ends of a kite is a symmetry line for the kite.
Given: ABCD is a kite with ends B and D.
Prove: Line BD is a symmetry line for ABCD.
Proof:
Statements Reasons
1. ABCD is a kite with ends B and D 1. Given
2. AB = BC, AD = DC 2. Definition of ends of kite (meaning)
3. Tri. ABC and ADC are isosceles 3. Definition of isosceles triangle (sufficient)
4. Let m be the perp. bis. of AC 4. A segment has exactly one perp. bisector
5. A' = C, C' = A 5. Definition of reflection (sufficient)
6. m contains B and D 6. The perp. bis. of the base = angle bis. of the vertex angle
(so it contains the vertex)
7. B' = B, D' = D 7. Definition of reflection (sufficient)
8. ABCD reflected over m is CBAD 8. Figure Reflection Theorem
9. m (Line BD) is a symmetry line 9. Definition of symmetry line (sufficient)
Notice that more than half of the reasons in this proof are definitions. This underlines how important definitions are to the study of quadrilaterals.
As for the other theorems in this lesson, the Kite Diagonal Theorem follows directly from lines 4 and 6 of the above proof. It makes the symmetry diagonal the perpendicular bisector of the other diagonal and so the diagonals of a kite are perpendicular, and the symmetry diagonal bisects the other one.
Finally, we have the Rhombus Symmetry Theorem. It states that a rhombus has two symmetry lines, as both of the lines containing its diagonals are symmetry lines. This follows directly from the classification of a rhombus as a kite. In texts that define kite exclusively, the theorems "The diagonals of a kite are perpendicular" and "the diagonals of a rhombus are perpendicular" are two separate theorems, often in two separate sections. But here we can easily see why the diagonals of both the kite and the rhombus are perpendicular -- because the rhombus is a kite! Also, since we are defining kite inclusively, we don't need any extra steps in the proof to ensure that our kite isn't a rhombus.
No comments:
Post a Comment