"The use of symbols in mathematics goes well beyond their appearance in notation for numbers, as a casual glance at any mathematics text will make clear."
In this chapter, the author describes the history of algebra. It's all about how we devised methods to solve for unknown quantities. Stewart begins by asking:
"How did algebra arise? What came first were the problems and methods. Only later was the symbolic notation -- which we now consider to be the essence of the topic -- invented."
The author informs us that the Old Babylonians were the first to develop methods of solving what we would now call "linear," and more particularly, "quadratic" equations. From an ancient stone tablet:
"'I found a stone, but did not weigh it. After I weighed out six times its weight, added 2 gin and added one third of one seventh multiplied by 24. I weighed it. The result was one 1 ma-na. What was the original weight of the stone?' A weight of 1 ma-na is 60 gin. In modern notation, we would let x be the required weight in gin."
(Of course 1 ma-na would be 60 gin -- these were the sexagesimal Babylonians, after all.) Stewart now writes an equation in modern notation:
(6x + 2) + 1/3 * 1/7 * 24(6x + 2) = 60
with its solution as x = 4 1/3 gin, but the tablet doesn't show how to find the answer. Later on, it does show the steps for solving a quadratic equation:
"Notice that the tablet tells its reader what to do, but not why. It is a recipe. Someone must have understood why it worked, in order to write it down in the first place, but once discovered it could then be used by anyone who was appropriately trained."
Of the example given, Stewart writes:
"The complicated numbers actually help: they make it clearer which rules are being used. To find them, we just have to be systematic."
The author now draws pictures to represent what the Babylonian discovered probably did. I won't show the pictures in ASCII, but I will write some of what Stewart does here. To solve x^2 + ax = b:
"Here the square and the first rectangle have height x; their widths are, respectively, x and a. The smaller rectangle has area b. The Babylonian recipe effectively splits the first rectangle into two pieces. We can then rearrange the two new pieces and stick them on the edge of the square. The left-hand diagram now cries out to be completed to a larger square, by adding the shaded square."
In other words, we are deriving the Quadratic Formula by completing the square. Stewart obtains:
x = sqrt(b + (a/2)^2) - a/2
"...which is exactly how the Babylonian recipe proceeds. There is no evidence on any tablet to support the view that this geometric picture led the Babylonians to their recipe."
Of course, it worked. Meanwhile, the author tells us about the word "algebra":
"The word 'algebra' comes from the Arabic al-jabr, a term employed by Muhammad ibn Musa al-Khwarizmi, who flourished around 820."
Stewart moves on to cubic equations:
"But until the development of negative numbers, mathematicians classified equations into many distinct types -- so that, for example, x^3 + 3x = 7 and x^3 - 3x = 7 were considered to be completely different, and required different methods for their solution. The Greeks discovered how to use conic sections to solve some cubic equations."
I wrote about cubic equations in my August 26th -- this was in response to a post by math coach Benjamin Leis, who in turn was referring to a Mathologer video. We can go back to those posts and video from two months ago to read more about a possible Cubic Formula. As for our reading of Stewart, let's jump right into the mathematical "duels" that used to take place:
"So, public mathematical combat was serious stuff. In 1535 there was just such a contest, between Antonio Fior and Niccolo Fontana, nicknamed Tartaglia, 'the stammerer.'"
He tells us how Tartaglia ultimately won the battle. At first, both contestants knew how to solve only one type of cubic:
"In a burst of inspired desperation, a week or so before the contest, Tartaglia figured out how to solve the other types too."
A few years later, the mathematician Girolamo Cardano wanted to publish Tartaglia's Cubic Formula:
"This was completely new, and of huge importance. So of course Cardano wanted quartic equations in his book, too."
And of course, Cardano published it, much to Tartaglia's dismay. Again, I won't repeat the entire story, but it's mentioned in the Mathologer video. Now let's pick up Stewart at algebraic symbolism:
"It took hundreds of years for today's algebraic symbolism to develop. One of the first to use symbols in place of unknown numbers was Diophantus of Alexandria."
But it wasn't until the Renaissance before notation barely recognizable to us nowadays appeared. The author describes the mathematician Francois Vieta:
"He did, however, use letters of the alphabet to represent known quantities, as well as unknown ones."
Symbols for mathematical operations began slightly before Vieta's time:
"William Oughtred introduced the symbol X for multiplication, and was roundly (and rightly) criticized by Leibniz on the grounds that this was too easily confused with the letter x. In 1557, in his The Whetstone of Witte, the English mathematician Robert Recorde used the symbol = for equality, in use ever since."
Stewart compares Renaissance to modern notation by looking at Cardano's work:
"He wrote qdratu aeqtur 4 rebus p:32, where we would write x^2 = 4x + 32, and therefore used separate abbreviations 'rebus' and 'qdratu' for the unknown (thing) and its square."
It was Rene Descartes -- as in the Cartesian plane -- who created most of our current notation. But the author returns to Vieta, who first called algebra "the logic of species," separate from arithmetic:
"The individual terms 2x + 3y and so on are themselves mathematical objects. They can be added, subtracted, multiplied, and divided without ever considering them as representations of specific numbers."
On that note, Stewart concludes the chapter as follows:
"So algebra took on a life of its own, as the mathematics of symbolic expressions. It was the first step towards freeing algebra from the shackles of arithmetical interpretation."
As usual, there several sidebars in this chapter. They describe the mathematicians Fibonacci, Cardano, and Descartes -- what algebra did for them, and what algebra does for us.
OK, let's finally get to the U of Chicago text.
Lesson 4-7 of the U of Chicago text is called "Reflection-Symmetric Figures." (This corresponds to Lesson 6-1 in the modern Third Edition.) This is what I wrote last year about today's lesson:
Section 4-7 of the U of Chicago text deals with reflection-symmetric figures. A definition is in order:
A plane figure F is a reflection-symmetric figure if and only if there is a line m such that r(F)=F. The line m is a symmetry line for the figure.
In other words, it's what one usually means when one uses the word "symmetry." Some geometry texts use the term "line-symmetric" instead of "reflection-symmetric." Some geometry and algebra texts use the term "axis of symmetry" instead of "symmetry line" -- especially Algebra I texts referring to the axis of symmetry of a parabola. Some biology texts use the term "bilateral symmetry" instead of "reflection (or line) symmetry" - in particular, when referring to symmetry in animals. As animals are three-dimensional, instead of a symmetry line there's a sagittal plane.
Indeed, it is this last topic that makes symmetry most relevant and interesting. Most animals -- including humans -- have bilateral symmetry. I once read of a teacher who came up with an activity where the students look for the most symmetrical human face. The teacher blogged about how students who are normally indifferent to geometry suddenly came fascinated and engaged to learn about the relationship between symmetry and human beauty. Unfortunately, this was more than a year ago, and I can't remember or find what teacher did this activity -- otherwise I'd be posting a link to that teacher's blog right here!
In the Common Core Standards, symmetry is first introduced as a fourth grade topic:
CCSS.MATH.CONTENT.4.G.A.3
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
Later on, symmetry appears in the high school geometry standards:
CCSS.MATH.CONTENT.HSG.CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Notice that if a reflection over a line carries a polygon to itself, then that line is a symmetry line. But symmetry lines for polygons formally appears in Chapter 5 of the U of Chicago text. Right here in Chapter 4, we only cover symmetry lines for simpler shapes -- segments and angles. The text reads:
"In the next chapter, certain polygons are examined for symmetry. All of their symmetries can be traced back to symmetries of angles or segments."
For segments, the text presents the Segment Symmetry Theorem:
A segment has exactly two symmetry lines:
1. its perpendicular bisector, and
2. the line containing the segment.
The text gives an informal proof of this -- as the mirror image of an endpoint, there can only be two possible reflections mapping a segment
But we also want to work with angles. The first theorem given is the Side-Switching Theorem:
If one side of an angle is reflected over the line containing the angle bisector, its image is the other side of the angle.
An informal proof: the angle bisector divides an angle into two angles of equal measure. The picture in the U of Chicago text divides angle ABC into smaller angles 1 and 2. Now the reflection must map ray AB onto a ray that's on the other side of the angle bisector BD, but forms the same angle with BD that AB does with BD. And there's already such a ray in the correct place -- ray BC. Notice that part b of the Angle Measure Postulate from Chapter 3 already hints at this -- the "Two sides of line assumption" gives two angles of the same measure, one on each side of a given ray. QED
The other theorem, the Angle Symmetry Theorem, follows from the Side-Switching Theorem:
Angle Symmetry Theorem:
The line containing the bisector of an angle is a symmetry line of the angle.
Earlier this week, I wrote that we'd be able to prove the Converse of the Perpendicular Bisector Theorem after this section. As it turns out, the Side-Switching Theorem is the theorem we need.
Converse of the Perpendicular Bisector Theorem:
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
Given: PA = PB
Prove: P is on the perpendicular bisector m of segment
Now I was considering giving a two-column proof of this, but it ended up being a bit harder than I would like for the students. But as it turns out, even though the U of Chicago text doesn't prove this converse, in Section 5-1 it gives a paragraph proof of what it calls the "Isosceles Triangle Symmetry Theorem," and the proof of this one and the Converse of the Perpendicular Bisector Theorem are extremely similar. After all, we're given that PA = PB -- so PAB is in fact an isosceles triangle!
Proof:
Let m be the line containing the angle bisector of angle APB. First, since m is an angle bisector, because of the Side-Switching Theorem, when ray PA is reflected over m, its image is PB. Thus A', the reflection image of A, is on ray PB. Second, P is on the reflecting line m, so P' = P. Hence, since reflections preserve distance, PA' = PA. Third, it is given that PA = PB. Now put all of these conclusions together. By the Transitive Property of Equality, PA' = PB. So A' and B are points on ray PB at the same distance from P, and so A' = B. That is, the reflection image of A over m is B.
But, by definition of reflection, that makes m the perpendicular bisector of AB -- and we already know that P is on it. Therefore P is on the perpendicular bisector m of segment
Let's think about what we're trying to prove here. We want the Converse of the Perpendicular Bisector Theorem -- and consider what I wrote earlier about the proof of converses. The proof of the converse of a statement often involves the forward direction of the theorem and a uniqueness statement -- and even though we didn't use the forward direction of the theorem here, we did use a uniqueness statement here. As it turns out, given two distinct points A and B, there exists only one line m such that the mirror image of A over m is B -- and that line is the perpendicular bisector of the segment
In this section, we found symmetry lines for simple figures such as segments and angles. But can we find symmetry lines for the simplest figures? As it turns out, a point has infinitely many lines of symmetry -- any line passing through the point is a symmetry line. But a ray has only one line of symmetry -- the line containing the ray.
Finally, does a line have a line of symmetry? This is exactly the answer to Question 25 of this section, in the Exploration/Bonus Section. A line -- considered as a straight angle -- contains more than one symmetry line. This is because any point on the line can be taken as the vertex of that straight angle. Since straight angles measure 180, their angle bisectors must divide them into pairs of 90-degree angles. Therefore, any line perpendicular to a line (straight angle) is a symmetry line of the given line. This is what I called the Line Perpendicular to Mirror Theorem. It implies that a line (straight angle) has infinitely many symmetry lines. (Of course, the line has one more symmetry line that I didn't mention -- namely the line itself.)
I included Question 24, even though it appears to mention corresponding and same-side interior angles formed by two lines and a transversal. But nowhere in the question does it mention anything about the two lines being parallel.
I left out Questions 16 and 17, which give the construction of an angle bisector. I finally plan on going to constructions sometime next week. But here's another video from Square One TV, where doctors have to perform a "bisectomy" on an angle. (Unfortunately, only the entire 30-minute show is available on YouTube -- the "bisectomy" doesn't begin until the 11-and-a-half-minute mark.)
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