Multiplicativity and Geometry

Klaus Hoechsmann

Our world is laid out in what we call space, and the first things we count or measure are inside it. The contact with drawing and (where possible) 3D constructions is extraordinarily helpful even for arithmetic. The famous Soviet curriculum had geometry as an integral part in all of its ten years, taking at least 20% of class-time from Grade 1 onward.

Here is a quote from the French mathematician C.-A. Laisant (1904): "I hope that, by the these few examples, I have succeeded in showing you that we can easily enlarge the field of investigations which the brain of a child is capable of pursuing much more than is generally imagined. To arrive at this result, a bit of material, which is very simple and infinitely variable, is nessary, as you have seen.

The very first element of this material is graph paper, a wonderful instrument which should be in the hands of anyone who does mathematics (whether in their living room, the common room of an asylum, or even all the way to the Ecole Polytechnique and beyond) and in general, anyone who does science. But above all, it is a wonderful instrument from a pedagical point of view, to give little children their first notions of form, size and position, without which this initiation is nothing but a sham."

The only facts from "synthetic" geometry required below are the one abot the angle-sum in a Euclidean triangle, and the equality of angles on the same circular segment. Later on, we might have to involve area.

Of course, the most obvious and plentiful are questions of scaling, map-reading, relative size on photos, geometric objects, etc. Here numbers are involved from the beginning. It is good to see how lengths, areas, etc., can be relative but angles are not (cf. "pie-charts"). The story of how Eratosthenes measured the earth also belongs here.

Two centuries ago, arithmetic and geometry were separate disciplines, but the foundational upheavals of around 1900 forged them into a single logical structure. For the naive observer, however, they are still different, the first having its source in counting, the second in measuring. A systematic development of mathematics would be possible on the basis of either, but today, the first one is almost universally preferred.

1. Multiplication.

Since the first mode is the orthodoxy of the day, we shall take the second point of view. The two are, of course, entirely isomorphic, and the learner usually has no compunctions about mingling them -- as can be seen older text books, like David Murdoch's Linear Algebra [?]. Since this is for beginners. we shall not use Artin's elegant treatment via translations and homotheties, but the more primitive method found in Hilbert and his modern disciple Hartshorne. In the presence of calculators (whether by permission or decreee), it might be a more convincing basis for "real" arithmetic to think that the number on the display is an imperfect rendition of a length than of a number other than itself.

The objects to be added, subtracted, and multiplied are line segments, one of which has been singled out (up to congruence) as the unit interval. Some back-benchers can now be heard muttering that hereby we have the whole rational numberline, and all that's missing is continuity. Yes, yes, we' ll take that approach some other time.

Leaving addition and subtraction as an exercise, we now have to explain how to find the product of the multiplicand b by the multiplier a, both being line segments. Here's how: among consenting adults, the picture on the left should not require many words. It is exactly the Definition No. 69 from le Brouet et Haudricourt Frères, which Michel Delord had to banish into a footnote, since it did not fit his narrative, to wit: multiplication ... aims to find a number, called the product which is to the multiplicand as the multiplier is to 1.

Obviously, we do not need the whole triangle for the multiplication: it is enough to know the angle indicated by the red dot. Nevertheless, it is convenient to keep right triangles in the discussion, if only because they are more directly related to Cartesian coordinates (which will come up later).

Two right triangles BCA and B'CA' are said to be similar if the lines AB and A'B' are parallel. Note that the two triangles on the right are not similar in this sense (i.e., as multipliers) although they are congruent, since CA = CB' and CB = CA'. In fact, they reciprocal, meaning that the rectangles they span become similar when one of them is rotated through 90 degrees, or equivalently, that their "multiplier angles" (here at B and B') are complementary.

To see that this has the accustomed meaning, we need to study the effect of performing one multiplication after another (say, by a and c, respectively). Instead of jamming both multipliers into the same quadrant, as shown on the far left, we shall display them on opposite sides of the numberline (this is one of the good reasons for holding back on the introduction of negatives). If we then multiply some line segment b on the one hand by a and on the other by c, we obtain a blue and yellow triangle POQ as shown on the right below. So far, there have been no succesive multiplications, just two separate ones yielding ab and cb, respectively.

Now comes a surprising move: we form the circumcircle of POQ, and find the point R where it hits the numberline (on which b was a segment). On the chords RP and RQ, we then find the "multiplier angles" of c and a, respectively, and thereby obtain the identity a(cb) = c(ab), whence both commutativity and associativity of multiplication.

Now, if a and c are reciprocal to each other, the two aforementioned angles are complementary, whence the angle POQ is a right angle, and therefore PQ a diameter. We conclude that, in this case, a(cb) = c(ab) =b. In other words: multiplication by the reciprocal of a multiplier undoes its multiplying effect.

Thus, in this setup, we have a multiplicative group. To get an additive one, it is better to wait a while. Subtraction a - b makes sense only if b is no larger than a. But the distributive law c(a - b) = ca - cb is an easy exercise, and for addition it is even easier.

2. Proportion and area.

The easiest way to get started on a geometric theory of proportion is with rectangles. It is important to note that initially, whenever we say "rectangle" we mean only those whose sides are lined up with a predetermined pair of horizontal and vertical directions (x-axis and y-axis, if you will). The removal of this restriction will come later. Two of these are similar, i.e., have the same proportion, if they can be slid into alignment like the blue one and the yellow one shown in Figure 2b. Those in Figure 2a are not similar because of the kink in the diagonals as we try to line them up.
If we (temporily) let [a,b]denote a rectangle with sides a and b, the diagram makes it fairly obvious that [a,b] and [c,d] are similar if and only if [a,d] and [c,b] have the same areas -- if you believe in "areas", that is. However, as Messrs. Delord and Grassmann like to point out, the latter live in a different world, so we should really get busy and work out ways of comparing and perhaps even quantifying them.

The beginner will have no such qualms, and will be happy if he understands the case where all the lengths are mutually commensurable (the bean-counters on the back-benches are grinning), For this "rational" case, Wu has produced about as clear an explanation as is possible in an elementary tract (Section 7 of http://www.math.berkeley.edu/~wu/ EMI2a.pdf). A very crude juxtaposition of "dimensional" versus "scalar" multiplication, is shown below.

One construction which is made particularly easy and attractive by this relationship between proportions and areas is that of the geometric mean. It is shown here on the left, together with its explanation. The mysterious "times" symbol on the last line means "area" -- whatever that means.

Remember that the geometric mean of two lengths a and b is defined to be the length h such that the square [h,h] has the same area as [a,b]. The diagram shows clearly that it is at most as large as the arithmetic mean (i.e., the radius of the circle).

A much more important fact about areas is, of course, the Pythogorean Theorem. With reference to the diagram above, it would say that the square on the green hypotenuse (call it d) has the same area as [a,c], where c = a+b. There are plenty of proofs for this by dissection or shearing, but one often finds the following argument using "ratios": a is to d as d is to c -- think: [a,d] is similar to [d,c]-- therefore [d,d] has the same area as [a,c]. What has been done here is to bend the angles whose respective legs are a,d and d,c into right angles. To check that this is permissible is our next task.

3. Proportion and ratio.

We wish to show that "similarity" is a property of the respective sizes of the legs of an angle, irrespective of the nature of the latter. Here is how we define it : the triangles ABC and ADE, where D and E are points on the line segments AB and AC, respectively, are said to be similar if the lines BC and DE are parallel. This agrees with what we have said when the angle at A was a right angle.
What needs to be shown is the following theorem: If the triangles ABC and ADE, as described above, are similar then so are ABC" and ABE", as long as AC" = AC and AE" = AE, with E" lying on the segment AC".

The picture on the right shows the situation in the commensurable case, which is the only one practical folks care for. They will then say that the "ratios" AB:AD, AC:AE, BC:DE are "equal", namely 13:8 in this case. We could use the same language (without the numbers) if we managed to prove the theorem.

Let be remembered that there are many non-arithmetic elementary ways of exhibiting incommensurability. The diagram on the left shows an isosceles right triangle with its tip folded down. If its sides were commensurable with the hypotenuse, the same would be true for the smaller yellow triangle (BC' = BC, and AC' would be measurable by the same footsteps).
If we don't wish to say that commensurability is a good enough assumption for practical purposes, we have 3 options: Eudoxus, Desargues, Dedekind. The last one is the standard approach to modern (standard) analysis, but we have opted against it. In older text-books, one can occasionally find the Eudoxan approach.
We shall here go for Desargues, because his theorem is entirely geometric and logically simple -- especially in the following affine version, which refers to the diagram on the right below.

Theorem: If the triangles ABC and A'B'C' are as shown, with sides AA', BB', CC' concurrent, and two pairs of sides (black) parallel, then the third pair (red) is parallel as well.

If you looked at it as representing a 3D configuration, it might appear obvious, because the yellow and blue triangular 'platforms' would then be on parallel planes.

Accordingly, the proof consists in the lifting of the basic triangle (shown lying on the ground) and then arguing in 3D. Of course, this presupposes that your plane fits into a space with certain simple rules of incidence, from which the theorem then follows. For details, click here.

This sequence of diagrams shows how similarity of rectangles (right triangles) is inherited by triangles with the same sides including a different angle, and finally allows ratios to be assigned to any collinear triples of points and compared as in the movies.

We can now multiply more freely, as shown on the left, and we can say that two triangles QRO and ABC are similar if and only if their angles are pairwise equal. However, as with congruence, it is important to distinguish two triangles like ORQ and ROQ.

Although we shall still pay particular attention to right triangles -- where one of the angles is fixed and the other two are complementary -- it is no longer necessary that the unit segment be a side of our "model" triangle, as it was at the beginning. As we shall presently see, it is more practical to designate the hypotenuse for this role.

4. Elementary trigonometry.

Hipparchus knew elementary trigonometry very well -- centuries before Al Khwarizmi officially invented algebra and even before Diophantus did so inofficially. He had no idea that the cosine of 120 degrees was negative, and yet would probably have scored high at the WMO. This suggessts that it might be possible to teach trigonometry early on, not necessarily as a separate item on the curriculum, and thus reinforce the learner's ability to deal with scalar mulipliers.

Elementary trigonometry has three ingredients: the Pythagorean theorem, similarity, and table which associates with every angle A certain multipiers related to it. The fist one, called tanA is the one we started with. A more important one involves the hypotenuse of the right triangle containing A and its complement B: it is called sinA or cosB. If the hypotenuse is 1, these are just the sides of the right triangle involved.

The numbers shown with the triangles on the right denote the sizes of the sides in appropriate units. If the quarter circle shown has radius 1, the sine of the angle at its base is 80/98, for instance.

Since angles are more easily measured than distances, trigonometry is the backbone of indirect measurement. For example, if the point C (referring to the diagram on the leftt) is visible from A and B, but inaccessible, one can measure the length AB and the angle at A to get the imagined segment PB by scaling AB with sinA; dividing this by the cosine of the easily compute angle PBC ("divide by" means "mutliply by reciprocal of"), one then gets BC, and finally obtains the desired height from this and the sine of CBQ.

On the other hand, if A and B are accessible from C but separated by the Taj Mahal, one can find the distance between them by first getting PB and PC from BC scaled by sinC and cosC, and then using Pythagoras to find AB. Since the object here is not to practise efficient geodesy but to become fluent in multiplicative numeracy, the Laws of Sines and Cosines can be left with Hammurabi, for now -- unless they are more that just rules. It is hard, for instance, to resist the temptation of presenting the diagrams below.

                     

The question which remains is how to get those sines and cosines. The answer is obtained by asking it in reverse: given a point on the unit circle how do you measure the angle that goes with it? We are not (yet) asking for radians, if the quarter circle were used as a unit we would be happy.

There are several ways of tackling this. For beginners, it is a useful exercise in "human program loops", using calculators. It is a race between two points P=(u, v) and Q=(r,s) on the unit circle. Though P starts high at P = (0,1), it is continually being cut in half to P'=(u'v'). This is done by scaling the angle-bisecting segment "(0,0) to (u+1,v)" back to the unit circle, and gives u' = sqr(1/2 + u/2) and v'=v / 2u' by easy algebra. Apart from P and Q, which will change during the race, there will be a string of binary digits b to measure Q.

For the human loop, it would be good to have two people: Jack to compute the new (u,v) for each succesive half angle, and Jill to turn (r,s) back by (u,v) if s exceeds the current v. In that case, she computes r'=ru + sv, s' = su - rv and records the binary digit b=1; otherwise she does nothing except to write b=0. Then Jack takes over again and around and around they go. It takes 10 binaries to make one decimal, so they might do this 32 times. Finally they will parse this string into 8 hexadecimals and then call in the decimalising crew.

This example suggests a constructive use of calculators. One could establish a hierarchy of buttons, which -- after thorough training (without calculators!) in the 4 basic operations -- would treat sqr before it gets to log and sin (log is similar to sin, but simpler). Each button would be taboo until its results can be obtained (by such human loops) from lower buttons.

5. The four quadrants.

Unfortunately, this writer knows nothing about the attitude of Descartes toward negative numbers, but he has seen Pascal deride them and d'Alembert obfuscate them, in print. This suggest, that they are not as easy as modern teachers believe, and that Landau [?] was wise to develop the whole system of positive real numbers (starting with Peano's axioms) before he tacked on the negatives.

Here is what d'Alembert wrote in Diderot's Encyclopedie:  Negative quantities in algebra are those which are affected by the sign -- and are regarded by several mathematicians as being smaller than zero. The latter idea is, however, incorrect, as we shall see in a moment. Then he goes on about the ratio of 1 to -1 and vice versa, even talks about the numberline where this so-called negativity is just a matter of direction, and finishes his first paragraph by saying that -3 presents absolutely no idea to the mind. His whole essay on the subject lacks the clarity one would expect of a competent mathematician.

Having run out of time and energy, I must leave the writing of this section to the reader's imagination. Suffice it to say, that the inclusion of negatives forces us back to rectangles lined up with the two axes. The pay-off is that we now get multipliers hanging like bats down from the x-axis, and lines which pass below them. For students having learned multiplication the geometric way, it will then be no mystery that (-a)(-b) = ab