Borovik A. Shadows of the Truth: Metamathematics of Elementary Mathematics

Подождите немного. Документ загружается.

Graph Paper and the Arithmetic of

Complex Numbers

Fig. 15.1. Graph pap er.

15.1 Graph paper

Some time ago I placed a question in my blog: who, when and where

printed the ﬁrst sheet of graphed paper—and what was the moti-

vation?

145

146 15 Graph Paper and the Arithmetic of Complex Numbers

This was what my anony mous readers told me. In this section,

I quote at length from the blog. He refers to the USA; in Br itain,

introduction of g raph paper in mainstream mathematics education

is associated with the name o f John Perry [168].

Graph paper, a math class staple, was developed betwe en 1890

and 1910. During this period the number of high school students

in the U.S. qu adrupled, and by 1920, according to E. L. Thornd ike,

one of every thr ee teenagers in Amer ica “enters High School”, com-

pared to one in ten in 1890. The po pulation of “high school age ”

people had also gr own so that the total number of people entering

High School was six times as large as only three decades befo r e.

Research mathematicians and educators took an active interest

in improving high school education. E. H. Moore, a distinguished

mathematician at the University of Chicago, served on mathemat-

ics ed ucation panels and wrote at length on the advantages of

teaching students to graph curves using paper with “squared lines.”

When the University of Chicago opened in 1892 E. H. Moore was

the acting head of the mathematics de partment. [. . . ] The Fourth

Yearbook of the NCTM, Signiﬁcant Changes and Tr ends in the

Teaching of Mathematics Throughout the World Since 1910, pub-

lished in 1929, has on page 159,

“The graph, of great and growing importance, began to receive the

attention of mathematics teachers du ring the ﬁrst decade of the

present century (20 th)”.normalsize

Later on p age 160 they continue,

“The graph appeared somewhat prior to 1 908, and although used

to excess for a time, has held its position about as long and as suc-

cessfully as any proposed reform. Owing to the prominence of the

statistical graph, and the increased interest in educational statis-

tics, graph ic work is assured a permanent place in our courses in

mathematics.”

Hall and Stevens A School Arithmetic, printed in 1919, has a

chapter on graphing on “squared pape r”. Some more notes on graph

paper can be found here.

The actual date of the ﬁrst commercially published “coordinate

paper” is usually attributed to Dr. Buxton of England in 1795 (if

you know more about this man, let me know). The earliest record

I know of the use of coordinate paper in published re se arch was in

1800. Luke Howard (who is remembered for creating the names of

clouds: cumu lus, nimbus, and such) included a graph of barometric

variations [193].

Notes on the History of Math Teaching and Math Books, http://www.

pballew.net/mathbooks.html.

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK

15.2 Pizza, logari thms and graph paper 147

The increased use of graphs and graph paper around the turn

of the century is supported by a Preface to the “New Ed ition” of

Algebra for Beginners by Hall and Knight. The book, which was

reprinted yearly betwe en the o r iginal edition and 1904 had n o

graphs appearing anywhere. When the “New Edition” appeared

in 1906 it had an appendix on “Easy Graphs”, and the c over had

been changed to include the subhead, “Including Easy Graphs”.

The preface includes a strong statement that

“the squared paper should be of good quality and accura tely ruled

to inches and tenths of an inch. Experience shows that anything on

a smaller scale (such as “millimeter” paper) is practically worth-

less in the hands of beginners.”

[. . . ] The term “graph paper” seems not to have caught on

quickly. I have a Hall (the same H S Hall as before) and Stevens A

School Arithmetic, printed in 1919 that has a chapter on graphing

on “squared paper”. Even later is a 1937 D. C. Heath text, Analytic

Geometry by W. A. Wilson and J. A. Tracey, that uses the phrase

“coordinate paper” (page 223, topic 153). Even in 1919 Practical

Mathematics for Home Study by Claud e Irwin Palmer introduced a

section on “Area Found by the Use of Squared Paper” and the n de-

ﬁned “paper accurately ruled into small squares” (pg 183). It m ay

be that the term square d paper hung on much longer in England

than in the US. I have a 1961 copy of Public School Arithmetic

(“Thirty-sixth impression, First published in 1910”) by Baker and

Bourne published in London that still uses the term “squared pa-

per” but uses graphs extensively.

15.2 Pizza, loga rithms and g raph paper

To illustrate our excursion into the history of graph p aper, it would

be timely to quote a story told by SB

. It does not ne ed much com-

ment.

One of my favorite stories to tell is how my parents taught me

logarithms on a pizza box. One night when I was in maybe sixth

grade, we were eating pizza f or dinner and I decided to share an

interesting fact I had picked up somewhere: “Did you know that

as the pi z za cools, the temperature drops in half every ten min-

utes?” Leaving aside the fact that this is a n oversimpliﬁcation (it’s

the diff erence between the temperature of the pizza and the air

that decays in this way, a nd I’ve no idea if the 10 minute ﬁgure

is correct), this little tidbit sparked a dinner-long discussion about

logarithms and exponential decay. My parents got out a marker

SB is female, American, a mathematics PhD student. A writer of a won-

derful blog.

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK

148 15 Graph Paper and the Arithmetic of Complex Numbers

and drew axes for time and temperature on the pizza box, then

plotted the temperature at ten-min ute intervals. When the points

were connected I could see that they didn’t form a straight line,

but rather a curve that got closer and closer to 0 and never quite

touched.

My parents then explained that when graphed on a special

kind of paper, the curve would become a straight line. My dad

got out a piece of log paper, with equally-spaced vertical lines and

strangely spaced horizontal lines, where every group of ten was

spread out at the bottom, but bunched up at the top of the interval.

My parents explained that on a normal axis, each line represented

one un it more than the one below. But on a logarithmic scale, each

line (actually, each group of ten lines) represented ten times as

much as the previous one. When we plotted our pizza temperature

vs. time graph on the log paper, sure enough, we got a straight

line! On the other hand, a graph th at would have been a lin e on

the original axes now became a curve th at looked very similar to

the original temperature graph.

Fig. 15.2. Log-log paper.

My dad also pulled out some log-log paper, where both axes

have a logarithmic scale. (I mus t b e the only person of my genera-

tion who’s ever s een real log-log paper—who needs it when we can

plot things so easily on computers these days?) I didn’t understand

why you would need logs on both axes—it wouldn’t straighten out

the temperature graph, and lines on normal axes would still be

straight on these (my parents weren’t about to try to explain expo-

nents and properties of logarithms in a mathematically rigorous

way to me at age 11!) Nevertheless, I thought it was in teresting,

and the odd-looking distribution of the lines had a kind of nice ﬂow

to it.

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK

15.3 Mu ltiplication of s quares 149

Now I don’t expect every parent to teach their kids about ad-

vanced math concepts at the dinner table. Most adults have never

learned even this much math, or learned it but forgot it as soon

as the ﬁnal exam was over. I was extremely lucky to have parents

with both the capability and the interest in math to teach me these

kinds of things just for fun. (It is fun—how wonderfully geeky is

it to draw mathematical diagrams on a pizza b ox? I wish I’d saved

it.) And I don’t expect every kid to be capable of understa nding this

while in elementary school; I was pretty exceptional . Bu t mathe-

matics doesn’t have to be hard, and it doesn’t have to be boring—I

ﬁrmly believe every high school student is capable of understand-

ing the concepts my parents taught me that night. More than that,

I believe every student is capable of having fun playing with ideas,

discovering patterns and trying to ﬁgure out why they work.

15.3 Multiplication of squares

Preparing once for a chat with my young friend DW I was thinking

about sums of two integer squares (an archetypal mathematical

problem: represen tation of nu mbers by fo r ms) and a remarkable

fact that

if n and m are sums of two integer squares ( let u s call them

bisquare numbers) then their product mn also h as the same prop-

erty.

What is the most natural way to explain this property to a child?

One may of cou rse use the Fibonacci identity

+ b

)(c

+ d

) = (ac − bd)

+ (ad + bc)

but it is not very co nvincing.

Actually bisquare numbers are exactly areas of squares with

vertices at nodes of graphed paper (with unit grid step)—this im-

mediately follows from the Pythagoras Theorem. And the mu ltipli-

cation property of bisquare numbers follows from a simple rule for

multiplication of squares drawn on graph paper (with one vertex

marked), see Figure 15.3.

EXPLAIN

Of course, this is multiplication of comp lex numbers—restricted

to the ring of Gaussian integers Z[i], the red square and the blue

square being Gaussian integer 3+2i and 2+i, correspondingly, their

product being 4 + 7i (check—it is the new big blue square!). What

becomes immediately o bvious is the addition rule for arguments:

arg(uv) = arg(u) + arg(v),

as well as the multiplicativity of absolute values:

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK

150 15 Graph Paper and the Arithmetic of Complex Numbers

Fig. 15.3. Multiplication of squares drawn on the squared paper.

|uv| = |u||v|

(which was our aim in the ﬁrst in stance, since |u|

is the area of the

square representing u).

Fig. 15.4. As Mi tchell Harris h as poin ted to me, the Fibonacci identity

+ b

)(c

+ d

) = (ac − bd)

+ (ad + bc)

has a simple graphical proof based on Pythagoras Theorem.

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK

15.4 Pytha gorean triples 151

15.4 Pythagorean triples

The sums of two square s are archetypal objects of number theory,

and, not surprisingly, they easily attract the attention of children.

Here is a testimony to that effect from JW

[. . . in the sixth form] I proved most of the easy results on expres-

sions of integers as sums of two squares for myself; I was fasci-

nated by algebraic relations between squares and, for example, the

half-angle formulae (expressing sin θ and cos θ in terms of tan(θ/2).

My friend and colleague Hovik Khudaverdyan explained to me

the right way of solving the classical Pythagorean Triples equation

+ y

= z

in integers. Of course, after change of variables

u =

, v =

we have to solve a slightly simpler equation

+ v

= 1

in rational numbers u and v.

Now draw the unit circle in the plane R

Then we observe that the line through po int (0, 1) and point

(u, v) has rational coefﬁcients and intersects the x-axis in a point

(t, 0) w ith rational coordinate t. Moreover, this projectio n estab-

lishes a one-to-on e c orrespond ence between rational points on the

circle (without the north pole (0, 1)) and the rational points on the

real line. This observation is fairly obvious in o ne direction,

(u, v) 7→ (t, 0),

and very subtle in another: the straight line through points (0, 1)

and (t, 0) intersects the unit circle

JW is male, a professor of mathematics and an editor of a research

mathematics journal.

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK

152 15 Graph Paper and the Arithmetic of Complex Numbers

+ y

= 1

in a rational point (0, 1); the coordinates of the second point of in-

tersection are the sec ond roots of quadratic equations with rational

coefﬁcients and are therefore also rational.

So we take t for a parameter. The equation of the straight line

through the point (0, 1) and (t, 0) is, obviously,

+ y = 1.

Hence we have to solve the simultaneous system

+ y = 1

+ y

= 1

which is fairly easy and quickly yield s

x =

1 + t

y =

− 1

1 + t

After expressing a ration al number t as a ratio o f tw o integers:

t =

we have, as a gene ral formula for Pythagorean triples (x, y, z),

(2pq)

+ (p

− q

)

= (p

+ q

)

;

that is,

x = 2pq,

y = p

− q

z = p

+ q

As simple as that. Of course, all that is just a birational parametriza-

tion of the circle by stereographic proj ection, a well-known folklore

fact frequently mentioned in introductory courses on algebraic ge-

ometry. But when I was a boy no-one explained that to me, and the

change of variable always appeared to be a mystery or, even worse,

an artiﬁcial man ipulation with symbols.

I was not the only one puzzled by change of variables in Pythagorean

triples. Listen to a testimony from RA.

RA is male, American, a professor of computer science.

SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19

 ALEXANDRE V. BOROVIK

15.4 Pytha gorean triples 153

I don’t consider myself a math emati cian, but your letter reminds

me of what I think of as my ﬁrst mathematical failure. I must have

been in my early teens. My math instruction was in my native

language, English. I always did well my math classes and was one

of the school “stars.” I wanted to ﬁnd a formula to generate all the

Pythagorean triples. I was very conﬁdent that would be able to do

it.

I struggled wi th it for a long time before giving up. What never

occurred to me at th e time was that the formula could have two

free va riables. I was looking for something with one! When I saw

the answer (and was surprised that others had even thought to

ask the question, whi ch shows what little contact I had had with

math at the time) I f elt embarrassed about my failure to think