High School Mathematics-Physics SMILE Meeting 1997-2006 Academic Years History of Mathematics |

**13 March 2001 Bill Colson (Morgan Park HS, Math)**

noted that tomorrow is **p Day** [3.14
--- get it?].
As an ** multicultural mnemonic** to remembering the digits of **p**,
he presented the following phrases, in which the number of letters in
each word form the sequence

French |
Que j'aime a faire connaître ce nombre utile aux
sages, immortelArchimède illustre inventeur quie de ton jugement peut priser la valeur. Pour moi ton problème est de pariels avantages ... |

Spanish |
Sol y Luna y Mundo proclaman al Eterno Autor del Cosmo |

English |
See, I have a rhyme assisting my feeble brain, its tasks
oft-times resisting. |

**10 April 2001 Porter Johnson (IIT)**

mentioned the **golden rectangle** ratio

**( 1 + Ö5)/2 =
1.61803... **

GOLDEN RECTANGLE

This number arises out of the ** definition** of a golden
rectangle; **specifically**,
that the ratio of
its height **h** {short side) to breadth **b** (long side) is
the same as
the ratio of its breadth **b** to the sum of its height and breadth
**(h + b)**:

or

If we define the "golden ratio" **x** as the long side** b**
divided by the short side **h**; i.e. **x = b / h**,
this
equation may be written as

This quadratic equation has two solutions, one positive and one
negative. The
**positive solution** is

This golden ratio can also be understood as the limit of ratios of
successive
number pairs in the **Fibonacci
Sequence**:

In particular, note that **89 / 55 = 1.6181818 ...** is fairly
close to
the limit. The sequence is generated from the first two entries **y _{1}
= 1** and

**y _{n+1} = y_{n}+ y_{n-1} .**

Let us assume that the ratio **y _{n+1}**/

The iteration formula

**y _{n+1} = y_{n}+ y_{n-1} .**

is equivalent to

At very large n, the ratios may be replaced by their limiting values to obtain this equation for the limit:

Thus the golden mean is the limit of the **Fibonacci Sequence**,
independently of the starting seeds **y _{1}** and

One may express any real number **uniquely** through its **continued
fraction** expansion [http://archives.math.utk.edu/articles/atuyl/confrac/]:

where the coefficients **a; b, c, d, e, ¼
** are
positive integers. If the number **A** is rational, the
continued
fraction expansion will terminate; otherwise it will go on
forever. We may
identify the number with its continued fraction: **A = (a; b,
c, d, e, ¼
)**. For the **golden mean** the continued fraction
has the
simplest form, in that the coefficients **a; b, c, d, e, ¼**
are all equal to **1**. That is,

The golden mean is related to **Penrose Tilings**; see the
website
http://mathworld.wolfram.com/PenroseTiles.html.
By terminating this continued fraction after various steps we recover
the ratios
of **Fibonacci numbers**,

**
**

The continued fraction for **e, the base of the Natural Logarithms**,
is
relatively simple [see
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#othercfsE]

**e = (2; 1, 2 ,1, 1, 4, 1, 1 ,6 ,1 ,1, 8, 1, 1, 10,
1, ... )**

On the other hand, the continued fraction expansion of **p**
is less elegant looking:

**p =
**

(3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, 1, 1, 1, 3, 1, 1, 8, 1, 1, 2, 1, 6, 1, 1, 5, 2, 2, 3, 1, 2, 4, 4, 16, 1, 161, 45, 1, 22, 1, 2, 2, 1, 4, 1, 2, ... )

The rational approximations are **3, 22/7, 333/106, 355/113 =
3.14159292,
...** . The last approximation is rather accurate, because the
next
number in the continued fraction, **292**, is rather large.
***** History of Mathematics *****

**07 May 2002: Hoi Huynh (Clemente HS) Division with Mirrors [***
but
without
smoke!]
Hoi** taped together two big mirror tiles [

**
Hoi** described what happened as the opening angle **q
= 360º/N ** becomes small, as **N** becomes large. In
the limit as the opening
angle **q** goes to zero, the number of
images, **N**,
becomes
infinite. ** It must be so!** Incidentally, the choice of **
360º **for
one revolution was made by the Babylonians; see
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_mathematics.html.
The website http://mathforum.org/library/drmath/sets/select/dm_circle360.html
contains several
explainations while another
contains the following information:

"Why, when, where and by whom was it decreed that there should be 360 degrees of arc in a complete circle?Hoi separated the mirrors, and placed them parallel to one another with the reflecting sides facing one another --- allegedly like in the hairdresser's shop! We could see an endless progression of images of an object placed between the mirrors. Very impressive,It was during the reign of Nebuchadnezzar (605-562 BC) in the Chaldean dynasty in Babylon that the circle was divided into 360 degrees. This was because the Chaldeans had calculated by observation and inference that a complete year numbered 360 days. The basis of angular measure for the mathematicians of Babylon was the angle at each of the corners of an equilateral triangle. They did not have decimal fractions and thus found it difficult to deal with remainders when doing division. So they agreed to divide the corner of an equilateral triangle into 60 degrees, because 60 could be divided by 2, 3, 4, 5 and 6 without remainder. Each degree was divided into 60 minutes and each minute into 60 seconds. If the angles at the corners of six equilateral triangles are placed together they form the angle formed by a complete circle. It is for this reason that there are six times 60 degrees of arc in the complete circle."

We then discussed the operation of reflecting mirrors for drivers of motor vehicles, which usually have an "anti-glare" setting, in addition to the regular one. These mirrors are actually wedge-shaped, and most of the reflection occurs from the silvered "back surface" of the mirror. However, a small amount of reflection occurs from the front surface, which is slightly tilted relative to the back surface. When a bright light falls on the mirror, the driver may switch to the "anti-glare" front surface reflection, in order to avoid being distracted by the light. Of course, the intensity of all images is reduced in this process, and the driver should switch back to the regular setting when the bright light is removed, to maintain visibility.

**06 May 2003**

**Porter Johnson** found some interesting entries on the **Islam
and Islamic History and the Middle East **website:
(1) **Arabic Numerals**
http://www.islamicity.org/mosque/ihame/Ref6.htm and (2) **Arabic
Writing and Calligraphy**:
http://www.islamicity.org/mosque/ihame/Ref3.htm.

In addition, **Ishaque Khan [IIT Chemistry]** has called
attention to the
stamp **Eid Greetings**, which was released about 2 years ago by
the ** US
Postal Service**. The words **EID MUBARAK **[Blessed
Festival: **Eid-al-Fitr **at end of **Ramadan **or **
Eid al-Adha** at the end of the hajj] are
written in continuous cursive Arabic calligraphy. For details see the
website http://www.usps.com/news/2001/philatelic/sr01_054.htm.

**23 September 2003: Walter McDonald [CPS Substitute and VA
Hospital Technician]
Fibonacci Numbers
Walter **began by writing some of the

fHe then calculated the ratios of adjacent_{2}= f_{1}+ f_{0}= 1 + 0 =1

f_{3}= f_{2}+ f_{1}= 1 + 1 =2

f_{4}= f_{3}+ f_{2}= 2 + 1 =3

f_{5}= f_{4}+ f_{3}= 3 + 2 =5

f_{4}/f_{5} |
3/5 | 0.600 000 000 |

f_{5}/f_{6} |
5/8 | 0.625 000 000 |

f_{6}/f_{7} |
8 / 13 | 0.615 538 615 ... |

f_{7}/f_{8} |
13 / 21 | 0.619 047 619 ... |

f_{8}/f_{9} |
21 / 34 | 0.617 647 058 ... |

f_{9}/f_{10} |
34 / 55 | 0.618 181 818 ... |

f_{10}/f_{11} |
55 / 89 | 0.617 977 258 |

^{. }. _{.} |
^{.}^{ }.
_{.} |
^{.
}. _{.} |

Limit |
- - - |
0.618 033 988 ... |

**Porter Johnson** pointed out that the the number **x**, the
limit of
the ratio of adjacent **Fibonacci** numbers, is the "simplest"
non-rational number ---** all "1's" in its continued fraction
representation**.. For more information as well as additional
references, see
his comments at the **Math-Physics SMILE** meeting of **10
April 2001**:
mp041001.htm.

Biographical information about **Leonardo Pisano (sobriquet
"Fibonacci") [**1170-1250??]
can be found at the** ST Andrews University History of Mathematics **
website [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html],
from which the following is excerpted:

"Fibonacci was born in Italy but was educated in North Africa where his father, Guilielmo, held a diplomatic post. His father's job was to represent the merchants of the Republic of Pisa who were trading in Bugia, later called Bougie and now called Bejaia. ... Fibonacci was taught mathematics in Bugia and travelled widely with his father, recognising ... the enormous advantages of the mathematical systems used in the countries they visited."

**Fascinating topic. Good job, Walter!**

**07 October 2003: Porter Johnson** touted the book **The
Golden Ratio and
Fibonacci Numbers **{Springer Verlag 1997: ISBN
981-02-3264-D). In this
invaluable book, the basic mathematical properties of the golden ratio
and its
occurrence in the dimensions of two- and three-dimensional figures with
fivefold
symmetry are discussed. In addition, the generation of the **
Fibonacci**
series and generalized ** Fibonacci** series and their relation to
the golden ratio
are presented.

**02 December 2003: Walter McDonald [CPS Substitute Teacher;
X-ray technician -- Veterans
Administration Hospital, North
Chicago]
Fibonacci Number Programs on HP 48G Programmable Calculator
Walter **passed around information on two programs to calculate the

**Why are the execution times so dramatically different?**
According to the handout, the **Program FIB1** calculates
intermediate values
**F(i)** more than once, while **Program FIB2 **calculates each
intermediate value **F(i)** only once. Thus, the time
required to
calculate **F(n)** grows **exponentially with n in FIB1**, and **linearly
with n in FIB2**. **FIB2 is obviously the "way to go"**,
whereas **FIB1** bogs down, even with a relatively fast **CPU**
[central
processing unit].

**Porter Johnson** pointed out that most currently
available programmable computers are at **least as sophisticated**
as the
first commercially available mainframe computers, e.g. **IBM 650**.
This
means that (1) they can perform **absolutely marvelous** tasks for
you
**if **you know how to program them but (2) when a particular
program **crashes**,
you generally have to reboot the system before it will run again.
**Note:** Programs that **crash** the computer will cause
it to **end up
in computer limbo land**, producing meaningless results! **In
fact, no
hand-held calculator is ever of any value, until and unless you know
how to use
it, and use it properly! The more sophisticated the calculator,
the easier
it is to crash it and get nonsensical answers!**

**Very interesting, Walter!**

**23 March 2004: Bill Colson [Morgan Park,
Mathematics} Euler's
Disk
Bill** showed us a carefully designed coin-shaped disk. Known
as

For additional details see ** The Official Euler's Disk**
website: http://www.eulersdisk.com/.
In particular, for information on the physics of this object see the
page http://www.eulersdisk.com/physics.html,
from which the following is excerpted:

"WhenEuler's Diskis spun, the disk contains both potential and kinetic energy. The potential energy is given to the disk when it is placed upright on its side. The kinetic energy is given to the disk when it is spun on the mirrored base. Euler's Disk would spoll (i.e., spin and roll) forever if it were not for friction and vibration. ... "

For insights gained by **Richard Feynman** concerning a plate
wobbling on
a table, see the following excerpt from his book **Surely You're
Joking, Mr
Feynman**: http://www.amazon.com/Surely-Feynman-Adventures-Curious-Character/dp/0393316041

**Euler's Disk** was named in honor of the famous mathematician
**Leonhard
Euler (1707-1873)** [
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html],
who maintained a life-long interest in puzzles. His puzzle
concerning the
crossing of the 7 bridges over the river **Pregel** in the city of **Konigsberg**
[
http://www-groups.dcs.st-and.ac.uk/~history/Miscellaneous/Konigsberg.html]
represented the beginning of modern topology. **Euler's**
beautiful equation **e
^{ip }+ 1 = 0 **connects the
five
most important numbers in mathematics --- two of which were invented by

An anonymous student once said **"infinity is where things happen
that
don't".**

**Thanks for these delightful math and physics insights, Bill!**

**04 May 2004: Don Kanner [Lane Tech HS,
physics]
Funnel Multiplication****
Don** recently learned of a novel method of multiplication, which he
illustrated with an example such as

numbers arranged in 4 | 5 | 6 | 7We could write out the multinomial product of decimal numbers

four columns of two ´ 9 | 8 | 7 | 6

--------=---------

product of numbers in same column |3 6|4 0|4 2|4 2|

sum of diag products; adjacent cols: 7 7|8 3|8 5|

sum of diag products; next col pairs |8 3|8 6|

sum of diag products; 1st and last cols: |8 7|

(now, add everything up) --------------------

(et, voilá!) |4 5|1 0|3 6|9 2|

´ [10

-------------- = -------------

10

10

10

10

For discussion of an ancient description of this
algorithm for multiplication, see the** Iowa State University
Department of
Mathematics** webpage ** Math
Night Module: Multiple Methods of Multiplication**
[
http://www.math.iastate.edu/mathnight/activities/modules/multiply/aboutmod.shtml]
from which the following has been excerpted:

"The history of mathematics in India is ancient. The Hindu Vedic tradition is an oral tradition of knowledge passed down in short verses, dating to before the invention of paper. The Vedas encompass a broad spectrum of knowledge, including the sutras (verses) pertaining to mathematics. In the early 20th century Swami Shri Bharati Krishna Tirthaji Maharaja claimed to have rediscovered a collection of 16 ancient mathematical sutras from the Vedas and published it in a book calledFor additional discussion of this and other ancient multiplication methods see also http://www.math.iastate.edu/mathnight/activities/modules/multiply/ and http://www.pballew.net/old_mult.htm.Vedic Mathematics.Historians do not agree on whether or not these were truly part of the Vedic tradition. If these sutras date back to the Vedic era they were certainly part of an oral rather than a written tradition. However, they are a novel and useful approach to computation: they are flexible in application and easy to remember. They can often be applied in algebraic contexts as well as in simple arithmetic. 'Vertically and Crosswise' [sic:URDHVATIRYAGBHYAM] bridges the gap between arithmetic and algebra: the algorithm is very similar to the standard algorithm used in the US and also is similar to the"FOIL" (first, outer, inner, last)rule used for multiplying binomials in algebra."

**Thanks, Don!**

**12 October 2004: Walter McDonald [CPS Substitute
Teacher]
Perfect Numbers and Triangular Numbers **

**Walter **passed around **Playthink 528**, taken from
the reference[**1000 PLAYTHINKS Games of Science, Art, &
Mathematics** by Ivan Moscovich; Workman Publishing [http://www.workman.com/]
2002; ISBN: 0-7611-1826-8 ], from which the following
information on **PERFECT NUMBERS** has been excerpted:

"A perfect number is the sum of all the factors that divide evenly into it -- including 1, but excluding the number itself. The first perfect number is 6, which is divisible by 3, 2, and 1, and is the sum of 1, 2, 3. So far; thirty-eight perfect numbers have been found. Can you work out what the second perfect number is?"The answer is1 + 2 + 3 = 6

"If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect."

... or ...If, for a positive integer

n, the number2is a prime, then^{n }-12is a perfect number.^{n-1}´ (2^{n}-1)

Perfect Number |
n |
2^{n-1} |
2^{n} -1 |

6 |
2 |
2 |
3 |

28 |
3 |
4 |
7 |

496 |
5 |
16 |
31 |

8128 |
7 |
64 |
127 |

33,550,128 |
13 |
4096 |
8191 |

As pointed out by **Bill Shanks**, a **triangular number
**is a number **N** that can be written as the sum of integers
from **1 through p**:

**Fascinating stuff! Thanks, Walter.**

** 26 October 2004: Don Kanner [Lane Tech HS,
physics] A
Book Review **

**Don ** touted the book **A History of Mathematical Notations**
by **Florian Cajori **[Dover Publication 1994], ISBN
0-486-67766-4. **Don** cited as an example that **Cajori**
described the origins of the signs** + and - **to represent
addition and subtraction, respectively. They were developed in
Germany during the last twenty years of the fifteenth century.
The + sign first occurs in print in a book by **Johann Widman (1489)**.
[See also **Widman's** biography: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Widman.html.
It is considered by experts to be a misprint for the symbol **vñ**,
representing **vnnd **or "**und**"-- "and" in modern German.
The **- sign** first occurs in print in a book by **H Brugsch**,
and it is given the name **minnes**. For a biography of**
Florian Cajori** see the website http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cajori.html.

**Thanks, Don**.

**29 March 2005: Don Kanner [Lane Tech HS,
physics]
Signs of the times + p a la mode
Don** described the history of the multiplication symbol

3 3 7 7 8The notation was then simplified by leaving off the vertical lines:

| | X | | X |

3 3 7 5 6

----- ... --------- ... ----------

9 4 9 4 8

2 1 4 2

2 1 4 0

9 3 5

--------- ---------

1 3 6 9 4 3 6 8

7 8The division problem

X

5 6

------

48

42

40

35

------

4368

3 2 3_{°}5 15

--- X --- = ----- = --

7 5 7_{°}2 14

"The European Renaissance brought about in due course a whole new mathematical world. Among the first effects of this reawakening was the emergence of mathematical formulae for p. One of the earliest was that ofWallis (1616-1703)

2/p = (1_{°}3_{°}3_{°}5_{°}5_{°}7_{°}...)/(2_{°}2_{°}4_{°}4_{°}6_{°}6_{°}...)and one of the best-known is

p/_{4}= 1 -^{1}/_{3}+^{1}/_{5}-^{1}/_{7}+ ....This formula is sometimes attributed to

Leibniz (1646-1716)but it seems to have been first discovered byJames Gregory (1638- 1675)."

**Don ** also mentioned the formula used by ** Francois
Vièta (1540-1603)**, for which the
following information appears on the** ST Andrews** website:

"Viète used Archimedes method with polygons of 6 ´ 2^{16}= 393216 sides to obtain 3.1415926535 < p < 3.1415926537. He is also famed as the first to find an infinite series for p."

**Don** is engaged in an investigation to see which of the
expressions converges most rapidly to** p**.**
Place your bets now!**

**04 October 2005:
Fred Schaal (Lane Tech HS,
mathematics)
Dubito Ergo Sum
**A student recently asked

**Fred **also asked the age of **The Naughty
Lady from Shady Lane**: http://www.theguitarguy.com/naughtyl.htm.
Aficionados of songs from the 50's will no doubt
recall that, in the ** 1954** song, "**She's only nine days old!**".
**Fred** has
started to have a regular **Google Session** in his math classes,
to address
such issues. **Thanks, Fred**