MTH 105, 10:30 am Class
- December 4, 1992 -
TERM PROJECT GROUP:
We chose, for our
Term Project, to investigate ways in which random numbers are produced.
“A phenomenon is
called random if individual outcomes are uncertain but the long-term pattern of
many individual outcomes is predictable.” Our Text, page 184.
The Nature of Modern Mathematics, a textbook by Karl J. Smith, published
in 1976, gives two tests for randomness:
a. Each digit should appear approximately the same number of times. This means
that, given enough such digits, the percentage of occurrence for each should be
close to 10%.
b. Are there any sequences of numbers? In naming 100 digits, we could expect a
random selection to have a sequence of several digits.
I. Tossing a Coin.
Coins have been tossed to make decisions, probably for as long as we have had
coins. The two-headed coin probably followed soon after. A coin toss may be the
most reliable way to choose between two equally desirable (or dreaded)
outcomes. Some of the randomizing influences acting upon a tossed coin are:
velocity, spin, wind speed and direction, and air resistance against the
different surfaces of the coin. Gravity, though it always pulls downward, may
combine with the others to have some effect on the result. One evidence that
coin-tossing produces a random result is that it may sometimes produce long
sequences of heads, or of tails.
II. Drawing Straws.
Drawing straws must also be a very old method of choosing an outcome, due to
its simplicity. There would be one straw for each person; all but one of which
would be the same length. One person would hold the straws while the others, in
turn, would draw a straw from his hand. The person drawing the
odd-lengthed straw would be chosen. As long as there is no cheating, and
the visible parts of the straws appear basically identical, It should be a
reliably fair way to choose who, of a relatively small group, will perform a
duty, receive an indivisible portion, or make some other decision which
requires the utmost impartiality.
Drawing names or numbers out of a hat, or from some other container, is a
common way of choosing from a group of people or options, and is akin to
Years ago, you used to see on television, the use of large drums, into which
postcards were put. The postcards were sent in by television viewers as an
entry in a contest of some sort. The large drum was made so that you could see
inside (so you could tell there was no trickery involved), and the drum was
turned to mix up the cards, to make it as impossible as possible to locate any
particular card, and then someone was blind-folded and led to the drum. At this
point, a door was opened in the drum and the blind-folded person reached into
the drum and pulled out a postcard, or more than one if there were more than
one winner to be chosen.
III. Rolling Dice.
According to the Academic American Encyclopedia, online with Prodigy, dice are
“... the oldest gaming instruments known to humankind, ....“ They can presently
be found in versions with different numbers of faces besides the original
(presumably) 6-sided dice. There are games that use dice with letters, numbers,
or dots, but all are used to represent a randomized outcome. We learned through
this course how dice for gambling houses are made to exacting specifications,
so that the randomness of the roll will not be skewed. As long as the roll of
the dice is fair, the gambling houses stand to make a bundle.
IV. Spinning Wheels.
Many board games use a wheel or circle divided into an appropriate number of
some type of
spinning pointer to select a number, or other outcome, by the position of the
pointer when the pointer stops spinning. These can be random, by making the
areas equal, or weighted, by varying the area of each segment. A famous example
is the Wheel of Fortune from television. We learned about the roulette wheel in
class. Some of us even remember the old game, Spin-the-bottle, which is a crude
use of a spinning wheel for choosing.
How many of us have picked up a daisy, or another flower, and started plucking
the petals off one by one, muttering something like, “ She loves me. She loves
me not. She loves me ... .” How did it turn out? Chances are, you
didn’t know when you started. Whether or not this is a good random
decision-maker depends on what mechanism in the flower determines the
oddness or evenness of the number of petals. If you knew (which we don’t) that
all daisies have an odd number of petals (or even), you would know whether to
start with “She loves me.” or “She loves me not.”
According to the Academic American Encyclopedia, “The earliest state lotteries
were organized in France in 1520. In 1680, England held a historic lottery to
raise funds for improving London’s water supply equipment. Spain developed the
Gordo, and Ireland, the sweepstakes. Lotteries were popular in the United
States ...“ until 1890, and until “1963, no government-sponsored lotteries were
held in the United States.”
Some lotteries used the spinning drum method as described in V. Lately, we seem
to prefer the little balls blowing around in a plastic chamber, then popping up
to the top as if chosen by the wind (which is believed to be pretty random)
with numbers printed on them. In gambling, the numbers are compared with the
numbers on a tickets held by the players to determine a winner. A lottery was
used to select the Selective Service Numbers of men to be called into the Armed
VII. Random Number
Various textbooks have tables of random digits. Our textbook has the famous (or
infamous) Table 5.1. It is very useful for classroom situations, in that it
allows an instructor to designate a situation and a starting place, and he will
be able to predict the outcome, and all students who follow the directions
carefully will get the same outcome. But this shows that the digits are not
truly random, because the outcome is predictable. And besides this, all
possible outcomes are not representable. Using Table 5.1, for instance,
it is impossible to represent an occurrence of more than three consecutive
identical digits. In a table of 100,000 non-duplicated 5-digit numbers, you
could not find represented an occurrence of more than nine consecutive
identical digits. Tables are useful, but they have their limitations.
VIII. The Digits in
the fractional part of Pi:
value of Pi has been approximated by a computer to 480 million digits
(according to the Academic American Encyclopedia). The digits, 0 through 9,
seem to be distributed randomly. If you ever needed a Table of Random digits,
the 480 million digits of Pi (so far known) would make a doozy of a table.
Here it is figured to 500 decimal places: Pi =
3. followed by
14159 26535 89793 23846 26433 83279 50288 41971 69399 37514 58209 74944
59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647
09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559
64462 29489 54930 38196 44288 10975 66593 34461 28475 64823 37867 83165
27120 19091 45648 56692 34603 48610 45432 66482 13393 60726 02491 41273
72458 70066 06315 58817 48815 20920 96282 92540 91715 36436 78925 90360
01133 05305 48820 46652 13841 46951 94151 16094 33057 27036 57595 91953
09218 61173 81932 61179 31051 18548 07446 23799 62749 56735 18857 52724
89122 79381 83011 94912
IX. Computer Random or RND Functions.
Some computer algorhithms use Pi and a seed number to generate a random number
between 0 and 1. This result is multiplied by the desired range of numbers and
added to the desired minimum, INTed (the fractional part removed) and 1 added
to give an integer in the desired range. The result of the first computation is
fed back into the algorithm as a seed number to produce the next random number.
Other computer algorithms may use a table, similar to Table 5.1, and a
seed number (sometimes from the TIME function) to produce random numbers.
Often you may choose between a random function that will create the same sequence of random numbers
every time the program is run, or an entirely new sequence each time the
program is run.
IN CONCLUSION, may we say that sometimes the oldest methods are the best. It is
hard to beat the randomness of a tossed coin, or the roll of a well-balanced
die. For otherwise uncomplicated decisions, these seem unexcelled, but if
we have to produce large numbers of random numbers, or simulate millions of
random outcomes, nothing beats a good computer.
Not having read any of John Steinbeck's works in the last 20 years [now 30], I
find myself at a disadvantage, since I would rather answer the man himself on
his opinions, than what I perceive them to be through a faulty memory, or
perhaps through an incomplete representation Nevertheless, I will respond to
the assertion, as represented, that our nation is at risk because the American
people are "selfish, self-centered, and incapable of handling most stressful
situations," and that "our children are reared under such lenient conditions
with little respect for themselves or others."
In regard to the assertion of our being "selfish, self-centered, and incapable
of handling most stressful situations," I would admit that it is a condition of
mortality that we are inclined to be self-centered, but would point out that
there are many who rise above this tendency. We tend to see the acts of
selfishness and greed, and failure to handle stress in the newspapers and on
television, but we have also seen (I believe, only the tip of the iceberg) many
instances in local or more general emergencies where selflessness is displayed,
beyond what most of us may think ourselves capable. I believe that such
greatness is in every one of us: not in America alone, but throughout the
In regard to the assertion that "our children are reared under such lenient
conditions with little respect for themselves or others," I would say this:
Many of our children are suffering from too little leniency, and many adults
are suffering because they, as children, suffered. Fear is not respect. Love
brings respect. It is true that we need to be firm with our children, but we
cannot gain their respect by hitting them because we can't get them to do what
we want them to do. We shouldn't even be able to respect ourselves as parents
if we have to resort to bullying. My wife and I are learning ways to raise our
daughter with love. We have not had to resort to hitting or bullying, though
the urge does come to us a times. It is a matter of education, and help from
the Lord to do that which we know to be right. Perhaps our nation is at risk
partly because parents have not known how to raise their children with true
respect: Parents must respect their children, who are a gift from God, and
children must respect their parents because of the example of patience and love
which they are shown.
In regard to the assertion that our nation is at risk, I believe that America is
a land belonging to Jesus Christ and is blessed to those who are obedient, and
cursed to those who are not. The nation is made up of individuals, some of whom
are obedient to that which they know [or understand] to be right, and some of
whom are disobedient.
"And if the time comes that the voice of the people doth choose iniquity, then
is the time that the judgements of God will come upon you; yea, then is the
time he will visit you with great destruction even as he has hitherto visited
this land" (Mosiah 29:27-39).
Yes, the nation is at risk, but it is also given a great opportunity. America is
the land where Adam and his family dwelt. America is the land that was
separated in the days of Peleg, the great-great-grandson of Noah, "for in his
days was the earth divided" (Genesis 10:25). America is the land to which some
of the scattered at the time of Babel were led (Book of Ether). America is the
land to which various pilgrims, exiles and wanderers have come, led by the hand
of the Lord, including part of the tribe of Ephraim of Joseph at the time of
the fall of Jerusalem (about 600 B.C.). America is the land to which a Jew,
Cristobal Colon (Columbus), was directed by the Holy Ghost (by his own
confession). America is and has been a land to which God has chosen to bring
some of all races and creeds, some in chains; most of them ignorant of their
calling. They came, and are still coming. We as a nation have a great
opportunity and a great responsibility. The world still looks to us to see what
we will do with it. God is watching, too.
Sawyer, Summer 1992
written as an Essay for my writing class at Chemeketa
This is where I will die: Sitting on dry leaves amid patches of sunlight and
shade, I lean my head back against this great tree and look up toward the
leaves which are lashing at each other in the breeze. Though a few of the
branches have long ago been hewn off by men and have healed over, the rest of
the branches reach toward the sky and intermingle with each other in the
I am sitting here a short distance from the top of a hill, that I might not be
easily visible to any who still search for me. I am the last of my people and
my enemies seek my life. I have wandered many years, avoiding all contact with
them because of their hatred; they would kill me if they found me.
They do not understand why they want to kill me; it seems right and natural to
them. It is because they were brought up hating my people, and as long as one
of my people lives, they will not be able to give up that hatred. I do not hate
them; I cannot. I do not fear them, nor do I fear that old enemy, Death, for I
fear only my God, and that is another kind of fear altogether.
My father, who led the armies of our people, refused to lead them when the
people desired to attack our enemies in revenge, after they had been driven off
from our cities. The people would not listen to his counsel that they humble
themselves and thank their God for deliverance, but took it as a matter of
pride that they were able to defeat their enemies. They began a war of revenge
against their enemies, and the enemy began to capture our cities, and our
people began to flee before them from city to city.
My father relented because of the state our people were in, and our people
eagerly looked to him to lead them to victory again. He had no hope of victory,
because our people had gone beyond their ability to repent; they did not give
up their hatred or pride.
It was a terrible thing for my father and I to lead our people in the final
battles with our enemies. We witnessed much carnage: our enemies in loincloths
hacking with scimiters and axes upon the bodies of our people, and they, though
in protective clothing, could not repel the attacks of so many of our enemy.
Finally, all my people, with their families, gathered to a hill much like this
one, and prepared for the attack. When the attack came, our people were filled
with terror. All but twenty-four of us, fathers, mothers, sons and daughters,
died in that battle, and we only survived because we were wounded and were left
for dead. And since that time, the other 23 have been hunted down and killed,
and I alone am left of my people.
I have buried the records which were given me to safeguard, and those which I
myself have written. They will be of great benefit to the descendants of those
who seek my life, and to all people, when they are recovered and read. Like the
leaves in the tree-top, the people will be stirred by them to great things.
I have had many dreams, or visions, from the Lord. I have seen the future of my
enemies; indeed, I have seen the future of this world. The people who seek my
life will suffer much, and also their descendants for many generations, but in
the end, the records which I have hidden up will be recovered and brought to
them. They will know of their fathers, and of their God. They shall be as the
branches of the tree that reach to the sky. They shall intermingle with the
other peoples and share with them their story, and the other peoples shall
share their stories with them.
When I first sat myself down here, being too old and too feeble to care for
myself any more, and having none to care for me, the birds brought food to me,
that I might not hunger. Then, when I began to fast in preparation for death,
the birds came, not with food but with their songs, to cheer me.
I have not eaten, nor have I been hungry, for many days now. I will close my
eyes as if to sleep; but it will not be sleep, but an awakening. It has been a
beautiful day. It is a good day to die, and a good day to begin new life.
- David E. Sawyer, July 11, 1992
written as an Illustration Essay for my writing class at Chemeketa
[or a Hindu or an Atheist or Anybody]
Become a "Mormon"?
attending a Church conference, a few years ago when someone came up to me and
told me that he had a friend visiting with him who wanted to know how a Baptist
could become a Mormon. She was a grandmotherly lady, and I told her my somewhat
memories are of playing in a sandbox in the basement of the Keizer Community
Church. It was my grandparents' church. My parents took me, and later my
brother John, there for Sunday School and Worship Services. At some point, when
I was still quite young, I became concerned about my eternal welfare and would
kneel at my bedside and pray for Jesus to come into my heart so I could be
"saved." After praying, I would climb into bed with an assurance that I was
"saved;" however, the next night I would feel I had to do it all over again,
and again. I finally decided to accept that I had done what was required, and
to consider myself "saved."
When I was
in the Fourth Grade, my family moved into the newly forming suburbs [three]
blocks east of Lancaster Drive. There were only two houses facing our street
(which was not paved). We had a neighbor, Rev. Jim Spillman, who was starting a
new church in town, the Willamette Baptist Church on the corner of 16th and "A"
Streets. He invited us to come. We stayed. We had three pastors after that who
helped guide me as I grew: John R. Turnbull, Jet S. Turner, and Warren L.
Fleischmann. They were all good men who loved the scriptures and loved the
people. John R. Turnbull, an elderly man, and a published author, who came here
from Canada, baptized me into the Willamette Baptist Church; Jet Turner had
been a missionary to Italy; and Warren was a fresh pastor, straight out of
Seminary. I respected and loved them all (and still do).
3 years at Salem Academy, a non-denominational Christian high school, rather
than going to the public high school (North Salem High). I consider it a great
experience, to have gone to school with others who shared the same spiritual
interests. After considering a career as an astronomer, I decided to become a
I went to
Judson Baptist College in Portland, Oregon, for 2 years and received a diploma.
During that time, my faith was both nurtured and challenged. I attended Lents
Baptist Church while I was going to school. At one time the college students at
the church planned a debate on the doctrine we called "Eternal Security." I was
the only one to volunteer to take the opposing view. I threw myself into
preparing for the debate. I researched all the arguments I could find for or
against the doctrine. I had been taking New Testament Greek so I examined the
scriptures from the point of view of the original language. I organized the
arguments by their logical validity and discovered to my surprise that the
soundest arguments indicated that the scriptures did not teach that doctrine,
which was a vital one in the Conservative Baptist group, anyway. The debate
never took place, probably due to lack of interest, but the important
comparison had already been made. I could not conscientiously become a minister
of a church which taught what I believed the Bible did not teach - I could not
just avoid the doctrine.
Oregon State University for half a year trying to continue my education, but
dropped out due to not having a real goal, not being able to get the courses I
needed, and financial problems.
got directed to the Salvation Army, probably by a friend. They had only a few
doctrines one had to accept and I believed in all of them, so I became a
member. I met my first wife, who was Catholic, and we were engaged.
Not being a student any more, I
began considering my options regarding the Draft. I couldn't picture myself
killing people - it went against everything I believed. I even considered going
to Canada. One day, I was reading in the Bible, in Jeremiah chapter 42, and
heard, not with my ears, but just as certainly, "Join the Army." There was no
doubt in my mind who had spoken to me, or what was meant by it, so I went down
to the Army recruiter, looked over their options and signed up for a Job on a
delayed entry program. I was in Basic Training a couple of months later. I
trusted that He who spoke to me would handle things, and indeed, I did not have
to kill anyone. I learned to repair printing presses, and then was sent to
there were no Salvation Army chapels, and I found the base chapels too
wishy-washy, though the chaplains were fine men. They had to cater to too many
different churches. I found a Pentecostal Servicemen's Center off-base that was
spiritually edifying for me. There, among other things, I was shown
scripturally that the gift of the Holy Ghost is not given automatically when
you believe in Jesus. They said that the gift of tongues was given as a sign
that you had received the Holy Ghost. I got off by myself and prayed for the
gift of the Holy Ghost. I went home from Okinawa discouraged because I never
spoke in tongues, though I heard others do it, and I believed the scriptures
said it was necessary to receive it.
to Oregon and the Salvation Army. I married my first wife in an ecumenical
wedding in the Salvation Army chapel. Two Catholic priests attended and one of
them co-officiated in the marriage ceremony. That was something.
One day, as
will happen, two Jehovah's Witnesses came to our door. I was interested in
listening to them, but my wife didn't want to, so we sent them away. After my
wife died, however - she died after we had been married barely 6 months - they
came again. I listened. They showed me that the doctrine of the Trinity was not
scriptural, that it was a later invention. I believed it. I discussed it with
my Salvation Army Captain (like a pastor), and with another friend who was a
Methodist minister, neither could give me any acceptable answers to this from
the Bible. I spent several months studying with the Jehovah's Witnesses, even
to going door to door with them.
they taught was that Jesus was resurrected without a physical body. I could not
accept that. The scriptures plainly say that Jesus showed himself to his
disciples, telling them (in the Witnesses' own translation) "See my hands and
my feet, that it is I myself; feel me and see, because a spirit does not have
flesh and bones just as YOU behold that I have: (Luke 24:39). He then proceeded
to eat a piece of fish as further evidence that he had a body.
over a couple of years, I attended several Pentecostal Churches and my friend's
Methodist church. I met and married my second wife. I attended a Seventh-day
Adventist church, which taught me the value of the Sabbath. Even though they
believed in the Trinity, I began to feel pretty good about that church. I was
also looking into the Worldwide Church of God which had a televised preacher.
They taught, among other things, about the Lost Ten Tribes of Israel.
time, I was working at the Y.M.C.A. in Salem. I was actually considering
starting my own church, teaching what I found the scriptures to say. One night,
one of the residents was sitting in the snack area at the "Y," and asked me
what I thought about the Book of Mormon. Everything I had ever been taught
about Mormons must have come to mind at once. I told him I thought that Joseph
Smith had been tricked by a devil disguised as an angel, that he probably
believed what he wrote, but that it was all a hoax perpetrated by the devil.
(You see, once in high school I had a chance to get my hands on a blue
paperback version of the Book of Mormon, and I prayed to God to show me what
was wrong with it: I opened it up to Mormon 8:12 which says, "And whoso will
not condemn it because of the imperfections which are in it, the same shall
know of greater things than these." I closed the book thinking, "The Bible
doesn't have imperfections," and didn't think about it again for 10 years).
responded with the statement that I would have to read it and ask God if it
were true to be sure. That stopped me. I had done the same thing with the Bible
back at Judson Baptist College. I had gotten the answer then that the Bible was
true, and have not doubted it since. I agreed with him, and said that I would
do that, for I was still looking for the Truth, and had not found it all in one
place yet, and even if this was false, it would be worth finding out for sure.
time, my son was born and we moved to West Salem, I was reading the Book of
Mormon and praying to know if it was true. I finished the Book of Mormon and
began reading the Doctrine and Covenants, a book of revelations received by
Joseph Smith in regard to the Church and the Gospel. I thought to myself, if
anything, this book will convince me it is not true because the "stranger"
doctrines such as plural marriage, etc., were in it. When I was almost
precisely in the middle of the book, I had occasion to be walking up to the
Mormon Bishop's home (I forget why) and suddenly I had thoughts and feelings
come to me which I cannot convey to anyone else. I think of it somewhat like
having one of those 500 piece jigsaw puzzles in a box and having the pieces
jump out of the box onto a table and assemble themselves correctly of their own
accord. All of the doctrines which I had wondered about for so long, and many
other details besides, suddenly came together into one unified whole, and it
was as if a voice spoke to me, comfortingly, saying, "Do not doubt, it is all
I went to
Albany, Oregon, where my parents were living and told them that I was going to
be baptized into the Church of Jesus Christ of Latter-day Saints (the Mormons).
They asked me to wait a year (hoping this craziness would pass). I said it
would not be right to wait, since I knew it was the right thing to be baptized.
They asked me if I would talk to their pastor, Warren Fleishmann - he had been
one of my pastors when I was a youth. He saw me immediately, no doubt as
concerned as my parents were. He had a two-sided mimeographed sheet with
information about the Mormons on it. He asked me questions. At one point he
quoted a verse from the Book of Mormon from his paper and asked me if it didn't
contradict the Bible. I asked him if he knew what the context was. He didn't. I
opened my Book of Mormon and read the entire chapter, then told him that I
didn't see any conflict with the Bible. We parted, neither having changed our
point of view.
baptized by the Bishop of the West Salem Ward. Since then it has been 17 years.
I have been through a divorce (my son's mother), a third marriage, a serious
auto accident, the death of my third wife, a fourth marriage, the birth of my
daughter two and a half years ago, and the loss of a job which I had worked at
for 16 years. I have no reason to doubt that I made the right decision when I
joined the Church. Because of the Gospel which I have received, I understand
the relationship between the members of the godhead, I have been baptized by
the authority of Jesus Christ, have received the gift of the Holy Ghost, been
ordained to the Priesthood (unpaid), have been privileged to give blessings to
others with the authority of that Priesthood, and been sealed to three of my
wives for all eternity (it's legal as long as they are not all alive at one
time). I have had the opportunity to do work in the Temple and help others to
have the same blessings.
shows that anyone can become a Mormon, that is, become a member of the Church
of Jesus Christ of Latter-day Saints. Everyone will not go through exactly the
same experiences which I have had, but they must have a desire for the Truth,
and have faith in God (which can be nurtured), believe that God will not give a
serpent for a fish or a stone instead of bread to those who ask him sincerely,
and be willing to act on that which God tells him/her.
In the last chapter of the Book of
Mormon there is a great promise: "And when ye shall receive these things, I
would exhort you that ye would ask God, the Eternal Father, in the name of
Christ, if these things are not true; and if ye shall ask with a sincere heart,
with real intent, having faith in Christ, he will manifest the truth of it unto
you, by the power of the Holy Ghost. And by the power of the Holy Ghost ye may
know the truth of all things" (Moroni 10:4, 5).
Sawyer, Summer 1992
my writing class at Chemeketa Community College.
cloud-shrouded peak in the cold rarified air, lofty and inaccessible - a
mountain spring bubbling from beneath a rock - a cotton cloud in an
otherwise clear blue sky - a pearl glinting in an oyster - the Evening Star on
a moonless night - an oasis in the desert - the full moon just before
sunrise - the last brown leaf on the tree.
struggling in a spider's web - an eagle riding the currents of the air - a
canary in a gilded cage - a wandering lamb - a spaniel left in a car while its
owners are shopping - a robin's egg in the grass.
weather-worn barn - a prison tower - a rope bridge over a chasm - a deserted
lighthouse on a jetty - the caboose of an empty train - a broken-down carousel
in a forgotten fairground.
wasting away in the grass - an errant balloon - the forgotten toy - a floating
bottle with a note inside - the last crust of bread.
mailbox - junk mail - a busy signal - reaching an answering machine - an echo -
receiving a cold shoulder.
A ship at
sea in a storm - a flat tire and no spare - a hitchhiker's thumb.
sleeping in a cardboard box in an alley - a child crying because she cannot
find her parents in a crowd - a condemned man blind-folded in front of a firing
squad - a football quarterback - a pioneer - a straggler - a dreamer of dreams
- a man or woman at prayer - a sinner - the self-righteous - a leper calling
Fear - Jealousy - unrequited love - having to always be right - divorce -
defeat - losing a friend - the death of a loved one -staring up from the mud
through barb-wire and tracer rounds - peace and quiet.
always are - What we never are.
Sawyer, Summer 1992
my writing class at Chemeketa Community College.