Natural Blogarithms

From an article by Mary Ann Bragg which appeared on CapeCodeOnline and was also printed in this month’s College Mathematics Journal:

TRURO — Voters narrowly approved one of four zoning amendments late Tuesday night at the annual town meeting. But town officials were still looking at the exact vote count on that article yesterday.

In a vote of 136 to 70, voters passed a new time limit on how quickly a cottage colony, cabin colony, motel or hotel can be converted to condominiums. The new limit requires that those properties be in operation for three years before being converted to condominiums.

The idea behind the zoning amendment is to slow the pace of condominium development in Truro and preserve more affordable accommodations for tourists, according to citizens proposing the warrant article.

Currently Truro does not allow condominiums complexes to be built outright in its zoning bylaws. Instead, property owners must build a cottage colony, cabins, motel or hotel first and then covert it to condominiums through a special permit.

The exact count of the vote — 136 to 70 —had town officials hitting their calculators yesterday. The zoning measure needed a two-thirds vote to pass. A calculation by town accountant Trudy Brazil indicated that 136 votes are two-thirds of 206 total votes, said Town Clerk Cynthia Slade.

But is it?  Is 136 a sufficient number of votes to be considered two-thirds of the total 206 votes?  Let’s check:

If you use the fact that and then proceed to multiply 206 by 0.66 you get 135.96.  There were 136 votes in favor which is  more than 135.96 so that means it passes, right?  If you think so, then you’d be WRONG!!! 

The main problem is the rounding.  In fact, or using repeated decimal notation, .  When you round, you are actually creating an error that, in this case, makes a pretty significant difference.

Think of it another way, lets compare 136 / 206 to 2 / 3.  First, just do it by decimal approximation:

My calculator cannot exactly represent either of these fractions but its accurate to 12 decimal places and I can clearly see that 136/206 < 2/3 so the vote should not pass.

Do you remember another way you can compare fractions?  Find a common denominator and convert each fraction, then compare. 

So, here we see that, again,

This second method of checking is even better than the first because there are no approximations involved.  We’ve confirmed, absolutely, that 136 votes out of a total of 206 does NOT constitute two-thirds.

Fortunately, a good citizen made an anonymous call in Truro, MA, to clear this up.  What perplexes me is that they decided they needed to let the State Attorney General’s office decide on the correct count. The mathematical explanation wasn’t good enough. Can you say quantitative illiteracy?

Read the entire story here.

I’ve not studied much of the history of mathematics but occasionally I read from a few books I have on my shelf on the subject.  When my mind is bogged down and I am unmotivated on my current projects, I pick up, say, Makers of Mathematics, by Stuart Hollingdale.

Today, I flipped open to the chapter on Leonhard Euler (one of my mathematical heroes) and learned (or re-learned) a few interesting facts about the man.

• Entered the University to study theology and Hebrew but his mathematical abilities attracted the attention of Johann Bernoulli who gave him a private lesson once a week.  He received his master’s at 17.
• His father greatly desired him to pursue his theological ambition’s but was convinced by Bernoulli that his son was destined to be a great mathematician.  Leonhard Euler remained a devout Calvinist all his life.
• At the age of 26, Euler took on the leading mathematical position at St. Petersburg Academy
• He and his wife had 13 children, only 5 of whom survived to adulthood.
• He lost sight in his right eye fairly early in his career, probably due to overwork.
• He spent 25 years at Berlin Academy and then returned to St. Petersburg at the age of 59 about which time he lost sight in his other eye.  The blindness didn’t stop him.  In fact, he completed a comprehensive analysis on the theory of the Moon’s motion.  All the complicated analysis was done entirely in his head.
• In 1771, his house burned down.  In 1776, his wife passed away.  He died in 1783 at the age of 76 still active to the end.
• All told, he published more than 500 books and papers during his lifetime, while a further 400 appeared post-humously.  It has been computed that his publications during his working life averaged about 800 pages a year.

For the record, the correct pronunciation of Euler is “oiler” not “yuler”.  That’s a minor pet peeve of mine.  It ranks right up there with folks that write my name as Scoot instead of Scott.   

Learn more about Leonhard Euler.

This week I was given the privilege of leading our campus Environmental Stewardship Bible Study.  Currently the study is walking through various sections of the Bible in order to provide a scriptural foundation for concepts in environmental stewardship.  It was my good pleasure to lead a study on Psalms.  While the actual Bible Study was driven by some excellent discussion, much of the material in the study below I was unable to get to.  This outline has been posted in our Blackboard classroom for the Bible study but I also post it here for those folks that have been following my previous postings on creation care.

Here are the two primary sources for the study (in addition to the Psalms themselves)

How to Read the Bible for All It’s Worth (2^nd Ed.), by Gordon D. Fee and Douglas Stuart (Zondervan, 1993)

Out of the Depths (3^rd Ed.), by Bernhard W. Anderson with Steven Bishop (Westminster John Knox Press, 2000)

Study Outline – Download PDF

In several lengthy discussions that I’ve had with friends and colleagues over my new interest in environmental stewardship, inevitably we begin discussing the stereotypes of environmental activists.  Being raised in West Texas my whole life, I haven’t really encountered very many individuals that would be classified as environmental activists, but I’ve certainly heard my fair share of disparaging political epithets: hippy, environmentalist wacko, tree hugger, ecoterrorist, econazi, etc.

If you have a personal conviction to care for God’s creation and feel that abuse to the environment is tantamount to desecrating a temple of God then you may be offended by such terms.  I wouldn’t blame you.  But there is also the very real concern that environmental activism might be concomitant with the deification of nature.  Some of the people I talk to have reminded me that the much of the theology of popular environmentalism comes from humanism, paganism and pantheism

As a friend stated on the last blog entry,

It’s too bad that over the last 2-3 decades, environmentalism has been appropriated by a mainly leftist political crowd and propped up as a kind of secular religion. Environmentalism is for everybody, and conservatives have a lot to bring to the table on this issue. Christians too.

I believe it is possible to respect God’s creation in the same way that we respect other temples of God without deifying them and treating them as a god, outright.  We are called to sanctify the church, the altar of God, and even our own bodies.

So, instead of beginning with the love of nature and its beauty as a motivation of environmental stewardship, I think the best place to begin is in Scripture.  After all, the love of the outdoors has not come naturally to the “indoorsman” city-boy that I am.  If I do end up an environmental activist (yikes, that still scares me) it will be as one who moved from environmental indifference to environmental concern as a result of my faith and not as one who started out concerned about the environment and added my faith to the reasons for that concern.

By the way, I am still deliberately avoiding any statement or position on the many hot-button environmental issues such as climate change, over-population, deforestation, etc.  My reasoning goes back to the fact that the starting place for creation care is the stewardship of God’s world out of respect and honor for Him. Concern for nature and the world God created seems to be one of our responsibilities as God’s children with or without a crisis on our hands.

Below are several passages that I’ve begun meditating upon that deal with humanity’s relationship to creation and creation’s relationship to God. Please note that relating any of these passages to environmentalism can only be done in an honest and accurate interpretation of Scripture.  The most basic of principles that I hold to when interpreting scripture is that a passage cannot be made to say something that was not intended by the original author.  It must fit into the context in which it is given as well as in its genre.  The application of Scripture to our lives must flow out of the original meaning to the original audience.

The following were collected and compiled by the Evangelical Environmental Network and Creation Care Magazine.  You can download the original document from the Evangelical Environmental Network. These will serve as a devotional guide for the next few entries on environmental stewardship.

Jesus Christ’s Relationship to All of Creation: Creator, Sustainer, Reconciler, Consummator, true Imago Dei, Heir of all things, Lord

• Colossians 1:15–20
• John 1:1–3
• Hebrews 1:2–3
• I Corinthians 8:6b
• Ephesians 1:10

Creation Declares the Glory of God

• Psalm 19:1-4
• Psalm 96:1, 11-12
• Romans 1:20-23
• Revelation 4:11

The Old Testament Proclaims God as Creator

• Psalm 104
• Genesis 1 and 2:1-4
• Isaiah 40:12, 26
• Isaiah 42:5-7

The Earth is the Lord’s

• Colossians 1:16b
• Hebrews 1:2
• I Corinthians 10:26
• Psalm 24:1
• Leviticus 25:23
• Deuteronomy 10:14
• I Chronicles 29:11-12
• Nehemiah 9:6
• Isaiah 66:1-2

Christian Love and Justice

• Colossians 1:20
• II Corinthians 5:14-21
• Philippians 2:4-8
• Luke 4:18-19
• John 13:34
• I John 4:7-8
• Matthew 25:34-45
• Luke 6:31
• Micah 6:1-4, 7-8
• Jeremiah 22:3
• Psalm 72:1, 12-14a
• Matthew 22:37-40

God Provide for and Desire’s Sufficiency and Contentment for the Rest of Creation

• Psalm 104:10-30
• Genesis 1:20-25
• Genesis 6:19-21
• Genesis 9:8-17
• Exodus 23:12
• Leviticus 25:1-7
• Job 38:39-41
• Job 39:5-8

The Interrelationship Between Humanity and the Rest of Creation

• Psalm 65:1, 9-13
• Psalm 104:13-15, 21-24
• Genesis 1:24-31; 2:1
• Genesis 2:7-9a, 15, 19-20a
• II Chronicles 7:13-14

The Rest of Creation Harmed by Humanity’s Sin

• Genesis 3:17
• Isaiah 24:5-6
• Hosea 4:1-3
• Jeremiah 4:18-28
• Revelation 11:18

God’s Future Kingdom: a New Creation

• Romans 8:19-23
• Isaiah 11:1-9
• Isaiah 35:1-2, 6
• Isaiah 55:12-13
• Isaiah 65:17-23
• Revelation 21:1, 5

During Calculus last week, we covered slope fields and Euler’s method of first order initial-value problems of the form

During class we demonstrated the use of slope field to perform a basic qualitative analysis. We utilized a couple of different online applets as well as Maple 13.

Visualizing Slope Fields and solutions
Another similar page, with zoom

Direction Fields – A Maple 13 Worksheet

Below is a short demonstration of how we set up a simple application of Euler’s Method in Excel 2007. Consider the simple initial value problem:

Recall that Euler’s Method is given by

Watch the demo:

This is just a quick and dirty implementation of Euler’s method in Excel but it gets the job done.

At Wayland, there has been an increased emphasis on the need to "go green".  I’ll admit to having inherited a very West Texas conservative viewpoint on the environment.  I’ll even go so far as to admit not giving it much thought at all and just taking for granted the natural resources that are available to me.

So, what is a West Texas conservative viewpoint on the environment?  Allow me to attempt to describe what I think it is and please don’t necessarily take the following as truth or even a fair representation of what I currently believe. And while this is by no means universal in this part the world, it’s certainly not uncommon.

If I can afford it, I can use it.  If it’s on my land, then I can do with it what I want.  The climate change issue is too controversial.  In spite of a vast scientific consensus among climatologists, the issue has become so politicized that I doubt the integrity of the scientists involved in the debate.  Is it warming, is it cooling, is it natural, is it man-made?  I don’t know because I haven’t paid close enough attention.  But if I allow myself to fall along party-lines, like I do on so many issues, then it may or may not be warming but it’s unlikely to be caused by man."

Of course, climate change isn’t the only issue.  There’s deforestation, there’s the massive extinction of animal life on the planet, there’s a significant decrease in the availability of fresh water, there’s pollution, and so much more.  If you listen to most environmentalists, we are doomed.  If you listen to the rest, we are at the very least at a crisis in human history.  The vast number of humans on this planet have reached the point that they are a geologic force changing the landscape, the ecosystem of the oceans, and the atmosphere.

Are these real issues?  Should I really be concerned?  I honestly haven’t decided yet.  But I have a renewed interest in finding out what I believe.

At Wayland, we’ve had a couple of guests to our campus within the last two weeks, both of whom are founders of the organization called Care of Creation.  It is their goal to make Christians aware environmental issues and to promote their involvement in helping to become stewards of the environment.  Also, at Wayland we have begun an Environmental Stewardship Bible Study in which several faculty and staff members are participating.  In preparation for participating in the Bible Study, for attending the talks by our guests, and for completing a two-day workshop on environmental stewardship, I’ve begun doing a lot of reading in this area.

I have chosen to use my blog to document my journey down this road.  I may very well end up where I started but most certainly, by the end, I will have made some decisions and will have reached place where I can reasonably defend my positions.  Here is a brief survey of some of the questions that have arisen in my thought process about which I hope to reach a conclusion:

• Does the Bible provide a foundation for environmental concern within the believer?
• There is clearly a call to exercise dominion over creation and a reasonable case can be made that we should not abuse and destroy God’s creation.  Where is the line between cultivation and abuse?
• Free markets are amazing for the development of new ideas to deal with societal and even environmental problems.  But they seem to be inevitably entangled with the problem of greed.  Free markets have polluted the environment and lead to the deposition of toxic chemicals into drinking water and caused major illnesses because of their disregard for people and their ultimate regard for the bottom line.  Is the free market the best and sole solution to potential environmental crises?
• Is the alleged environmental crisis important enough to be a central ministry of the church or do we need "to keep the main thing, the main thing"?

Here’s something I already know:  We are called to be stewards of many things as followers of Christ: stewards of our lives, stewards of our talents and gifts, stewards of our families, and stewards of our finances. God is creator and this world was deemed "good" by Him.  We should take care of this Earth, a gift that He gave us to live on.  The resources of this world are a gift and we are to be stewards of the Earth just as much as of our lives, family, finances and homes.

There is much, much more to come.

I have a book on my shelf that I’ve probably had for close 15 years.  Every once in a while I pull it off the shelf and tell myself that I should read it and try it out.  The book is called “List  Your Self: Listmaking as the way to Self-Discovery”.  The subtitle says that it is “A provocative, Probing and Personal Expedition Into Your Mind, Heart, and Soul”

Seems to me that this is not only great fodder for a blog but also for a social network like facebook.  While many meme’s that go around annoy me to no end, there have been a few that have allowed me to know a little more about my online friends (mostly friends from days gone by).  That has to be one of my favorite things about Facebook and Twitter, getting to reconnect with old friends and keeping up with all my current ones.

So here it goes, List #1

List the Biggest Turning Points in Your Life

• 1985: In fourth grade, I was recruited to participate in a UIL contest called Number Sense.  This definitely started me down a path toward becoming a mathematician and an educator.
• 1985: Also the year that I accepted Jesus Christ as my personal Lord and Savior.  Life would be totally different without Him.
• 1994: Summer I traveled to Bakersfield, CA, to work as a summer missionary.  Met Si Budagher and made a lot of good friends, a few of whom would eventually come to Wayland.
• 1994: Senior year in high school when I decided to go to Wayland for college.  That was a major life decision considering I met my wife there and now plan on teaching there for as long as they’ll let me.
• 1995: After a year as a Religion major, I missed math too much and decided to add mathematics as a major.
• 1995: Also the year I met the most intriguing, hilarious, intelligent, witty, and beautiful woman that I have ever laid eyes on.  I took only a couple of months to learn that I would have to spend the rest of my life getting to know her.
• 1998: Decided to go to Texas Tech to get my graduate degrees in Mathematics.  I decided very early that I wanted to teach undergraduate mathematics.
• 1998: Marriage.  Almost nothing turns your life around more than this.
• 2000: Children. NOTHING turns your life around more than this.  Emily came in August and being a father has become the most fulfilling and challenging job of my life.
• 2000: Wayland hired me as a Mathematics Instructor. 
• 2003: Along comes Timothy and we now have both a boy and a girl.  The family is seemingly complete.  It is the family that both Lori and I had always planned on.  We’re most likely done having kids at this point.
• 2004: Along comes Zachary and God reminds us that his plans are better than ours.  Life is different and better than we ever imagined.  Our family is just not complete without the life of the party, Mr. Z.
• 2005: Completed the Ph.D.  My formal schooling after 24 years is finally over.  Hard to believe.
• 2007: With all my life goals seemingly met: family, Ph.D., teaching at WBU, I get restless and wonder if I shouldn’t try full-time research for a while.  So I do and after 6 months in a bioinformatics post-doc with Dr. Wilkins in Lubbock, I know that I’m cut out to be an educator.  Research will have to become my hobby.
• 2008: After commuting for a term, we moved back to Plainview for financial reasons.  It’s tough at first but eventually, it becomes clear that this was the right decision.
• 2009: We buy the house of our dreams.  Having saved up and having Lori working full-time (without sending any of the kids to day-care) we are in the perfect place to buy our dream house. 
• 2009: I have been promoted to Associate Dean and begin in August in a new role as an administrator.  Still waiting to see what becomes of this turning point.

That’s a lot of turning points. Not all of them were major changes but most were definitely course corrections in life.

I’m curious, from both the readers of the blog and from my friends on facebook, what are some of your major turning points?

Last week, a friend on Facebook pointed me to a story that seemed like it would be of interest to mathematician like myself but upon reading the story, there were a number of quirky details in the story and some important details missing.  It made me suspicious.

STOCKHOLM (AFP) – A 16-year-old Iraqi immigrant living in Sweden has cracked a maths puzzle that has stumped experts for more than 300 years, Swedish media reported on Thursday.

In just four months, Mohamed Altoumaimi has found a formula to explain and simplify the so-called Bernoulli numbers, a sequence of calculations named after the 17th century Swiss mathematician Jacob Bernoulli, the Dagens Nyheter daily said.

See the rest of the news story here.

Now, I’m no number theorist but I did take a course in Special Functions where the topic of Bernoulli numbers came up.  I was not aware of any 300 year old problem so I did some hunting and couldn’t find out what problem had been solved.  I also couldn’t find any mathematical news source citing the major development.

Thanks to Keith Devlin, we have a little more insight to the story.  In his MAA article this week, he wrote,

So I dug around on the Web for more details. There were a lot of news stories about the topic, but they all said more or less the same as the article I had already seen. Eventually, however, I found a Swedish news Website with an English-language story that was close to the source (Uppsala University).

"Swedish teen tackles centuries-old numbers challenge" was the headline. The story began, "A 16-year-old Iraqi immigrant in central Sweden has single-handedly figured out a formula with Bernoulli numbers that is normally reserved for much more seasoned mathematicians, earning him praise from professors at prestigious Uppsala University." Ah. Much more believable.

The reporter went on to explain that Altoumaimi, the young high school pupil, had developed some equations involving the Bernoulli numbers. When his school math teachers were unable to tell him whether what he had done was correct, the student contacted a professor at Uppsala University, who, after examining his work, declared that it was indeed correct. Not new, however. As the story continued,

"While it’s not the first time that someone has shown such Bernoulli number relationships, it’s highly unusual for a first year high school student to make his way through the complicated calculations, according to Uppsala University senior maths lecturer Lars-ke Lindahl."

I feel better knowing the truth behind the story.  Devlin was able to find the source I couldn’t find in my own digging.  Thanks, Keith.

The four numbers A, B, A+B and A-B are all prime.  The sum of these four numbers is

A) Even
B) Divisible by 3
C) Divisible by 5
D) Divisible by 7
E) Prime

Source: 2002 AMC 10/12B #15

I was having a problem when using Camtasia Studio to do a screen capture of an algebra lecture.  On my laptop, the capture works just fine but on my desktop it was very jumpy.  For those who don’t know, I use a Wacom Tablet, Microsoft OneNote and Camtasia to produce a series of videos for our online Algebra sequence. 

Whenever I would begin recording, the system would slow down enough that the writing on the screen was broken and hard to read.  The sound capture was fine, it just seemed that the CPU was not able to keep up with capturing the video on the screen and allow me to write smoothly.  The laptop, where it works fine, is a faster processor but with the same amount of memory.  I’m not certain how the video adapters compare.

I first discovered the problem several months ago and had been switching back and forth ever since.  However, today I took initiative and attempt to solve the problem one more time and came across a tip I had not considered. 

The Solution that worked for me:  Reduce the color depth from 32 bit to 16 bit.  For the types of videos I am doing, that makes makes no discernible difference and now it is as smooth on my desktop as it is on my laptop.

Some other tips for increasing the capture rate were found here: http://tinyurl.com/arua4b

Here’s a short sample: