Proof by Photo: Please see Figures 1 and 2. (Okay, a few examples don’t constitute a proof, but there photos get close.)

Figure 1--Mr Yellow M&M speaking to Las Vegas tourists as they pass by. I snapped this photo as I walked along the LV Strip.

Figure 2--The famous Las Vegas Welcome Logarithm sign on South Las Vegas Blvd.

In community college mathematics classes, logarithms are typically introduced in an intermediate algebra class just after the topics of exponential functions and inverse functions. They arise as a solution to the inverse function of the exponential function. A common introduction goes like this.

Problem: Find the inverse of , where .

Solution: For convenience, replace with .

becomes

Interchange and . (Standard method of finding an inverse function.)

becomes

Solve for .

At this point in intermediate algebra, we have no way of doing this, of solving for .

We wish to get an expression for but we have no algebraic method to do so.

We know an expression for exists because the original exponential function, , is one-to-one. See Figure 3.

Figure 3--The exponential function is always increasing so it is one-to-one. It therefore has a inverse.

Not only do we know the exponential function’s inverse exists, we know what it looks like. Graphs of inverse functions can be produced by revolving the graph of the original function about the line . See figure 4.

Figure 4--The graph of the inverse of the exponential function.

One-to-one functions have inverses. Whether we can find them or not is a different story. Sometimes we can and sometimes we can’t.

Since we have no algebraic way of solving for , we devise an expression for . We devise the logarithm. But we will devise the expression in a visually descriptive way.

In the expression , we define to be .

In words, we say is the logarithm, base , of the number .

The letters “log” represent an abbreviation for the word “logarithm.”

So even though we don’t have an algebraic way for solving for , we are going to define to be .

Let’s look carefully at to see exactly what it means.

In , there are 3 letters, all of which represent numbers.

We know what the value of is. It is the number raised to the number .

What we are saying now is that the value of is the number of the number .

But what is ? What is a logarithm?

Logarithms are Exponents

In the expression , is the exponent. The expression we devised for represents an exponent.

We defined as a logarithm. So logarithms are exponents.

The logarithm notation is visually descriptive.

In the expression , the base of is the number . Look where is placed in . It is written lower than all the other letters. It is sort of in the basement.

Logarithms are exponents. Exponents reside on bases. (For example, in the expression , the exponent 3 resides on the base 5.)

How to think about

When you see , think “this thing is an exponent.” When you see , you are looking at an exponent.

Ask yourself: What exponent would I put in the number to get the number ?

Let’s look at some examples.

Some Examples

1. What number does represent?

Think: The expression is an exponent. Exponents go on bases.

The base here is 2 (the lowered number).

Ask: What exponent would I put on 2 to get 8? By our experience with exponents, we know this number to be 3.

Thus, . It is true that .

2. What number does represent?

Think: The expression is an exponent. Exponents go on bases.

The base here is 10.

Ask: What exponent would I put on 10 to get 100? By our experience with exponents, we know this number to be 2.

Thus, . It is true that .

3. What number does represent?

Think: The expression is an exponent. Exponents go on bases.

The base here is 10.

Ask: What exponent would I put on 10 to get 75? Our experience with exponents doesn’t tell us. We don’t know. In such cases, we either leave the expression as it is, or we use calculator with the Log on it.

Doing so with my TI 84 calculator, I get that

Thus, , to 4 decimal places.

This means that . We’ll say approximately equal because we rounded to calculator value to just 4 decimal places.

Listening to Lisa talk about people’s diverse views of Heaven I got to wondering a few things about Heaven. I wondered, Where was Heaven? How far was Heaven from my house? What do people do in Heaven? What do the souls in Heaven look like and what do they do there? What is the average temperature in the Heaven? The last question, the temperature question, involved a measurement and was, I thought, the only one to which I might be able to find an answer. (It turns out I could not, but I did come up with an answer to a related, still very interesting question.) Being familiar (and only vaguely so) with the Christian religion, I narrowed my question to be What is the average temperature in the Christian Heaven?

The Average Temperature of Heaven

In my searches through Bible literature I could not find a direct answer to this question, but along the way to the answer I decided, through some straight-forward computations, that the temperature on Earth could be higher than the temperature in Hell and people would still be healthy, happy, and prospering.

Earth – Hotter than Hell

If you believe in the Christian concepts of Heaven and Hell, then proof of this fantastic statement can be found in the Bible. Here is how it goes.

We’ll use three facts, two as stated in The Bible, and one from the well-known law from Earthly physics, the The Stefan–Boltzmann law.

Bible Facts

We’ll take the two Bible facts from two sources, the King James Version and the New International Version.

Here are the verses from both versions followed by an interpretation of those verses. (found at COLLECTIONS Commentaries, Word Studies, Devotionals, Sermons, Illustrations Old and New Testament).

Isaiah 30:23-26 can be interpreted to tell us how much light will be on Earth when Christ establishes His Millennial Reign upon the earth.

Isaiah 30:23-26, The King James Version

The Book of Isaiah

[23] Then shall he give the rain of thy seed, that thou shalt sow the ground withal; and bread of the increase of the earth, and it shall be fat and plenteous: in that day shall thy cattle feed in large pastures. [24] The oxen likewise and the young asses that ear the ground shall eat clean provender, which hath been winnowed with the shovel and with the fan. [25] And there shall be upon every high mountain, and upon every high hill, rivers and streams of waters in the day of the great slaughter, when the towers fall. [26] Moreover the light of the moon shall be as the light of the sun, and the light of the sun shall be sevenfold, as the light of seven days, in the day that the LORD bindeth up the breach of his people, and healeth the stroke of their wound.

Isaiah 30:23-26, the New International Version

[23] He will also send you rain for the seed you sow in the ground, and the food that comes from the land will be rich and plentiful. In that day your cattle will graze in broad meadows. [24] The oxen and donkeys that work the soil will eat fodder and mash, spread out with fork and shovel. [25] In the day of great slaughter, when the towers fall, streams of water will flow on every high mountain and every lofty hill. [26] The moon will shine like the sun, and the sunlight will be seven times brighter, like the light of seven full days, when the LORD binds up the bruises of his people and heals the wounds he inflicted.

An Interpretation We see that the curse on the earth and over Jerusalem is being removed by the Lord because: The rain is falling and causing good growth of the crops. The yield of the ground is rich and plenteous. The livestock are grazing in a roomy pasture. Every loft mountain and high hill has streams running with water. The wicked have been slaughtered and removed from the earth. The light of the moon is as the light of the sun, and the light of the sun is seven times brighter than normal.The fulfillment of these promises concerning the land of Israel has in preparation of Christ’s Millennial Reign already been occurring. When Israel came into the land in 1948 and was declared a nation again, the land was deserted and considered completely worthless and uninhabitable, and no one had wanted the land. Today, however the Lord has blessed the land and it produces much agriculture and fruits, and much of it has even been reforested. This is the first desert land upon the earth which has ever been refurbished, everywhere else in the world desert has always been constantly advancing.

It looks like things will be pretty nice on Earth. Except, as announced, but hidden, in Verse 26, it will be hotter on Earth than it is in Hell.

The Temperature on Earth as Given by Isaiah 30:26, the Mathematics

Our clue as to the temperature of Earth is in verse 26, The moon will shine like the sun, and the sunlight will be seven times brighter, like the light of seven full days…

Light is electromagnetic radiation. According to verse 26, the amount of radiation received by the Earth is (will be)

7 times 7 = 49 times as much radiation as that contributed by the moon.

So, the Earth will receive 1 (moon) + 49 (suns) = 50 times its current amount of radiation.

Now we’ll employ the Stefan-Boltzman Law. This law expresses, mathematically, the relationship between amount of radiation and temperature.

The Stefan-Boltzman Law states that the energy radiated by a radiator (the Earth in our case) per second per unit area is proportional to the fourth power of the body’s absolute temperature. Using

to represent the new temperature of the Earth,

to represent the original temperature of the Earth,

we get

We can solve this equation to obtain the new temperature of the Earth. We will solve for .

So, the new temperature of the Earth will be times the original temperature of the Earth. The temperature of the Earth has been established to be K.

Then,

The new temperature of Earth is, or will be, K. This Kelvin temperature translates to C and F. Geez! F is really hot.

The Temperature of Hell

We cannot get the temperature of Hell as precisely as we did the newer temperature of the Earth. But we can get an interval estimate for it. The interval comes from information contained in The Book of Revelations.

Revelations 21:8 tells us
But the fearful, and unbelieving, and the abominable, and murderers, and whoremongers, and sorcerers, and idolaters, and all liars, shall have their part in the lake which burneth with fire and brimstone: which is the second death.

Revelation 21:8 (New International Version)

The Book of Revelation

But the cowardly, the unbelieving, the vile, the murderers, the sexually immoral, those who practice magic arts, the idolaters and all liars—their place will be in the fiery lake of burning sulfur. This is the second death.”

Brimstone is another name for sulfur. According to the New World Encyclopedia, sulfur melts at 239.38 °F (388.36 K, 115.21 °C). This means that the temperature of Hell must be at least 239.38 °F. Also, the boiling point of sulfur is 832.3 °F (717.8 K, 444.6 °C). At temperatures higher than 832.3 °F, sulfur boils and turns to a gas and is no longer sulfur. This means that the maximum temperature of Hell is 832.3 °F .

Now we have that the newer temperature of Earth is F and the maximum temperature of Hell is 832.3 °F.

So there it is, expressed in the Bible — Earth will be hotter than Hell.

Since people have lived on the Earth, it has never been F. Because the current temperature of Earth is not 876 °F, we can conclude that the curse on the earth and over Jerusalem has not been removed by the Lord. Otherwise, we have a contradiction. So it must be sometime in the future that the curse on the earth and over Jerusalem will be removed by the Lord.

But even at F, Isaiah 30:25-25 states that all people will be healthy and prospering.

So now I wonder if the bodies of people suddenly change to tolerate this new, high temperature, or if they will evolve to tolerate it.

I wonder, too, about the temperature of Heaven. I think Heaven must be cooler than Hell, else people would burn there. So maybe an upper bound on the temperature of Heaven is 832.3 °F, the boiling point of brimstone.

Or, maybe Isaiah got the whole number thing wrong.

Or, maybe I got the numbers wrong. It can happen. One time in a calculus II class, I presented a very nice derivation of the the coordinates of the center of mass of a thin sheet, but then realized I got them exactly backwards. . Geez. I am glad I am not in charge of temperatures.

I found quite a few articles on the internet relating the temperatures of Heaven and Hell. All that I found conclude that Heaven is Hotter than Hell. They do so by interpreting Isaiah 30:26 as giving the amount of radiation contained in Heaven. I think that the previous lines, 23-25, tell us that the amount of radiation is the amount received on the Earth, not the amount contained in Heaven. That’s my take, anyway.

I called my Polish friends from my college’s mathematics department and asked them to meet for breakfast at MiMi’s, a nice local Polish Cafe near the school, to discuss this wonderful event.

I made a short video to capture our joy. In the video,

1. Jim Vilchuck appears on the left. Jim is real smart guy who likes to talk in detail about both the coefficient of determination and the coefficient of variation. Start any discussion with him and within 5 minutes he’ll have you asking about . You can’t help yourself. Soon you will find yourself asking Jim about other coefficients. He has a remarkable talent.

2. Sven Svenkoski appears in the center of the picture. Sven, at the age of 10, while practicing some topological probability homework problems, discovered a submarine lost for 30 years some 80,000 ft down on the Atlantic Ocean floor. Sven’s discovery came just in time as the crew of 150 sailors was just about out of oxygen. When we started discussing Copernicium, Sven suggested the heaviest element was being named after Copernicus because it was Copernicus who wrote the 1969 hit song, He Ain’t Heavy, He’s my Brother. Sven has an Erdos Number of 1!

3. Craig Allenski appears on the right in the picture. Craig is one of the smartest people in the department. I remember going to him when I had questions about how to derive and when presenting the topic Center of Mass in calculus. For some reason I still don’t understand, one morning Craig beat me up in the school parking lot.

See the guys discussing Copernicium in this video clip. The guys discussing Copernicium.

We had a nice breakfast, spent about 2 hours talking about mathematics, then went our own ways.

It was raining pretty hard the morning of our breakfast and as I hurried to my car from the restaurant, I got pretty wet. I was still pretty wet when I got home some 10 minutes later. In the bathroom while blow drying my hair, I saw something that just astonished me. See Figure 1.

Figure 1 A bottle of Nutrium

Do you see this? Nutrium. An element I had never heard of. Not only had I not heard of it, but the Dove company was using it in soap! Now I am pretty good at keeping up with the Periodic Table, but somehow I had completely missed the discovery of Nutrium. But there it was with its electron configuration right there on the bottle of Dove soap. See the Periodic Table picture.

The Periodic Table

My house is pretty much like most people’s house, I have the standard living room, kitchen, bedrooms, bathrooms, and hallways leading to other rooms likes laboratories and engine rooms. See Figure 2.

Figure 2 Main hallway in Denny's house

I put the slide containing the Nutrium under the microscope to get a visual of this element. See Figure 3.

Figure 3 Soap sample

An Astonishing Discovery

Among all the other particles in the sample, I was able to detect the Nutrium atom. Figure 4 shows an individual Nutrium atom.

Figure 4 The Nutrium atom

I zoomed into one of outer electrons. Figure 5 shows the zoom magnification at 10 kabillion times the atom’s original size. Look at the astounding electron at the top of the image! Do you see it? That electron looks like a guy in a 60’s surf music rock band! What the heck, I shouted!

Figure 5 Nutrium atom with the Ellis electron characteristic

I connected the microscope to speakers so as to get an audio of the Nutrium atom.

Speakers connecting to the microscope

The Nutrium Atom – See it and Hear it Here

What a surprise! You have got to see and hear this element. Be ready to adjust the volume on your computer’s speakers.

Here it is, The sight and sound of Nutrium

So how long, Don T asked me, will it take until my range is back to 100%?

Assuming that he, that would be Don T, was down to 20% his full range of motion, we calculated as follows.

We let represent Don’s full range of motion.

Then, 20% of full range can be represented by . Its true. Think about it this way and just write what you say.

20% of current range.
The word of translates into mathematics as times .
Since the current range is , 20% of is represented as , or just .

You know, I think I will use the star symbol rather than the multiplication dot to indicate multiplication. With decimal points and multiplication dots, there are too many dots.

We called the day of injury Day 0.
The amount of motion on Day 0 is .

Now, we will let represent the total amount of motion at the end of Day .

Then, on Day 0, the total amount of motion is

We called the first day following Day 0, Day 1.
A 1% increase in range is represented by .

Since the current range is , a 1% increase is represented as .

Then, the amount of motion at the end of Day 1 is the range of motion on Day 0 plus the 1% increase in that range. That is, the total amount of motion at the end of Day 1 is

.

We can simplify a little by factoring out the that is common to the two terms on the right side of the formula. Doing so gives us

We can use exponents to related the day number and the total amount of motion at the end of that day.

At the end of Day 0, the total amount of motion .

At the end of Day 1, the total amount of motion .

We called the first day following Day 1, Day 2.
A 1% increase in range is represented by .

Since the amount of range at the end of Day 1 is , a 1% increase is represented as .

Then, the amount of motion at the end of Day 2 is the range of motion at the end of Day 1 plus the 1% increase in that range. That is, the total amount of motion at the end of Day 2 is

.

We can simplify a little by factoring out the that is common to the two terms on the right side of the formula. Doing so gives us

You may be able to see the general form, but if not, we’ll make one more iteration.

At the end of Day 0, the total amount of motion .

At the end of Day 1, the total amount of motion .

At the end of Day 2, the total amount of motion .

We called the first day following Day 2, Day 3.
A 1% increase in range is represented by .

Since the amount of range at the end of Day 2 is , a 1% increase is represented as .

Then, the amount of motion at the end of Day 3 is the range of motion at the end of Day 2 plus the 1% increase in that range. That is, the total amount of motion at the end of Day 3 is

.

We can simplify a little by factoring out the that is common to the two terms on the right side of the formula. Doing so gives us

Now we have it. We can see the general form.

The total amount of motion at the end of Day is given by

We can use this formula to determine how many days it will take to reach 100% range.

100% of the total range is represented by , or .

We want to find the number of days, , it takes to get to .

Set equal to in the formula and solve for .

Divide both sides by .

The ’s divide out (cancel), and we are left with

Divide both sides by 0.20.

Which gives us

To solve for we will take the natural log of each side.

In decimal form, days.

Now, Donut Head’s doctor told him his wrist would take about 60 days to get back to 100%. So, that constant 1% increase in range of motion must be incorrect or he started with more than 20% his original range. Let’s assume he really did start with 20% his original range.

Then the problem is, given 100% range of motion in 60 days, that is, given , find the rate at which the range of motion increases. Can you do it?

Can you solve for ?

You don’t need logs.

I got

Since I posted that article, I have been thinking about connections between mathematics and music. I started wondering if songs might have equations associated with them and if so, what the graphs of those equations would look like. The thought didn’t seem that strange to me since sheet music looks somewhat graphical. As you look at a song’s sheet music, dots rise and fall, just like one sees when looking at the graph of discrete data points.

Because my niece Randi Jean loves Beatles music and because she can exert great influence on cosmic events, I decided to try to graph a Beatles song. I bought a book of Beatles sheet music and decided to see if I could create a graph of the song I’ll Follow the Sun.

Randi Jean listening intensely to a Beatles tune

You can see a youtube version of I’ll Follow the Sun as performed by Paul McCartney at

http://www.youtube.com/watch?v=LfO1nbCX0g8&feature=related

The Process

1. I copied the music for the song on my printer.

2. The song is in 4/4 time and the shortest note in the song is an 1/8 note. So, I decided to count 8 beats to each measure.

3. From wikipedia I copied a nice picture of notes so that I would have an idea of each note’s frequency value. See Figure 1.

http://en.wikipedia.org/wiki/Musical_notes

Figure 1 - Note frequency vs note name

4. Using Excel, I made a list of the notes and their corresponding frequencies. See Figure 2. Click on the picture to see it larger.

Figure 2 - Notes and their frequencies

5. Next to each note, I wrote in red ink the note’s frequency in Hertz (Hz).

6. Then next to each note I wrote in blue ink the number of beats the note required to be played.

Figures 3 and 4 show the a cropped version of the sheet music and my notation. Click on the picture to see it larger.

Figure 3 - Verse 1

Figure 4 - Refrain

The following steps show how I created the lists of data, the scatter diagram, the coordinate system, the piecewise function, and the graph of the data using the T-Nspire. They are just a how-to-do-this type of thing. If you want to skip it, scroll down to the The Graph.

7. I accessed my copy of the new TI-Nspire software that emulates the Texas Instrument’s TI–Nspire calculator.

8. Using Nspire, I first created a list of values. The first column in my list is the beat number, and the second column, the note frequency. See Video 1.

Video 1

9. Then, still using Nspire, I created a scatter diagram of my data. See Video 2.

video 2

10. Still using the Nspire, I created a piecewise linear function of the data in the beat vs Hz lists. See Video03.

Video 3

11. Video04 shows a summary of steps 8-10.

Video 4

12. Video 5 shows a minor change in the horizontal axis, one that makes the graph look a little better.

Video 5

13. The following four figures shows the four piecewise functions I constructed to create the graph.

Function, 1 produces verse 1.

Function 2 produces verse 2.

Function 3 produces the first part of the refrain, and

Function 4 produces the last part of the refrain.

The Graph

Figure 5 shows a black and white version of the graph of I’ll Follow the Sun when plotted as a piecewise function.

Figure 5

Figure 6

I don’t know yet how to label axes on the Nspire. I will work on that.

I think what I will try next is to graph another Beatles song, or even a few more Beatles songs of approximately the same size, (number of measures) and compare them to each other. I am not sure just yet how I will compare them, but my colleague Joe suggested I use a subtraction method to see how various differences compare with each other.

December squared = (the 16th) times (09), or better yet,

I think this is a fun relationship and will give such dates the name Sari Dates. So, of course, we need to know how many Sari Dates there are.

I counted 40 of them. I listed each month and just counted. Here is how.

In general we want to count the number of occurrences of the event

Let’s narrow it down to this century.

I am just going to make a straight count.

Let = the month number
= day number
= year number

So, we are looking for how many times between 2000 and 2099 that

Not all numbers work. There are some restrictions on the values , , and .

Because represents a month, it must be between and including 1 and 12. That is,

Because represents a day, it must be between and including 1 and 28, or 1 and 29, or 1 and 30, or 1 and 31.

Because represents a year in this century, it must be between and including 0 (2000) and 99 (2099).

To make the count, start with the square of the month then find all the factors of that number that meet the

Let’s count.

January:

The only factors of 1 are 1 and 1. So,

Then for January, we get only one date, 01/01/01

February:

The factors of 4 are 4,1; 2,2. So

Then for February, we get 3 dates: 02/02/01, 02/02/02, and 01/01/04.

March:

The factors of 9 are 9,1; 3,3. So

Then for March, we get 3 dates: 03/09/01, 03/03/03, and 01/01/09.

April:

The factors of 16 are 16,1; 8,2; 4,4. So

Then for April, we get 5 dates: 05/16/01, 04/08/02, 04/04/04, 04/02/08, and 04/01/16.

May:

The factors of 25 are 25,1; 5,5. So

But this cannot happen because we can’t have a day of 25!

Then for May, we get 2 dates: 05/05/05, and 05/01/25.

June:

The factors of 36 are 36,1; 18,2; 9,4; 3,12. So

But this cannot happen because we can’t have a day of 36!

Then for June, we get 6 dates: 06/18/02, 06/09/04, 06/04/09, 06/03/12, 06/02/18, and 06/01/36.

July:

The factors of 49 are 49,1; 7,7. So

But this cannot happen because we can’t have a day of 49!

B

Then for July, we get 6 dates: 06/18/02, 06/09/04, 06/04/09, 06/03/12, 06/02/18, and 06/01/36.

August:

The factors of 64 are 64,1; 32,2; 16,4, 8,8. So

But this cannot happen since we can’t have a day of 64!

But this cannot happen since we can’t have a day of 32!

Then for August, we get 5 dates: 08/16/04, 08/08/08, 08/04/16, 08/02/32, 08/01/64.

September:

The factors of 81 are 81,1; 9,9. So

But this cannot happen since we can’t have a day of 81!

Then for September, we get 2 dates: 09/09/09 and 09/01/64.

October:

The factors of 100 are 100,1; 50,2; 25,4; 20,5; 10,10. So

But this cannot happen since we can’t have a day of 100!

But this cannot happen since we can’t have a year of 100! That you would be the year 3000.

But this cannot happen since we can’t have a day of 50!

Then for October, we get 6 dates: 10/25/04, 10/20/05, 10/10/10, 10/05/20, 10/04/25, and 10/02/50.

November:

The factors of 121 are 121,1; 11,11. So

But this cannot happen since we can’t have a day of 121!

But this cannot happen since we can’t have a year of 121!

Then for November, we get 1 date: 111/11/11.

December:

The factors of 144 are 144,1; 72,2; 48,3; 36,4; 24,6; 18,8; 16,9; and 12,12. So

But this cannot happen since we can’t have a day of 144!

But this cannot happen since we can’t have a year of 144!

But this cannot happen since we can’t have a day of 72!

But this cannot happen since we can’t have a day of 48!

But this cannot happen since we can’t have a day of 36!

Then for December, we get 7 dates: 12/24/06, 12/16/09, 12/18/08, 12/12/12, 12/09/16, 12/06/24, and 12/03/48.

To summarize, letting SD stand for Sari Date, I counted:

January: 1 SD
February: 3 SDs
March: 3 SDs
April: 5 SDs
May: 2 SDs
June: 6 SDs
July: 2 SDs
August: 5 SDs
September: 2 SDs
October: 3 SDs
November: 1 SD
December: 7 SDs

Then,

Thank you very much. This century, I count 40 Sari Dates.

Figure 1 Cousin Dave's Fashion

Figure 2 SJ to Irvine

Okay, so I am expecting to drive 380 miles. As I get ready to leave my house, I check my gauges to ensure all is well with my Jeep and the Universe. See Figure 3.

Figure 3 - Odometer reads 000

The gauges show that all is well.
Gas: Full. Good
Battery: Charging. Good
RPMs: Since I am stopped they are 0. Good
Speed: Since I am stopped it is 0. Good
Oil: Got positive pressure. Good
Water temp: Since I just started the Jeep, its low. Good
Universe Condition” Now. Good

Off I go.

I drive nonstop for 220 miles down Interstate 5 to Hwy 58 and make a stop for gas at the Buttonwillow gas station zone. See Figure 4.

Figure 4 - Map of SJ to Buttonwillow

About a mile before the off ramp I check my gauges and something is not quite right. See Figure 5.

Figure 5 - Gauges in Buttonwillow station

My Universe gauge reads above “Now!” Why? What could it mean? I think about it as I cruise to the off ramp, then it comes to me. I have been traveling at 40 mph for 220 miles. When I got on Hwy 5, I thought the speed limit sign said “Maximum Speed 40.” Apparently I misread the sign as the max speed is actually 70. Now I understand why so many cars were blowing by me with occupants waving at me. I didn’t really think I knew that many people. Checking the Universe chapter in the 2009 edition of the Physicists Handbook, I see that if one travels too slow for too long, basic physical laws are twisted, affecting the age of space/time. Geez! I feel lucky to have gotten to the Buttonwillow stations when I did.

Now with a full tank of gas and a clear knowledge of appropriate speeds, I am back on Hwy 5 headed to Irvine through LA.

I arrive at destination house Point B. I pull the Jeep up in front of the house and, as I always do, check my gauges. Everything looks good. I check my mileage, expecting to see 380. But the odometer reads 378. Okay. That’s okay. The difference is negligible and I can account for the 2-mile difference by believing that my odometer is not calibrated exactly with the MapQuest satellite.

Then I realize I am experiencing a surprising event. My odometer reads 378 and the house I am visiting has street number 378! For proof, see Figures 6 and 6.5.

[

Figure 6 - Odometer reading in Irvine is 378

Figure 6.5 - House with street number 378

What is the likelihood of that happening? I don’t know, but I liked it and I went to work determining that probability. At first I thought the probability of a match between my odometer reading and the house number would be very small. But maybe it is not. With a good suggestion from my friend and colleague Jim Vilchuck, I thought as follows.

I know by MapQuest and experience that destination Point B is about 50 miles south of LA, making it somewhere between 350 and 400 miles from San Jose. I know my odometer reading will be somewhere in this range. The sample space then consists of the 51 numbers between and including 350 and 400. The probability of randomly selecting one of them is 1/51. So, I could reason that that probability of my odometer number matching the house number is 1/51. So a match should happen about 1 time about 51 visits. The mileage is likely to vary each trip as I make different stops/turns along the way.

But Irvine is closer to 50 miles south of LA than 1 mile south. So maybe the probability of a match is even higher than 1/51. We could say that destination Point B is between 375 and 385 miles. This makes a sample space of 11 numbers and gives a probability of a match of 1/11. That is a surprise result, at least to me.

But now here is an even more surprising event. After being in Irvine for 2 weeks, I make a visit about 14 miles away to a location in Newport Beach. After a nice visit, I leave Newport Beach to drive back to San Jose. See Figure 7.

Figure 7 - Newport Beach to SJ

Seven hours after leaving NB, I arrive home in SJ. I pull into my garage and check my odometer. It reads 377.6. Surprise! See Figure 8.

Figure 8 - Odometer reads 377

Even though I left Southern California from a location different from the Irvine location, the travel distance was still 378 miles. Now it looks likes the number 377 is not the same as the number 378. After all, the last digit of the first number is 7 and the last digit of the second number is 8. But the two numbers are actually the same, 378. Using the well-known Craig Allen Theorem of Equal Numbers, 7 = 8, for large values of 7. Hence, 377 = 378.

Proof:

Now, of course the question is what is the probability that the Jeep travels 378 miles from San Jose to Irvine and matches the house’s street number of 378 and then travels back to SJ from a different starting point and also travels 378 miles? I don’t know. But right now I am going to Jack-in-the-Box to get tacos and a large diet coke. Maybe I’ll work on this problem later.

Figure 1 Las Vegas to San Jose

In the last few months I have had a few problems (about 8K bucks worth) with my Jeep, so when I drive it now, I keep my ears open for sounds that might indicate approaching danger and my eyes on the Jeep’s instrument panel gauges for indications of imminent vehicle
failure and/or the end of the universe. See Figure 2.

Figure 2 Jeep gauge panel

Just this last August, 2009, I left the Flamingo Hotel in Las Vegas at 2:30 AM to make the drive back to San Jose. About 50 miles out and in the desert, I started getting a clanking sound at about 70 mph. I slowed down to 65 mph and the clanking stopped. But then it started again at 65 mph. I slowed to 60 and it stopped. Then it started again at 60 mph. This was not good, I thought. I am in the middle of desert at night and my car is going to break down and strand me. I will die a horrible death, my bones being picked clean by buzzards near Zyzzx Road. I don’t know much about scavenger type birds, but this is what I imagined. See Figure 3.

Figure 3 Zzyzx Road near Baker, CA

I get chills just thinking about it even now.

My Jeep did not break down and I was able to limp home without incident. It turns out the problem was in my transmission. I forget now what the part was called, but I do remember it cost me 22-hundred bucks to get it and install it.

Staring just this last July, 2009, the “check engine” light on my Jeep’s instrument panel began illuminating. It first illuminated when I was in Los Angeles putting gas in my car to start the 265 mile drive to Las Vegas. Just what anyone would want to see, the check engine light. What light could be worse? Of course the check engine light must be Jeep’s way of politely saying, “impending and inescapable engine explosion.” See Figure 4.

Figure 4 Denny's Jeep exploding

I drove to Vegas anyway. I was nervous the entire trip, but I still went. Then, with the light still on, I drove around Vegas for a few day, then drove the 520 mile back to San Jose.

My mechanic checked for the cause of the light. His diagnostic machine announced the cause of the problem as, let’s just call it Mistaken Cause 1. He fixed it. Three hundred miles later, the check engine light pops back on. The diagnostic machine states the cause of the problem as, let’s call it Mistaken Cause 2. To make a long story shorter, the cause was not found until after $1200 worth of Jeep parts. The Jeep dealer finally correctly determine the cause to be an 80¢ fuse.

Now I am always nervous about the ability of this car to operate in a way that will not leave me stranded in the middle of the Mojave Desert in the 150 degree summer heat or the biting winter cold.

When I drive my Jeep now, I am constantly aware of noises and smells and clunks and jerks. I keep a close eye on my instrument panel gauges.

So Friday on my drive back to San Jose from Las Vegas, I am on Highway 58 passing the Desert Sage apartments near Edward’s Air Force Base and the town of Mojave, and I check my water temperature gauge. I expect to see the needle in the standard operating zone just slightly to the left of 210. But its not! Oh no, it is just slightly to the right of 210. It is never there, it is always just to the left of 210. I am sure my engine is overheating and ready to explode. See Figures 5 and 6.

Figure 5 The bad zone and the good zone

Figure 6 The Desert Sage Apartments near Mojave

To delay engine explosion, I back off on my speed from 65 to 60. Ahh, the temperature gauge needle moves to the left back into the standard operating zone. All is well. I speed back up to 65. The needle moves back to the right in the engine explosion zone. Oh no, I am 300 miles from home and my car is about to ignite into flames.

But now I have another problem. I am having trouble reading and making sense out of the water temperature gauge. Check out the increments on both the fuel and oil gauges. See Figure 7.

Figure 7 Fuel and Oil and Water temp gauges

I think they read as one would expect. Each increment is the same. That is, the scale is linear. In the case of the fuel gauge, each increment is 1/8 of a tank of gas. In the case of the oil gauge, each increment in 10 psi. Count them. The oil pressure starts at 0, then with markings at every 10 units, goes to 40 psi then to 80 psi. Forty psi is right in the middle between 0 and 80 psi, right where we would expect it to be. These scales are linear as they indicate constant change.

But look at the increments on the water temperature gauge. What the? The increment lines are evenly spaced, visually giving us the idea that the gauge is set up linearly. But look closely at the numbers. Visually 210 is halfway between 100 and 260.

If the scale is linear between 100 and 210, then each increment represents an increase of 27.5 degrees.

(210 – 100)/4 = 27.5

If the scale is linear between 210 and 260, then each increment represents an increase of 12.5 degrees.

(260 – 210)/4 = 12.5

So not each tick mark represent an integer value water temperature. The tick marks represent:

100
100 + 27.5 = 127.5
127.5 + 27.5 = 155
155 + 27.5 = 182.5
182.5 + 27.5 = 210
210 + 12.5 = 222.5
222.5 + 12.5 = 235
235 + 12.5 = 247.5
247.5 + 12.5 = 260

This looks like a crazy way to design a gauge to me. There must be some logic to this as I want to believe that Mr Jeep would not be joking around with me while I am in the middle of the desert.

Now I worried about high water temperatures and crazy gauges. I am in the middle of the desert and I am about to drive uphill into the Tehachapi mountains. I am sure my water temperature indicator needle will move even farther to the right, my car will overheat and my engine will explode, taking me out as well as all the cars around me.

It didn’t happen. I eased the Jeep up the mountain, the needle bounced back and forth between OK and Explode and I made it to the summit. Once there, I could drive without putting much of a load on the engine, keeping the water temperature in the safe zone. That went well.

When I got down the mountain and into Bakersfield, the needle stayed in the standard zone and all was well.

Driving north on Hwy 5, I tried to figure out why the needle moved to the right when I was in the desert. The outside air was cool and I was at a high elevation. Both these features I thought would keep the water temperature low. But if the Jeep really is running well and is not experiencing mechanical problems, there is something else going on. I am thinking about the gas laws and I will explore that this week.

The Harmonic Series in Music
Much of what I know about harmonic series in music comes from the many guitar lesson books I have read as well as from the information provided by Wikipedia at

http://en.wikipedia.org/wiki/Harmonic_series_(music)

Here is how I understand it. Strings on a guitar are fastened to two nodes, one called the nut, which is located at one end of the fretboard, and one called the bridge, which is located on the body of the guitar. A node is a fastening point at which little or no vibration takes place. See Figure 1.

Figure 1 Guitar nodes

When a guitar string is plucked, it oscillates simultaneously at many different frequencies.

The Fundamental and its Overtones

The lowest of the frequencies at which a string vibrates is called the fundamental tone. The fundamental tone is the tone produced as the string vibrates over the full length of the guitar, from the bridge to the nut. See Figure 2.

Figure 2 The fundamental tone

Each of the other frequencies is a whole number (the whole numbers are 0, 1, 2, 3, 4, 5, … ) multiples of the fundamental frequency. Symbolically, if we represent the frequency of the fundamental tone with the letter f, then the frequencies of the other overtones are represented by 2f, 3f, 4f, etc.

Check out Figures 3, 4, and 5 to see graphical representations of the first, second and third overtones.

Figure 3 The first overtone

Figure 4 The second overtone

Figure 5 The third overtone

There is wonderful demonstration of the fundamental tone and the first five overtones at Phil Tulga’s “Music Through the Curriculum” website

http://www.philtulga.com/harmonics.html

Harmonics
The terms harmonics and overtones are often used interchangeably.

The Harmonic Series in Music

Figure 6

In stringed instruments, such as the guitar, the harmonic series in music theory is the set of integer multiples of the harmonic.

Symbolically, we have

A frequency is harmonic if it is an integer multiple of the fundamental frequency. See Figure 7

Figure 7

For example, if the fundamental frequency of a plucked string is 100 Hz, then

Figure 8

The Harmonic Series in Music

In music theory, a harmonic series is the set consisting of the fundamental tone (fundamental frequency) and all its overtones. See Figure 9.

Figure 9

When taken all together, the human ear perceives the harmonic series as a single tone. I think of this as adding all the tones together produces a single tone.

The Wavelengths of the Harmonic Series
If you look back up to Figures 3, 4, and 5, you can see that

Figure 10

So, the ear perceives the sum of the fundamental and all the overtones as one single tone. If we think in terms of wavelengths we generate the sum illustrated below.

The Harmonic Series in Mathematics

The harmonic series in mathematics is illustrated below

This series diverges. That is, it does not sum to a finite value. As terms are added, the sum just gets bigger and bigger with no bound. Put another way, given any number at all, we can go far out enough in the series to obtain a number larger than the number we are given. Here is a nice proof that the harmonic series diverges. To believe the proof, you have to believe that

gets bigger and bigger and bigger, and never adds to a finite value. The proof shows that the harmonic series is even larger than the ½ series.

Proof by Comparison

Because the sum of the infinitely many ½ terms diverges to infinity and the harmonic series is larger than the ½ series, the harmonic series must also diverge to infinity.

Relating the Harmonic Series in Music and the Harmonic Series in Mathematics

Now, finally we get to it. The harmonic series in mathematics diverges. If we think in terms of wavelengths, the harmonic series in music is the same as the mathematics harmonic series. This means that the harmonic series in music also diverges.

But we noted that the ear perceives the harmonic series as a single tone. That seems to indicate that the series adds to a finite value. What! But we know it does not. I think what happens is that once we are out far enough in the series, the wavelengths are so small (say like 1/500, 1/1,000,000), that we do not hear the sounds they generate. We hear only the sounds that correspond to maybe the first few wavelengths. Theoretically, there are infinitely many tones. We hear only some. But practically, there are probably not infinitely many tones. The string and the guitar are physical objects and as such there are likely only a finite number of overtones. It would be hard to believe there could be infinitely many nodes along a guitar fretboard.

I think, too, that part of the problem I was having in relating the two series was one of terminology. In English as a noun, the word series means a group of related or similar things arranged in some order or succession.. As a noun, a series is also defined as a sequence of related things. Thus, the set of harmonics can be viewed as a series. In mathematics, however, the set of harmonics is a sequence. It is a set in which the individual elements are listed in a particular order. A series in mathematics is the sum of the elements of a sequence. So as a noun, the words series and sequence are used interchangeably. In mathematics they are distinct terms. I was thinking of series in the mathematical sense, as a sum of terms.

When I started this article I really did not know how what the relationship between the harmonic series in music and the harmonic series in mathematics would be. But by writing about it and drawing the pictures, I had to think about it. I think I have my thoughts about a connection between the two resolved.

What’s more, I resolved it without once referring to my cousin, let’s call him “Dave,” and his fascination with white socks and sock heights. This week I will think hard about a connection between the harmonic series and white socks and the affect of sock height on series convergence.

I generated the pictures of the overtones using the free graphing calculator provided by the nice people at

http://rentcalculators.org/stheli.html

I captured the graphs with screen shots using Snapz Pro X by Ambrosia Software.

http://www.ambrosiasw.com/utilities/snapzprox/

Try it now. Try it a few times. Then we’ll break it apart and see how it works.

The game starts by asking you to pick a two-digit number.

Each two-digit number is composed of a digit in the one’s place and a digit in the 10’s place. The digit in the 10’s place is ten times the value of the ten’s digit.

For example, the two-digit number 47 is composed of two individual digits, 4 and 7.

The digit 7 occupies the one’s place and indicates the presence of seven ones.

The digit 4 occupies the ten’s place and indicates the presence of ten fours.

Thus, the number 47 represents 4 tens and 7 ones.

We don’t normally think of 47 this way, but it can be expressed as

Now think about representing a two-digit number in the following way.

Let be the digit that goes into the one’s place. See Figure 1.

Figure 1

In our example, represents 7.

The digit that goes into the ten’s place is represented by since whatever number we choose for , putting it in the ten’s position indicates there are of those numbers.” The word “of” translates to the arithmetic operation “times.” So we have . In our example, the 4 in the ten’s place indicates there are tens present.

Figure 2

So our two-digit number can be represented as . In our example, can be represented as . See Figure 3.

Figure 3

So, step 1 in the regifting game asks us to choose any two-digit number. All two-digit numbers can be represented by .

Step 2 asks us to subtract both the first and second digits from our number. The first digit is and the second digit is , so we now have .

In our example using as our number, we get .

Combine like terms.

Now we are getting a clue. The is just the number multiplied by . That means it is a multiple of .

Multiples of 9 are:

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99

Now look back at the 10-by-10 square board in step 2 of the game. Notice that all the “multiples-of-9” squares have the same gift! Clever.

The names in the squares change each time the game is played, but the same names always appear in the squares numbered.

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99