Additional Math Pages & Resources

Monday, August 31, 2009

Predicting the future

It started when I closed the garage door by leaning out the front door and pressing the button on our ancient Genie remote control.

My wife asked,

How does that work? Is it by radio? How come the various signals in the air don't bump into each other?

That's a tough bunch of questions. The short answer is yes, the remote sends a scrambled radio signal to the garage door opener/closer. The opener sees that signal. It recognizes one push as an instruction to open the door and two pushes as an instruction to close the door. It turns on the motor for a moment, and the door closes.

The signals don't "bump into each other" most of the time because government agencies make sure all manufacturers of transmitters put their radio signals into the right places "in the air".

Or in more technical terms, they allocate the frequency spectrum.

Here's the US allocation chart  and also the UK allocation chart (be ready to ZOOOOM)

Sheesh. Can you believe those charts? A simple list was all I needed.

It seems like there are millions of frequency bands. Here are the main ones:
Designation Frequency Wavelength
ELF extremely low frequency 3Hz to 30Hz 100'000km to 10'000 km
SLF superlow frequency 30Hz to 300Hz 10'000km to 1'000km
ULF ultralow frequency 300Hz to 3000Hz 1'000km to 100km
VLF very low frequency 3kHz to 30kHz 100km to 10km
LF low frequency 30kHz to 300kHz 10km to 1km
MF medium frequency 300kHz to 3000kHz 1km to 100m
HF high frequency 3MHz to 30MHz 100m to 10m
VHF very high frequency 30MHz to 300MHz 10m to 1m
UHF ultrahigh frequency 300MHz to 3000MHz 1m to 10cm
SHF superhigh frequency 3GHz to 30GHz 10cm to 1cm
EHF extremely high frequency 30GHz to 300GHz 1cm to 1mm

A global group called the International Telecommunication Union manages these definitions.

Some frequencies are used by the military. Others are considered international because they are used by satellites, aircraft and other non-local, non-national devices. The rest of the frequencies are allocated by agencies (FCC in the US, Ofcom in the UK, etc.) who monitor usage and figure out ways to maximize use and revenue from each frequency band.

Ofcom recently did an experiment where they loaded antenna arrays on top of regular vehicles and drove all around the UK, monitoring how crowded the frequencies were. After scanning hundreds of pages, I almost gave up trying to find a simple way to explain the issues. But I will give it a try.

The regulatory agencies need to
  • Decide what wireless services people are using
  • Predict how people might use those or more services in the future
  • Determine which technologies / frequencies each service uses
  • Figure out how quickly the demand might grow for each kind of wireless
  • Determine the geographic distribution of the users: rural wide open; cities full up
  • Guess how quickly the technology might change and get more efficient at using its space
  • Decide which frequencies will be full first
  • Decide if other frequencies can give up space (military? satellite? aircraft?)
  • Reallocate
Sounds highly complicated. Beyond my math skills, and beyond our ability to do an accurate prediction ...

But in doing this research I found something that would be really cool to have - a full-spectrum jammin truck. You can drive it around and block virtually all frequencies across all spectrums. All electronic services come to a halt.

I can confidently make this prediction - They won't sell it to us!

Friday, August 28, 2009

Make Money Making Money

The US Mint makes a profit minting new quarters for collectors. It's such a nice profit there's a special word for it - Seigniorage

It's a benign sounding word from old Anglo-French not used these days outside the economics profession. Originally, it referred to the right of the feudal lord, the "seignior," to coin money for use in his realm. Central governments didn't exist. Each seignior had the right to coin money so commerce could rise above bartering. There was something in it for the seignior as well. Since it cost him less to make the money than the money was worth, he made a profit just through coining and printing. The more currency he made, the richer he became ... James Goldborough

How does this work at the US Mint?

Many of the quarters are immediately taken out of circulation and put into collections. It's like the Post Office selling stamps that you put into an album. They don't have to deliver a letter for the 44¢ you spent. In the same way, the government mints quarters (at a cost less than 5¢) which collectors "buy" and hide away.

The 50-State Quarters program started in 1997 and resulting the minting of 34,797,600,000 quarters. More or less. At least 100,000,000 people started collections. Our hoarding has given the Mint a profit of about $4.6 billion dollars.

From these numbers can you calculate how many quarters have been taken out of circulation? Let's see...

Minted 35 billion quarters at 5¢ each for a cost of $1.76 billion and a street value $8.8 billion.

The profit of $4.6 billion divided by 20¢ profit per quarter means 23 billion quarters are safely stashed away! That's 66% or 2/3 of the quarters.

The 6 US territories [did you know there are territories aligned with the US?] are getting "their" quarters right now, in a program authorized for 2009. That's the District of Columbia, Puerto Rico, Northern Mariana Islands, Guam, US Virgin Islands, and Samoa.

This program is expected to clear another half-billion $ for the mint!

Not wanting our enthusiasm to slacken, Congress has authorized the Mint to begin a 56-coin "Beautiful National Parks" series in 2010. Rather than 4 states per year with the state quarters, this time every 10 weeks another batch of quarters will be released.

Will this flood of coins tire out collectors? Will they kill the goose that lays the golden eggs? Will the US sink under the weight of coins stashed in our bedroom jars?

Only one piece of wisdom can come out of all this story of profit by minting:

Don't try this at home, Seignior!

Thursday, August 27, 2009

Solar, Lunar & Terrestrial Time

A few days ago I stated that most of our units of time are related to the regularly-observed movements of heavenly bodies.

Today let's have a look at these units. Let me add at this point that I am not going to explain the Citizen Astrodea watch shown here.

And we'll ignore a common unit called the millisecond or New York Second  - the interval of time between a traffic light turning green and the New Yorker behind you honking his horn!

  • Second - 1/60th of a minute; the time required for 9,192,631,770 microwaves to be emitted by a chunk of cesium at sea level at absolute zero temperature and zero magnetic field. This has been defined as our fundamental unit of time.
  • Minute - 1/60th of an hour, 60 seconds; also a unit of rotational movement (angle) of the earth
  • Hour - 1/24th of a day, 60 minutes, 3600 seconds
  • Day - time required for the earth to rotate once around its own axis; 24 hours; 86,400 seconds. We have defined 2 types of days - Sidereal Days (a single rotation, viewed from a distant star) and Solar Days (a single rotation viewed from our own star!) TERRESTRIAL
  • Week - based solely on religious or cultural factors
  • Month - time required for the moon to orbit the earth; approximately 30 days LUNAR
  • Year - time required for the earth to orbit the sun; approximately 365 days SOLAR
  • Decade - a group of 10 years; varies from 3652-3653 days in length
  • Century - a group of 100 years
  • Millenium - a group of 1000 years
  • Epoch - a period of time defined by the movement of a celestial body leaving one point in an orbit and returning to it later.
As with many issues we investigate in this blog, of course there are weird time exceptions:

Leap Seconds account for 2 milliseconds per day slowing in the earth's rotation. Since we started using the Cesium clock, seconds are stable. But since the earth is slowing we have to adjust the time occasionally. A "leap second" is added to or subtracted from our clocks by the International Earth Rotation and Reference Systems Service.

Equation of Time reflects irregularities in the sun's path and the rotation of the earth. Time on a sundial and time shown on a clock can vary across a range of 30 minutes throughout the year.

Terrestrial Time is essential for astronomers. Because time is related to rotation, the distance of the astronomer from the center of a planet will change the time (for example, sea level vs the top of Mauna Kea in Hawaii). To resolve this, astronomers use a constant figure - International Atomic Time + 32.184 seconds. Another alternative is to utilize Geocentric Coordinate Time, which is the time at the center of the earth.

You can see some time variations displayed on a very few, carefully-constructed computer clocks. Click on the link to see an example. To make this one work for your time zone, uncheck the RUN box, select the city closest to you (or input your coordinates) and check RUN again.

Wednesday, August 26, 2009

A square root

It started innocently when Alton Brown, my favorite TV cook, did a show on mushrooms. Alton said the mushroom is the spore-bearing fruit of an underground fungus. He went on to say the underground part could easily extend over an acre in size.

My house sits on a lot that is 1/3 acre. I asked my wife,

If that mushroom was growing here, how much area would it take up?

Sounds like a question for the blog, she replied. With a yawn.

Now how can I figure this out with the least amount of work? I started by dredging up a drawing of our property. I don't know the scale, but it looks like this:

Since it's not strictly rectangular or square, calculating the area can be a bit difficult. But I really don't want to know the dimensions, I just want to see what a mushroom might look like compared to my yard.

So here's my next step. Make a rectangle with the same area as my yard, without having to do any measuring or calculating.

I drew a rectangle (green) and overlaid it on my yard diagram (red). I compared the size visually and made the rectangle as close as possible. Then I created some triangles to make sure I was right on.

Triangles 1 and 3 represent the area where my rectangle is outside (larger than) the original lot boundaries.

Triangles 2 and 4 represent the area of the original lot that my rectangle does not cover.

Those triangle pairs (1 & 2) and (3 & 4) roughly offset one another, so my green rectangle does represent the area of my lot.

I know the lot is 1/3 acre, so all I have to do now is draw a square or circle 3 times the size of the rectangle. How do I do this in the easiest way?

It's a simple job to calculate the area of the rectangle. I could have used a ruler to measure the rectangle, but I had this drawing in Adobe Illustrator. The software told me the numbers are 506 x 128 pixels.

That means its area is about 65,000 square pixels. Notice that I don't need to convert to feet; I just need my one-acre shape to be 3 times the size of the rectangle. I need about 200,000 square pixels.

Using trial and error, I mentally I think 5 x 5 is 25; that's too big. 4 x 4 is 16; that's too small. I need a square about halfway in-between. 450 x 450 in size. I quickly make a square and here it is!

Wow! That's one big square root.

This much land area (in Uganda where it rains a lot) can produce 4,560kg or 10,000 lbs of mushrooms a year!

PS - yes, I could also have said to Illustrator, Scale this up 3 times. But where's the fun in that?

Tuesday, August 25, 2009

No reverse gear.

There are many perspectives from which we can discuss the nature of time. Here are a few:

1. Time is integral to the structure of the universe. It’s one of 4 dimensions, where events occur in sequential order. It can be shown on a time line. Time travel may be possible.


2. Time is NOT a container or fluid through which events and objects move, nor does it flow. Time is an intellectual idea. We use it, together with space and numbers, to sequence, organize and compare events. Time is neither an event nor a thing, is not measurable and cannot be traveled.


3. Time is a cyclical, repetitive thing, endlessly repeating in a great cycle. A snake chasing its own tail. There is no linear movement in time, no progress, no upwards or onwards.

These three conceptions are religious and philosophical, as well as scientific. We can each chose a favorite. In fact we each have to choose, because time is social.

If you are on your own (no boss, no spouse, no kids) or on vacation, you don't need a watch. Time is more or less irrelevant. You do what you wish, when you please.

But add another person, and suddenly time is necessary. It allows us to compare the duration and sequence of events. Time makes it possible to say to each other,

This takes longer than that
and Which one happened first? and I want my dinner now!

A unit of time is usually defined as a number of cycles of a regularly moving physical body (a pendulum, planet, moon or star). We watch this moving item - either together or separately - and synchronize ourselves in time.

I'll see you the day after the next full moon, ok?

Watching heavenly bodies is fairly easy. They usually appear in the sky over the earth for a few hours each night. But not always. Weather can hide them, and they aren't visible from some locations. And they are not really all that regular in their movements.

So we learned to make swinging pendulums and water clocks - then clocks and watches of all sorts. They let us share time at night, in bad weather, on the seas and in the air.

So time is a social way to compare the length and sequence of events. What else?

I don't know about you, but time travel has always seemed like an appealing thing to me. As long as there is a reverse gear so I can get home!

I'll write more on this at another time - I'm late ...

I used to own this very Rolex. Click it to see what time it is.

Monday, August 24, 2009

Russian Studies, Welding and Math

How do we predict how people will grow up? What they will do with themselves? What jobs they will enjoy?

Back in 1969 the Washington State Pre-College Test suggested I could have future success in Russian Studies or Welding.

"academic prediction in Washington State is unique in that it is not directed at the admissions needs of colleges but at the decision-making needs of the average high-school graduate ... having to make choices against a background of aptitude and training which may spell success for some choices and failure for others ... " and so on.

I ignored their suggestions and went to college in sunny California. I studied Speech, with a minor in Latin American Literature. Having no career thoughts in mind, I continued on to graduate school. However, I quickly got bored studying medieval culture and its transfer via oral tradition.

I starting thinking about getting a job, and getting married. I soon did both. Publishing and Laurie have been my companions for 35 happy years.

Publishing is almost like school - you learn something and write about it before the deadline. But it's better than school, because you get paid. And when you grade/edit other people's papers, you still get paid.


But here I am today, explaining probability and prediction to teachers and students.

Prediction means to say before [it happens]. There are many ways to go about making predictions:

If we could really predict things we would be rich. But despite monumental efforts to improve our predictive abilities, we still can't reliably predict natural disasters, movements in the stock markets, software development schedules, sports scores, etc.

Nor can we predict who will love, tolerate or hate math.

In Excel Math we introduce a problem-solving process known as trial and error. You start with one possible number, try it in an equation, and if it doesn't work, try the next possible value.

This works in real life too.

I just read an article headed Fail Early and Fail Often. It argues for listing your failures after your name, instead of a list of your academic degrees.

Anyone out there looking for a work? Try listing your failures on your resume or CV. I hope this approach will get you a job.

If it doesn't, you can always try something else. Like Russian Studies, or Welding. Or Math.

Friday, August 21, 2009

Spiral or Helix?

Wikipedia defines spiral as a planar (or flat) curve, shaped like a watch hairspring, expanding outwards at one end.

Here are some good and bad watch hairsprings. The bad one doesn't work.

A helix is a three-dimensional coil that runs along the surface of a cylinder. Spiral is often used (incorrectly) in place of helix, as in spiral binding or spiral staircase. Here are Excel Math Teacher Editions which have coil "spiral" bindings, and an image of a "spiral" staircase.

Math is a language of counting, measurement, shapes and calculation. A language with precise definitions and specialized terms. It's not just numbers. Ok, you might concede, but still ask, Why this focus on spirals?

We have limited attention spans. We can only take so much before we grow tired and look out the window or check our mobile phones. We can only trudge straight up a staircase for so long before we get tired. Look at this subway escalator!

Math can be like that at times, but it doesn't have to be arduous. Variety helps.

A week ago I did a tea post. Yesterday I did another. There were two posts on airplanes. Once I wrote about coins. I'll probably get back to money later. I like tea, and I like money. Math helps me understand both tea and money.

Math is a lot like this blog. The same concepts come up again and again, in different ways. How you learn math depends on the situation, and your teachers.

We can't really teach coins thoroughly without having explored fractions - because coins are fractional parts of a larger unit of currency. We can talk about probability, but we can't express it numerically unless we understand ratios and decimals, such as "1 in 50 chance of a jackpot".

What does all this have to do with this math blog?

Excel Math exposes kids to math in a spiraling fashion. We don't want to bore them but we do want them to learn. And we want it to stick. So they practice. Even mediocre athletes practice, and good ones are always at it! One thing after another, not the same thing all day.

Likewise, we move on to more complex things, then back to prior concepts for a brief refreshing.

If we carefully tweak our spiraling so concepts aren't tangled (like the bad hairspring) or dauntingly linear (like the escalator) we can help kids experience math in an enjoyable way. Math should wind and unwind around and through their world, lifting them to higher planes of understanding, allowing them to integrate math into daily life ...

Oops. Got carried away.

I predict this won't be the last time I write about spiraling, or tea, or watches. Those subjects will come around again. Smoothly.

PS - I went to Helix High School. Really.

Thursday, August 20, 2009

Sold by weight not volume

When you see that phrase you might think of opening a new, but half-empty package of potato chips, or a box of breakfast cereal that seems mostly air. I am wondering about a box I received today from Thunderbolt Tea in Darjeeling, India.

I had ordered 7 different kinds of tea, all Darjeeling varieties. When I opened the box I was amazed to see the various sizes - what a range of packet sizes there were! Time for some math!

First we needed precision measurements, so I got my Central Tools digital caliper and a postal scale. The numbers in orange are weight (grams) and in black are dimensions (millimeters). The final number in green is a calculation of the volume of each packet.

Notice the weight only varies between 100-105 grams. That's good, because I ordered 100 grams of each variety.

(I am including the packaging because it's light, consistent between batches, and because tea can be very messy when spilled on the counter.)

The cubic volume was determined by multiplying height, width and depth in mm and then converting to liters. A liter is equal to 1,000,000 cubic mm.

The first example is 115 x 105 x 68 = 821,100 cubic mm, or .8211 liters.

Here are the 7 different
packet volumes in order of
largest to smallest:


The largest package is 2 1/2 times the volume of the smallest, for the same weight. Astonishing!

If we want a more precise comparison we can determine the relative density of the teas. Density is the amount of mass (grams) per unit of volume (liters).

We divide 1 by the volume of the tea to find out how much tea is required to fill a one liter container. Using the .4200 sample, we divide it into 1 and learn that 2.38 packages of this tea should fill a liter container.

105 g x 2.38 = 250 grams of tea per liter.

Since water is 1000 grams per liter, this tea is precisely 1/4th the density of water!

Now what I really need to know is, How do I measure to make a cup of tea?

Wednesday, August 19, 2009

How bright are we?

Today we ask "How many light bulbs cost how much to run?"

I read an article about saving energy by changing light bulbs. It said, "First step is to count your bulbs".

Since I have no idea how many bulbs are in my house, I'll go check while you count yours. Remember, count the bulbs, not the fixtures!

Fill in the numbers below. Add or take away rooms as necessary. Don't forget the bulbs in closets, stoves, refrigerator, freezer, microwave, dryer, garage door opener, etc. Ignore any battery-operated lights.

Here's the count for my 50-year-old house:

02 Entry
12 Kitchen
08 Dining room
08 Living room
12 Family room / Office
03 Bedroom 1
03 Bedroom 2
04 Bedroom 3
04 Bathroom 1
04 Bathroom 2
04 Bathroom 3
02 Laundry
08 Garage
03 Porch
04 Patio
01 Garden / Yard

82 Wow!

In my house, about half are fluorescent lamps. But we'll ignore that and just multiply each bulb by 60 to get average wattage. Let's imagine they are all on (yes, unlikely, but this is just for fun).

60 x 82 = 4920 watts ≈ 5000 watts or 5 kilowatts

If we ran the lights for an hour, it would consume about 5 kilowatt hours of electricity.
At the lowest rate in San Diego, the electricity cost would be calculated like this:

The baseline rate is 13¢ per kwh or 5 x $.13 = $.65
The highest rate is 34¢ per kwh or 5 x $.34 = $1.70

Let's calculate in our heads - come on, you can do it. First we round the numbers slightly for convenience since we just need a reasonable estimate. Besides, our meters may not be all that accurate (subject for another post!):

$.65 = 2/3 of a dollar
$1.70 = 1 dollar + 2/3 of a dollar, so 5/3 of a dollar

Low Rate (24 x 2/3) = (48/3) ≈ $16
High Rate (24 x 5/3) = (120/3) ≈ $40

[How easy was that? Next we'll use Associative & Distributive Properties to shift things around to make the math go faster.]

Low rate (24 x 7 x 2/3) = (8 x 7 x 2) = (56 x 2) ≈ $112
High rate (24 x 7 x 5/3) = (8 x 7 x 5) = (56 x 5) ≈ $280

Low (365 d x 24 hr ÷ 12 mo x 2/3) = (365 x 4/3) = (122 x 4) ≈ $488

[Let's check our work (30 1/2 days x 16) = (30 x 16) + 8 = $488 We're okay so far.]

High (365 x 24/12 x 5/3) =(365 x 2 x 5/3) = (365 x 10/3) = 3650/3 ≈ $1217

[Checking ... (30 1/2 days x 40) = (30 x 40) + 20 = $1220 Close enough.]

Low (365 x 24 x 2/3) = (365 x 16) = (3650 + 2190) ≈ $5,840
High (365 x 24 x 5/3) = (365 x 40) ≈ $14,600

Wow! Turn off those lights!

PS You can check with your calculator now. I haven't used mine til now. Honest.

My calculator says the annual amounts are $5694 and $14892
Mental math says the annual amounts are $5840 and $14600
If we add up our estimates and compare to the calculator, we have 20586 ÷20440 = 1.007
We are within one percent, overall. Isn't math fun?

Tuesday, August 18, 2009

More Airplane Math

Monday I was at Jackson, WY airport. We had these conditions:

Altitude 6450 feet Runway 6300 feet max Temperature 75° F Barometric Pressure 30.2

These factors interact. Here's a chart I constructed from a variety of sources, to try to illustrate the details. These are not exact, but I'm sure no pilots will plan a flight with this, as they will be calculating a much more precise figure known as Density Altitude.

The orange dotted line shows a full load take-off at sea level and 70° F requires about 4500 feet of runway. Now look at the green and raspberry dotted lines. You can see as the altitude increases so does the minimum runway length.

The yellow and yellow/green lines show the maximum loads at 6500 feet altitude for different temperatures. Jackson has a short runway, and we need a margin for error. The red spots show how the maximum passenger capacity changes with temperature.

One quick rule of thumb for a small plane is 200 lbs change in payload equals 500 feet of runway.

Now why does all this math matter?

When we checked in, we learned that due to weather and altitude, capacity had to be restricted to 55 passengers. The airline had booked more than 70, expecting a few no-shows. But more than 70 had already checked in, so at least 15 people would miss the flight.

We didn't. In 2 million miles of flying, this has happened to me before. I quickly asked to change to an earlier flight, and we got home with no problems.

Isn't math useful?

Monday, August 17, 2009

Airplane Math

You can't be a pilot without also being good at math. Numbers, numbers, numbers.

Pilots worry about how much weight an aircraft can carry, how long a runway it needs to take off and land, how far it can fly and how fast it can fly.

For instance, let's look at weight. The weight of an aircraft is the total of:

1. Plane (operational weight empty, includes pilot)
2. Fuel (usable fuel & reserve fuel)
3. Passengers & cargo (payload)

The plane I flew on this weekend was a CRJ700 regional jet.

1. Plane Empty = 43,500 lbs
2. Maximum Fuel = 19,500 lbs
3. Maximum Payload with full fuel = 10,500 lbs or with less fuel = 18,800 lbs
4. Maximum Take-off Weight = 72,750 lbs

Maximum Take-Off Weight = Plane Empty + Fuel + Payload.

Notice that Fuel and Payload are variables we can adjust to stay under the Maximum Take-Off Weight. Now that we know this - we can ask "How many passengers can this plane carry?"

The FAA tells us to use these weights:
  • Women 179-184 lbs. (including clothes & carry-on)
  • Men 200-205 lbs. (including clothes & carry-on)
  • Checked bag 30 lbs.
Let's assume equal numbers of men and women, with one bag per person. How many can we carry and still stay under Maximum Take-Off Weight? Here's one way to find the answer:

180 + 200 = 380 ÷ 2 = average 190 lbs per person then add 30 for a bag to get 220 lbs.

Maximum payload of 18,800 ÷ 220 = 85 people.

Since this plane seats 70 plus a crew of 4, it looks like we have a bit of excess capacity. Of course flight distances can be limited if we can't carry a full load of fuel. And we have to consider that
  • If we have bad weather or delays we need more fuel so we can't carry as much payload.
  • If the weather is hot we need more runway to take off or less payload.
  • If the airport is above sea level it also means more runway or less load.
Let's stop there. Whew! Math can be hard mental work.

Thursday, August 13, 2009

Space Junk - how much is too much?

Can math answer just one simple question? How much is too much?

"How much is too much?" generated 295 million responses in .23 seconds on Google.
  • How much drinking (alcohol or caffeine or water) is too much?
  • How much exercise (running or training) is too much?
  • How much homework is too much?
  • How much junk in orbit around Earth is too much?
Math can help you make decisions. Let's look at that space junk question.

1. Define the terms.
Space junk is stuff we have sent into space from Earth
. Space junk is a nice catchy title. It means man-made objects larger than 4 inches in diameter that are orbiting earth. We could also say "Orbital Debris" if we want to be more precise.

2. Can we establish a baseline for the activity or situation?
Yes. There was no space junk at all before October 1957 when Sputnik went into orbit.

3. Can we get reliable information on the subject?
We probably can - at the
NASA Orbital Debris Office.

4. What is the current state of affairs?
We are still sending stuff up into orbit, collisions and explosions are breaking it into smaller pieces, and some pieces are falling back to Earth. Most burn up and a few hit the ground. The population has increased from 0 in 1957 to 9000 in 2000 to 13,000 in 2009.

5. What does the trend look like (change from base state to current state, over time)?
We could draw a chart - try it at home.
We can say "At this rate within 50 years there will be millions of pieces of space junk"
"There are 80 trillion cubic acres of space and only a few pieces of junk. It will take a million years to fill it up."

6. Now what should we do about space junk?
"We ought to start sending Earth junk up too."
"We should stop sending up satellites. We should bring it back. We should send it further out. We could charge people to blow all the big stuff into tiny bits (it would be fun), etc."

Can we answer "How much Space Junk is too much?" and "What should we do about it?" No, not yet. We need a lot more information. And some consensus.

This decision-making approach is reasonable when there is a clean starting point, it all happens up in space and we are not personally or emotionally involved.

It's much harder when the question is "How much homework is too much?"

Wednesday, August 12, 2009

Money's not just for buying things

A few days ago I asserted that math was "A language of counting, measurement, shapes and calculation. A language with precise definitions and specialized terms."

TV chef Alton Brown can't bear precious space being occupied by a device that's good for only one thing. I agree. So I don't want any (more) things around that only have one function. We say "Death to Uni-Taskers!"

Yesterday we looked into the cost of a cup of tea. I needed a scale to measure the weight of both dry and liquid tea.

At The Mighty AnsMar (home of Excel Math) we have 3 scales. The first is connected to our postal meter (0-5 lbs) and the second stands alone (0-10 lbs). The third is in our warehouse (0-200 lbs) and isn't suitable for tea.

Sadly, the two smaller scales disagreed on the weight of a small item. Disagreed by about 30 percent even after zeroing and resetting. How could I do an accurate measurement?

I needed a reference weight - like in old Westerns, where the clerk puts a brass weight on the scale to balance the pile of gold dust.

But I didn't have one. If I went to the post office, weighed something and labeled it - would I keep it handy? It would be a Uni-Tasker. No. I had to find an item that was consistent, readily available, and whose weight was well documented. Coins.

I thought a one-ounce weight would be useful for calibrating the scale. But none of our coins weigh an ounce. I would need several. A simple group of coins. Too complicated to stack 4 dimes, 3 pennies and a nickel, for example.

At the US Mint I learned that this Louis Braille Dollar Coin is August's Coin of the Month.

I didn't even know there was a Coin of the Month. Or a Braille coin. After looking at all the ways I could spend money on money, I got back on task.

A little division, multiplication and metric/standard unit conversion indicated that I needed quarters.

Knowing an ounce was 28.349 grams, I confirmed the choice of a quarter using a handy comparison table on Wikipedia.

A quarter weighs 5.67 grams so 5 quarters must equal 28.35 grams. Five quarters equal one ounce. Come on, do it with me in your head 5 x .07 = .35 and 5 x .6 = 3 and 5 x 5=25.

VoilĂ ! My standard weight is readily available, always replaceable and easy to stack on the scale. And how about this? A quarter's weight equals a teaspoon of most teas ≈ 5 gm!

Money. Not just for buying things, it's for precisely counting, measuring and calculating. It's for math.

PS - Death to Uni-Taskers.

Tuesday, August 11, 2009

How much is a cup of tea?

This is the kind of question a mathematician really likes.

Here's why:
1. Tea is sold in grams or ounces or in teabags
2. Tea is measured for use with a teaspoon, by fingers - "a pinch", or by shaking out
3. Tea is consumed from tea "cups" . Here are a few common American tea "cups"

So using our math skills, we simply:
1. take the price we paid for the tea (per unit of weight)
2. measure out a small pile of tea (the measure of volume)
3. add hot water, wait 3-5 minutes and then remove the leaves
4. pour into your favorite "cup" of any size
5. drink while pondering this blog topic

Drink and do it again why trying to remember. What was the question?

Aha, it was How much does it cost?

The answer - It depends. So make a chart, with graphs (not using Blog software either!)

[click the chart to see a larger version]

Extra credit questions:
1. How much water is retained by the tea (lost from the cup) when teabag or leaves are removed?
2. Is brewed tea heavier than the same amount of plain water?
3. My tea cup weighs 5.5 ounces empty and 18.6 ounces full of tea (without leaves). How many liquid ounces of tea does it hold?

Monday, August 10, 2009

Why become a mathematician?

Great question, eh? Well, it is if you are thinking about becoming a mathematician. I am not. My job is publishing books about math, although I do get asked to divide the check at restaurants.

But I asked this question of my local mathematician, George. He gave me two reasons why he took this career path.

1. George thought that mathematics would be definitive, precise, and free of ambiguity. Sadly, he was wrong. "The deeper you get into a subject, the more vague it becomes," he sighed.

2. Math looked like it would be fun. Not so much the calculations, which can be tedious, but the people.

"Some REALLY STRANGE people do mathematics!" George enthused (not meaning himself, of course). He went on to list a few ...

George's Bio: I obtained a PhD in mathematics from Carnegie-Mellon University with an emphasis in Continuum Mechanics. I worked at the Space and Naval Warfare center for 44 years. My principal job was mathematical modeling of Sonar transducers and arrays. This involved elasticity theory, electrodynamics, acoustics, and numerical analysis. I was a co-developer of a program (CHIEF) that numerically solves the Helmholtz integral equation to predict acoustic radiation or scattering from arbitrary-shaped bodies.

No, I am not making this up. I don't understand it either.

Mathematicians are not on every corner. Ask around. See if you can find a mathematician. Odds are, you can't.

About 45,000 research-oriented PhD degrees are conferred each year in the United States. Under 1000 are in mathematics. That's ok because there are only about 3000 jobs available if you want to do heavy-duty math research. Most of them are with the Department of Defense or NASA. If you can get one of those jobs you can expect to earn about $100,000.

See, math can be fun. And lucrative.

Sunday, August 9, 2009

Can Math be fun?

A definition for FUN is "activities that are enjoyable or amusing."

Look around on the Internet - search for math and fun - you find endless silly questions and puzzles.

Today my question is NOT "Can adults create math puzzles and exercises that will amuse children?"

The question IS "Does mastering basic math skills enable people to enjoy solving real math problems throughout the rest of their lives?"

I think so. Yes.

For many people, reading is effortless, until they come to a category of literature they've never tackled before - poetry, the Medicare Handbook, Instructions for IRS Form 1040, Russian novels. But even if that novel is a hard slog, the ability to wade through slowly is much better than not reading at all.

Math may never be effortless but compared to the alternative - math illiteracy - it's great.

Bicycling across town, especially a hilly town like San Diego, can be lots of hard climbing, mixed in with the fun of coasting downhill. Compared to walking - it's great.

Cooking can be hard, hot and tiring - ask a chef! - but it's certainly a rewarding skill to master.

So is math.

Saturday, August 8, 2009

Is Math really a language?

Some languages are better than others when it comes to discussing certain concepts.

If I want to talk about land animals, such as the American Bison (buffalo) I might be able to use the Lakota language, from the Sioux tribes. If I chose instead to focus on discussing salmon and their journeys to the sea and back, I'd be able to use Salishan, from the tribes along the Pacific Northwest coast. Gaelic might work too, as Scotland = salmon (and golf).

Any of these would be more useful than Pama-Nyungan, an Aboriginal family of languages from Australia, where neither buffalo nor salmon were common.

Now back to math. Because there are millions of homes at risk of foreclosure right now, any discussion of their value will involve specialized math terms used in economics and accounting and probability.

But if I am marketing this real estate, I might be interested in expressing the square footage of the homes, the acreage of the lots, etc. Geometry would be the "language" I'd use.

If language is a system for encoding and decoding information so it can be shared, then math indeed can be a language. A language of counting, measurement, shapes and calculation. A language with precise definitions and specialized terms.

As I live only a few miles from the Mexican border, I sometimes need to speak a bit of Spanish. Even if it's only used in short sentences now and then, it's extremely helpful to be bi-lingual. Otherwise you are always wondering "what am I eating?" and "where's the nearest restroom?"

So it is with math.

Friday, August 7, 2009

Why learn Math? I've got a calculator ...

Ask your calculator how much bark you need to cover the back garden. Can it tell you?

Enter this into a spreadsheet: will a 650 x 32 tire will fit on your 26 inch mountain bike rim?

Can you determine if you have enough fabric to make a costume by counting on your fingers?

Should I have a medical test, or am I more likely to harm myself by testing than by getting a disease?

Are three 6-oz. cans of tomato sauce enough to make marinara sauce for 20 people?

Is .02 cents per kilobyte of data to my mobile phone a good deal, or will I go broke in a week downloading my favorite television shows?

No matter how computerized we get, we still need to think mathematically.

I know I am a bit of a nerd but that's how I think. What do you think?