Skewered Contest!!!

Yay!  We have a new contest.  The rules are a little different from previous DON contests, but you can still win a free copy of How Many Licks?  Here's how it works. I’m posting a numbers question below. To enter, come up with an answer and send it to “aaron at aaronsantos period com.” If I think your answer is the best, I'll mail you a free signed copy of How Many Licks? Submit your entry on or before March 15, 2012 for consideration. Don't worry…I won't spam you or share your email with any third parties. Here’s the question:

I was reading an Asimov book the other day and came across something called Skewes' number.  At one time, Skewes' number was the largest number ever to appear in a mathematical proof.¹  It would be impossible to write Skewes' number out in the conventional way, but using exponents we can write it as

^10101034.

This is an exceedingly large number.  It is vastly bigger than the number of grains of sand in the Earth.  In fact, it's vastly bigger than the number of grains of sand in the solar system.  In fact, it's vastly bigger than the number of atoms in the solar system.  In fact, all of these physical quantities are extreme underestimates of Skewes' number.  What physical quantity comes closest to Skewes' number?

 Note: It must be something physical like a molecule, atom, or grain of sand (i.e. not something abstract like a number appearing in a mathematical proof.)

[1] It has since been replaced by other large numbers.  For more info, see Graham's number and Moser's number.

Alton Brown Wants to Burn Your House Down

I'm a big fan of Food Network's resident gastrophysicist, Alton Brown.  Mr. Brown excels at explaining food science to lay people in fun and entertaining ways.  With this in mind, I'm kind of sad to call him out on an egregious error in one of his recipes.  The recipe, "Plain Brown Popper", gives the following instructions:

"Toss the popcorn with the olive oil, salt, and jalapeno seasoning mix in the paper bag. Fold the top of the bag over and staple the bag twice to close. Place the bag in the microwave and microwave on high for 2 minutes to 3 minutes..."

Come again?  Staple?!  As in a metal staple?  In a microwave?! Perhaps staples are too small to do any damage.  How bad is it to put a metal staple in a microwave?

Microwaves work by generating an oscillating electric field that vibrates the electrically polarized water molecules inside.  Humans observe this enhanced molecular movement as an increase in the temperature of the food.  However, water molecules aren't the only things that are getting pushed by the electric field.  Any mobile electric charges will vibrate as well, including the electrons in any conductive metal that found its way into the microwave.  Charges in the metal can move around in  a way that will make the electric field even bigger.  If the field is large enough (~3×10⁶ V/m), sparking will occur, which can start a fire.

If there's any electric field inside a metal, it will push the electrons from one side of the metal to the other.  In fact, the field will keep pushing electrons from one side to the other until the electric field created by these electrons exactly cancels out the original field.  This happens very quickly because electrons move very fast.  Electrons cancel out the electric field in metals so quickly that we can often assume the electric field inside a conductor is always zero.  How many electrons would it take to cancel out the 2×10³ V/m field created by many microwaves?  Well, if you take a plate and put charge on the opposite faces, you'll find the charge on a surface is given by

total charge = (permittivity of space) · (electric field) · (area),

where the "permittivity of space" is a constant equal to 8.854×10^-12 F/m.  Plugging in 0.5 mm by 1 cm for the area, you get a total charge

total charge = ( 8.854×10^-12 F/m) · (2×10³ V/m) · (0.5 mm × 1 cm)

= 1×10^-14 C.

That's a small charge, and it doesn't create much of an electric field when spread out over broad side of a staple.  However, when the same charge aggregates on the sharp end of the staple, the electric fields can get quite large because the area is so small.  If sharpened, the points might have and area of 0.05 mm by 0.05 mm,

electric field = (total charge) / [ (permittivity of space) · (area) ],

= (1×10^-14 C) / [ ( 8.854×10^-12 F/m) · (0.05 mm × 0.05 mm) ],

= 4×10⁶ V/m

That's more than large enough to generate a spark. The electric field just outside the staple will depend greatly on how sharp the surface is, with sharper objects being more likely to spark.  With the possible exception of pets and small children, a staple might be the worst of all possible things to put in a microwave.^1,2,3


[1] Being a scientist, I had to test whether or not putting staples in a microwave was actually harmful.  Mr. Brown, you owe me one microwave.
[2] Special thanks to Melinda Keller, who directed me to a very nice article in The Physics Teacher titled "Microwave Mischief and Madness", by Heather Hosack, Nathan Marler, and Dan MacIsaac.
[3] Immediately after posting this, I was struck by the realization that the quasi-static approximation I've made might not be legit.  I think I'm right on this (my microwave was incinerated after all), but feel free to call me out on it if I'm making a mistake.

Ballparking: Practical Math for Impractical Sports Questions

I have a new book.  It comes out in a few months.  You should buy it.  It will make you smile.

Fuzzy Numbers

We're halfway through January, which means most people have probably given up on their New Years resolutions.  For those who are still committed to going to the gym every day and losing that last three pounds, I salute you, but let me give you one bit of warning: Don't trust the nutritional labels on backs of packages.  If you're counting calories, you may find yourself disappointed.  One of the most difficult things to impress on students is that real world numbers aren't exact.  There's always some degree of uncertainty.  When you look on the back of that energy bar and it says "200 calories", it's probably not exactly 200 calories.  Perhaps it's 201 calories.  Or 197 calories.  Or 234 calories.  Just how far off is it?  In a science lab, you would never report a real world number without providing some error range.¹  It may seem like I'm being a pedantic stickler here, but real world numbers are incomplete without an error range.  Saying "This chocolate covered bacon cake is only 137 calories", is not as truthful as saying "This chocolate covered bacon cake is only 137 calories plus or minus 6,723 calories" when there's an enormous error range.  Just how far off are nutritional labels?

To investigate how far off the reported numbers on a nutritional label are, I first bought a package of Mini Clif Bars.²  Given the limited amount of equipment at my disposal, it would be difficult to measure the number of calories in each bar directly.  However, it seems reasonable that the number of calories in a bar would be proportional to its weight, so you can estimate the caloric fluctuations by measuring the fluctuations in the mass.  Each bar had a listed weight of 28 g and was said to contain 100 calories.  The package contained 18 bars with weights ranging from 28.2 g to 33.3 g.  The bars had a mean weight of 30.7 g and a standard deviation of 1.3 g.  That's a 4% fluctuation away from the mean, which itself was almost 10% larger than the listed value.  Scaling these fluctuations up to the listed value of 100 calories, we find that the actual number of calories for a Mini Clif Bar is likely closer to 109.5±4.5 calories.

A histogram for the different Mini Clif Bar masses (in grams)
shows significant fluctuations away from the mean.

Let me be clear.  I'm not insinuating that the good people at Clif Bar & Company have devised a dastardly plot to plump up casual exercise enthusiasts.  If anything, I'm kinda happy I got more food than I paid for.  My only complaint is that nutritional labels don't include a range detailing how imprecise or "fuzzy" these numbers are.  No matter how careful a manufacturer is, there are always the fluctuations that will creep in, and we would do well acknowledge this fact.

[1] The term "error range" is a bit misleading because the word "error" implies a making mistake. "Oops...I really didn't mean to drop that now bloody hatchet down your underwear.  Please forgive my error."  In reality, error ranges have nothing to do with making mistakes.  All real world measurements have some degree of uncertainty, so perhaps it's better to think of them as "uncertainty" ranges  rather than "error" ranges.

[2] Admittedly, a rigorous study should include more than one box and a variety of different food types, but doing one sample is enough to illustrate my point.  Plus, I like Mini Clif Bars.

It's That Time of Year Again

It's that time of year again.  The holidays are here, and you can't flip through the channels without eventually hitting a Christmas special.  We watched Miracle on 34th Street yesterday, and I was struck by the scene at the end where all the mail addressed to Santa gets shipped to the courtroom where Kris Kringle is on trial. One of the clerks in the mail room says "there must be about 50,000" letters to Santa Claus that they have to get rid of.  How many letters get delivered to the court room?

In the movie, 21 duffel bags full of letters are brought in and dumped on the judge's desk.  Judging by how the men carry the bags, they certainly weigh between 2 and 200 lbs, so we can reasonable assume there are about 20 lbs of letters to Santa in each bag.  According to the United States Postal website, a standard letter weighs less than 3.5 ounces.  Assuming a 1 ounce letter, we can easily compute the number of letters delivered,

# of letter = (21 bags) × (20 lbs/bag) × (16 ounces/lb) / (1 ounce/letter)

= 7000 letters.

At about 7000 letters, the mail clerk was correct to within about an order of magnitude.  Not too shabby.

A Couple of Updates...

Fermi Lives at Harwoon Union

It's been a busy semester.  A couple of months ago, I received a very nice email from Lisa Therrien and several other teachers at Harwood Union Middle School in Duxbury, VT.  Apparently Fermi fever is catchy, because Ms. Therrien is teaching her students how to make order of magnitude estimates.  I got invited to listen to several of her students describe their estimations over Skype.  It was a very enjoyable experience, and I can truly say that Harwood Union Middle School has some very creative young mathematical minds.  I only wish more teachers incorporated this style of thinking into their math and science classes.


Estimation as a Skeptic's Tool

In addition to Harwood Union School, I also got invited to speak for Ann Arbor Science & Skeptics in October.  It was a great group, and we discussed how estimation can be used as a skeptic's tool. Here's a few of the cool estimates that were suggested by the group:

• How much hair throughout the entire world is grown in one day? (Enough to cover about 10 square miles.)
• What was Forrest Gump’s average speed as he ran across the country? (About 5 cm/s.) 
• If the Sun disappeared, how quickly would the temperature drop on Earth? (About 0.1°C per century if you’re talking about the average temperature of the whole Earth, or about 20°C per day if you’re talking about just the surface temperature.)

The only downer of the trip for me was that I had to admit my secret shame.  I, physicist Aaron Santos, have never actually read a Carl Sagan book.  It's a problem that I will hopefully be remedied in the near future, but for now, feel free to throw your rotten tomatoes.

Shark Blood

"Nom nom nom nom nom!"

I've been seeing this commercial a lot lately, so I got curious.  How sensitive would a shark's nose have to be in order to detect a drop of blood from one mile away?

Let's say you took a drop of blood and dumped it in the ocean.  Because of diffusion, that drop would eventually spread out in all directions, much like a drop of red food coloring in a glass eventually spreads out and makes the entire glass red.  If the blood spreads out equally in all directions, then it will eventually cover a hemisphere 1 mile in radius.  This means the blood will be distributed over a volume of

V = 2 π R³

=2 · (3.14159) · (1 mile)³

= 2.6×10¹⁰ m³.

A drop of blood is about 1 mL.  This means the density of molecules in a 1 mile radius drops by a factor of about

(1 mL) / (2.6×10¹⁰ m³) = 3.8×10^-17.

That's very tiny.  I'm not exactly sure what part of the blood the shark is supposed to smell, since there are many components that make up blood.  Let's assume he's smelling the red blood cells.   A healthy red blood cell count is in the range 4.6-6.1 million per μL = 5×10^-9 per mL.  If the blood spreads out uniformly over this distance, there will be

(3.8×10^-17) · (5×10^-9 per mL) = 1 red blood cell every 5 m³.

That's a very tiny amount.  I highly doubt a shark, or any other creature, could detect something with this small a concentration.  Besides, the MythBusters already busted this myth during shark week. 


Bonus Shark Blood Question #1: Even if a shark could detect concentrations this small, how long would it take a red blood cell to travel 1 mile?

Bonus Shark Blood Question #2: Where does a shark's blood come from if blood comes from bone marrow and sharks are cartilaginous fish?  (I know...this is not a numbers question, but I've always been curious.)