Really Big Numbers

What is the biggest relevant number? This is actually a huge and complex question, attributed both to the limits of our technology and scientific understanding, as well as our imagination.
Let’s start with the estimated number of particles in the universe. The current estimate is in the range of 10^78 and 10^80. For those of you rusty on your exponential notation: these numbers can be written out as a 1 following by 78 zeros or a 1 followed by 80 zeros respectively. That’s huge, right? Well it depends on how you look at it. For counting; yes. To write out? No. I have already written more than 100 characters in this posting!
Then comes the famous googol=10^100. This number is bigger than the number of particles in the universe, so why would we need it?
Well, if you can let go of thinking of numbers as being only for counting purposes, it is easy to see why a number as large as a googol is both relevant and important. Take Cryptology. Some decades ago Number Theory was considered to be only the poetry of mathematics: beautiful, pure, and totally non-essential. Then three mathematicians (look up the RSA Encryption patent) rocked the world by patenting a public-key encryption system that is used by everyone to send information through the internet. The “public-key” part is what is so amazing. Unlike the secret codes of World War II and earlier years that could be cracked by seeing the codes encoding formula, a public-key cryptosystem make the code formula public! This is so that anyone can send there credit card information to Amazon without being worried about someone intercepting it and decoding it. How is this possible?
Well every coding system has to be a one-to-one function: that is, each input goes to one output (coding), and each output goes to one input (decoding). The genius of RSA is that the function that encodes uses a modular function that has an inverse that can only be found by factoring extremely large numbers. These large numbers must be carefully chosen to be the product of two also extremely large and “unobvious” prime numbers. It is a curious fact that while it is not very hard to find large prime numbers with computers, it is exceedingly difficult to factor large numbers that have large (like a googol) prime factors. Even as computers get faster, the prime generators are always ahead of the factoring algorithms, and encryption is “safe.” Safe, that is, until one of two things:
1) It has not been proven that there is no “fast” factoring algorithm (for the computer science literate “fast”=able to be completed in Polynomial-time). Watch the movie sneakers to see a fictional account of what could be the result if this happened now.
2) Quantum computing becomes standard. With quantum computing factoring large numbers could be very fast and encryption would no longer be safe. But quantum encryption could then be born, which would be safe but also unreadable by non-quantum computers… L

So there you go! If you are interested in even larger “relevant” numbers that lack the “important” adjective, look up Graham’s number.
http://en.wikipedia.org/wiki/Graham

Also a Googolplex=10^10^100

If you are interested in creative new ways of writing numbers that are big in a way that is incomprehensible to most people, look up Conway’s Arrow notation

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

Nonwords: Nonsense and Science

We may begin with the word nonword, since only a few dictionaries list it as a word (but Webster’s does, as “a word that has no meaning, is not known to exist, or is disapproved”). I will begin by focusing on the third listed meaning.
Some words that were disapproved a century ago are now legitimate and familiar: accountable, answerable, donate, jeopardize, practitioner, presidential, reliable, and contact (as a verb). There is a good reason for these words to become standard; they are useful! And yet some nonsensical nonwords continue to fester in conversations among the less-informed. The two main culprits are irregardless and preventative. Using these two words is especially unjustified as they both have simpler equivalents: regardless and preventive. Though many dictionaries include preventative (and sadly, Microsoft word does not underline it in red), preventive is found 5 times more often in print.
Irregardless, on the other hand, is totally unacceptable because it confuses the meaning of regardless, the word that actually means what the speaker intended. This provides a “teachable moment” as education jargon-mongers love to refer to. Next time you hear the word irregardless, politely interrupt the speaker and ask “Excuse me—do you mean regardless.” She or he will probably pause as they realize that irregardless sounds like it means the opposite of regardless, even though that is what he or she really meant. The entire conversation may stop here.
Regardless, nonwords should be avoided. Some come from trying to make a word that is already an adverb into one by adding an unnecessary ly, as in doubtlessly, seldomly, thusly, and muchly. One can only laugh at (and perhaps feel sorry for) people who come up with the following the abominations uncatergorically, unmercilessly, and unrelentlessly.

There are also interesting scientific studies of nonwords. Contrary to conventional wisdom that had scientists believing for a while that it takes longer to process meaning of letters embedded in words, and longer still for letters in words in sentences, the contrary is true. In fact, words in sentences do not even need to be spelled correctly to be read. For words of 4 letters or more, all you need is the first and last letters to be correct, and the other letters may be shifted without keeping you from discerning the words. Don’t believe it? Read the passage below:
I cdnuol t blveiee taht I cluod aulaclty uesdnatnrd waht I was rdanieg.
The phaonmneal pweor of the hmuan mnid, aoccdrnig to a rscheearch at Cmabr igde Uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoatnt tihng is taht the frist and lsat ltteer bein the rghit pclae. The rset can be a taotl mses and you can sitll raed it wouthit a porbelm. Tihs is bcuseae the huamn mnid deos not raed ervey lteter by istlef, but the wrod as a wlohe.
Amzanig huh? yaeh and I awlyas tghuhot slpeling was ipmorantt!

But spelling still is important, as it takes longer to read the scrambled passage that an unscrambled one. There are still many unanswered questions about what is really going on when we “read.”

Batastic

Bats. This posting will summarize what I learned about them in Malaysia. For those of you I have already talked to, I apologize for any redundancies.
First of all, to dispel a common myth, Bats are not blind. In fact, only the smaller insectivorous bats (microchiroptera) use the famous echolocation to navigate and hunt. Megachiroptera do not. The order Chiroptera that bats are classified under means “hand-wing” in Latin.
So we were only studying the microchiroptera suborder. We caught them using something called a harptrap that looks just like a harp. It is made by connecting horizontal metal bars with vertical and taught strands of fishing line spaced apart about 2 inches. The bats crashes into the trap, gets tangled and slides down into the bag below. They can’t walk or even crawl, and don’t have enough space to fly out, so they stay there, and are never hurt at all by the process (except sometimes these giant ants would invade the traps and kill them).
After we caught the bats we would “process” them ,which could be as little as reading the band of a recapture, and as much as weighing, banding, measuring bone-lengths and taking tissues samples of new bats. Some bats were very tame and could be easily held and petted, and others would bite themselves if they couldn’t bite you. Hipposideros Cervinus, the bat that we caught the most of was especially feisty. Here some pictures below.


Okay, so back to bat-facts. Most insectivorous bats eat half their body weight in a night of hunting. This is a lot more than most mammals. Considering that many weigh only about 3-14 grams that may not sound like much. Then consider that some bat caves in Texas have an estimated 10 million bats, with each one eating half of its body weight in insects each day. That’s quite an efficient pesticide! It’s free, too. Unfortunately, chemical pesticides are still used, and since bats eat so many of the poisoned bugs, bioamplification is killing them too. Less bats means more pesticides are needed to kill the extra bugs, and the circle becomes a spiral with and unhappy conclusion for bats. This begins to answer the question of what the purpose of a bat-conservation project could be anyway. Bats are also important for pollinating a lot of fruit trees in the old world.
It turns out that bats were the original hosts of the SARS virus too. They didn’t give it humans directly though, or the outbreak would have been a long time ago (as the variations in the virus in bats suggests that they have had it for awhile). No, it was our own fault, as we facilitated the transfer from bat to civet (a cat sized mammal) in the markets of china. These animals don’t interact at all in the wild, but they are both delicacies in parts of China, so they were unnaturally forced into close proximity of each other and humans. Host-hoping to the civet made the virus that gave us SARS. Bats couldn’t give it to us before.
Bats are in fact one of the oldest mammals. They are second in diversity only to rodents, having 1115+ different species! They are the only flying mammals too. It is remarkable how much they evolved so long ago.
Their most famous adaptation is of course how they use sonar to echolate for both navigation and hunting (though in general more for the latter). Like most animal attributes, this has carefully evolved to suit the needs of different bats in different environments. For instance, bats that hunt out in big open space have fewer obstacles to avoid, but also lesser density of prey. Thus they use a low frequency call that attenuates less when it travels far. They also use a long call because then it is easier to notice a subtle change in its broad range.
The bats we studied in the rainforest, on the other hand, have the opposite situation: lots of prey and lots of obstacles. Thus they use short-duration and high frequency calls to a get very detailed reading (it is tempting to say “picture” instead of “reading” here, but it is not fair to assume that they really see it the same way we do when we use radar and sonar imaging) of their surroundings. So precise in fact that they can identify a type of insect by how fast it flutters its wings or how soft its skin is (which is important, because some bugs taste better or are easier to eat than others). They do this by sending out and receiving calls at an amazingly fast rate. Amazing because their calls are so loud (at 120 Decibels!) that they would deafen themselves as they called if they did not lift the cochlea from their ear drum simultaneously. But then they have to pull the ear back together to hear the call on the way back. They can flex the muscle that controls this action about 300 times a minute, which means that they can make just as many calls. This is the fasting-moving muscle in any mammal. The diagram below shows more.

Much more can be said about echolocation (and flight, which I apologize for not talking about, but otherwise this posting would be ridiculously long), and I plan to teach about it to a much great depth in my Precalculus class this year (if interested, ask me and I’ll send you that information when it becomes available).
So remember, bats are important! The vampire bats are only in south America, and very few people each year are hurt by them. The professor that lead our group, Tigga Kingston, starts a professorship at Texas Tech this Fall and can be easily tracked down there if you want to learn more.

Hint for 4-segment problem (Shaky solution revisited)

For any of those of you working of the problem of finding the probability of forming any quadrilateral from a segment being randomly broken into 4 pieces, below is a hint. I will post the full solution later.
Reread the earlier posts if this hint doesn't make any sense to you.

Poetry and Math: moments and removable discontinuities

Below is one of my favorite poems.

This Much I Do Remember:

It was after dinner.
You were talking to me across the table
about something or other,
a greyhound you had seen that day
or a song you liked,

and I was looking past you
over your bare shoulder
at the three oranges lying
on the kitchen counter
next to the small electric bean grinder,
which was also orange,
and the orange and white cruets for vinegar and oil.

All of which converged
into a random still life,
so fastened together by the hasp of color,
and so fixed behind the animated
foreground of your
talking and smiling,
gesturing and pouring wine,
and the camber of your shoulders

that I could feel it being painted within me,
brushed on the wall of my skull,
while the tone of your voice
lifted and fell in its flight,
and the three oranges
remained fixed on the counter
the way stars are said
to be fixed in the universe.

Then all the moments of the past
began to line up behind that moment
and all the moments to come
assembled in front of it in a long row,
giving me reason to believe
that this was a moment I had rescued
from the millions that rush out of sight
into the darkness behind the eyes.

Even after I have forgotten what year it is,
my middle name,
and the meaning of money,
I will still carry in my pocket
the small coin of that moment,
minted in the kingdom
that we pace through every day.

--Billy Collins

A moment is such a hard thing to define. Does it have any length, or only position relative to moments before and after it?
A removable discontinuity in mathmatics and a moment in time have some similar attributes. Take the function graphed below that has a R.D. at x = 2.


We see that even thought there is a small circle marking the discontinuity at (2,4), actually the point has no dimension and thus the discontinuity has location but not length.
If you have taken math past Algebra II, the picture above is probably (I hope!) somewhat familiar, but maybe not all that interesting. Recall this part of the poem:
"Then all the moments of the past
began to line up behind that moment
and all the moments to come
assembled in front of it in a long row"

Too bad mathematics courses so rarely try to build on such beautiful connections like this.

Lumps in the Batter of Mixed Metaphors

This entry will actually cover three topics: defining metaphor, mixed metaphors, and dormant metaphors.
Before discussing the use and abuse of metaphors, it is worth clarifying the definition of metaphor. A metaphor is a figure of speech that directly calls one thing something else, or says that it is that other thing. In grade-school our teachers did a great job of helping us distinguish between simile and metaphor, the former using like or as and being explicit, the latter being implicit. Unfortunately, analogies are not contrasted with metaphors enough.
An analogy is a similarity (or a comparison based on a similarity) between two things that are otherwise dissimilar. While similes, metaphors and analogies all extend meaning through correspondence, typically metaphors and similes give elegant depth to a description, while analogies give practical understanding. This is why analogies are a central tool in teaching and explaining.
Yet too frequently people incorrectly call analogies metaphors. Usually these infractions begin with the phrase “A good metaphor for … is …”.
For example, “A good metaphor for the stages of intimacy with someone is rounding the bases to score a run in baseball.” This is actually an analogy, and an overworked and generally lame one at that. Try to fill the … in the phrase with an example of a metaphor. Semantically this construction is perilous. A safer approach is to start “A good metaphor is …”. Just say the metaphor—then describe it.

Unless being used for purposeful humor (like the title of this posting), mixed metaphors are embarrassing mistakes that leave readers either confused or laughing at (instead of with) the writer. One of my favorites is in a speech by a scientist who referred to “a virgin field pregnant with possibilities.”
Clichés can also spin out of control: “They’ll be watching everything you do with fine-toothed comb.” Sorry, combs can’t watch anything.
A truly amazing foul up is when two expressions are mixed together incorrectly and the result takes on a whole new (and unintended) meaning beyond the confused parts. The website http://www.stuntmonkey.com/metaphor/ calls this a triple wammy. The following is my favorite: “We’re starting from ground zero.” This confuses “starting from square-one/the beginning/zero”, with “building from the ground-up,” to result in dealing with a nuclear attack. Remarkable!

Finally, a dormant metaphor is somewhat like a Freudian slip. It occurs when the literal meaning of the metaphorical language infringes on its extended purpose. This comes from poor contextual placement. For instance, consider this sentence taken from a law journal: “This note examines the doctrine set forth in Roe v. Wade and its progeny.” Since progeny literally means offspring, this is probably not the most appropriate place to use it more loosely referring to a result or product.
On a subtler note, the vehicle (i.e. literal sense of the metaphorical language) should harmonize with the tenor (i.e. intended metaphorical sense) of a metaphor. For instance, “the internet superhighway” makes sense because both highways and the internet are modern technological achievements. “The internet super-trail” falls flat.
However, sometimes the discord is purposeful and effective, as in “concrete jungle”.

Please send in comments with some of your own favorite uses and abuses of metaphors.

Shaky Solution Resolved

Okay, so the solution to the triangle problem is shaky because there are multiple ways of forming the same set of segments. Consider breaking a line segments into three different sized pieces: one short, one medium, and one long, and call each one a, b, and c respectively. There are six possible arrangements of these three segments to make the original:
abc, acd, bac, bca, cab, and cba
Fortunately, the equilateral triangle solution offered before also gives six arrangements for any set of three distinct segments. For example, in the picture shown below each dot gives six identical sets of three segments (well, approximately identical, since I am no graphic artist).

So it was not enough that the sum of the segments is constant and that there were arrangements for all possible sizes; there had to be the same multiplicity of arrangements, which, conveniently, there was.

A more straightforward solution can be achieved using simple algebra and graphing. Let the arbitrary original segment have a length of 1 unit. Let the first of the three segments it is broken into have length x, the second have length y, and the third have length (1 – x – y).
All possibilities are given by the area in the solution to the three inequalities x > 0, y > 0, and x + y < 1 (superfluous are x <1 and y < 1 because x + y < 1)


Okay, so now we consider what region of this area makes a triangle possible, or, as is common in probability problems, what makes a triangle not possible. To make a triangle none of the segments can be longer than the other two put together. In this problem, where we start with a segment that has a length of 1 unit, we cannot let any of the pieces be longer than 1/2 a unit, or it will be longer than the other two combined, and a triangle cannot be formed. This gives the following three inequalities for what each piece can be:
x < ½, y < ½, and x + y > ½ (from the inequality for the third side: 1 – x – y < ½) This gives the region in yellow in the picture below:

And once again it is clear (but now even clearer) that the probability is 25%. Funny how a similar picture results from a completely different process…

There is another creative approach to solving this problem that I found. It involves bending the segment to make a circle and considering which placements of the two breaking points on the circle-segment will work. I’ll post this sometime in the future.

Like this problem? Try the next one that it leads to. If you break a line segment into 4 pieces, what is the probability that you can make a quadrilateral?
Still having fun? How about if you break a segment into n-pieces?
I’ll revisit these.

A Beautiful but Shaky Solution

There is something so good feeling about finding a simple and elegant solution to a difficult problem in mathematics. Indeed, it is most impressive in any field when someone can make abstruse concepts understandable to people who are not experts in the field (Richard Feinman is famous for this).
If the solution or explanation is faulty or incomplete, however, the supplier has done a disservice that can outweigh whatever good might have come giving the correct answer. It is still more frustrating is if the wrongly-reasoned conclusion happens to lead to the correct result. This allows the misconception to be perpetuated.
A great example is the following “solution” to a famous math problem. It does work, but there are some dangerous assumptions (or overlooked details) that can lead someone astray if they tried to use a similar method for a different problem. Here it goes.

Question: If you randomly break a line segment twice, what is the probability that you can make a triangle from the three resulting segments?

Solution: Start by inscribing an equilateral triangle inside a larger equilateral triangle as shown below:



Now pick a point anywhere in the big triangle, and draw three segments connecting the point to each side of the big triangle at perpendicular angles. Wherever you pick a point, the sum of the drawn segments is always the same (for instance, sum=sqrt(3), if each small triangle's side length is 1 unit) so they can represent three segments we get from randomly cutting some segment twice (as stated in the original question). Look below:


If the chosen point is in the center equilateral triangle (like the yellow dot), then you can make a triangle from the three segments. If it is in one of the other three small trangles (like the blue dot), then you cannot make a triangle from the segments. Since the center triangle’s area is 25% of the total area, the probability of being able to make a triangle is also 25%.


Okay, so it is true that the probability is 25%, but can you find it by alternate means? Can you explain what is shaky about this solution?

I’ll post other solutions in one week.

Nuanced Intensifiers

The English language has many wonderful intensifiers. This gives the able speaker or writer the ability to be acutely specific. This is why the word very is so dreaded—it has no flavor. Just like flavors, though, intensifiers can taste bad if combined improperly.
A word that comes to mind is highly. Highly is frequently substituted for a simpler adjective to give some loftiness (figuratively speaking, that is) to another adjective. “Highly intelligent” is perhaps the most common and clichéd, but at least it makes sense. Examples of less logical uses abound. For instance, one hears of “highly unmotivated students” (especially these days). How about “heavily unmotivated students”? Doesn’t that sound a bit more accurate? Sadly, a Google search retrieves 514 hits for “highly unmotivated,” and only 12 for “heavily unmotivated”.
Another example involves discounts. “Highly discounted” is an especially illogical and unappealing advertisement. Once again, “heavily discounted” rings much truer. Fortunately, businesses have caught on to this one as “heavily discounted” gets 495,000 hits compared to only 191,000 for “highly discounted”.
Here are some other nuanced pairs of intensifiers to consider:

particularly and especially
quite and very
considerably and significantly
severely and tragically