## Archive for November, 2012

### Upside-Down Tooth Numbers

Alex and Eli know that 15 ÷ 3 = 5, that 63 ÷ 7 = 9, and, given a little time, could figure out that 104 ÷ 8 = 13. That’s not bad for five-year-olds.

As we were discussing sharing a treat the other day, we naturally happened upon a situation in which it would be good to know what 1 ÷ 2 is.

Alex suggested, “Two.”

I explained that while the order of the numbers in multiplication doesn’t matter — for example, 2 × 3 = 3 × 2 — order does matter with division. I used the word *commutative*, but I also tried to explain it with plainer language, too.

I took a rectangular piece of paper and ripped it in half. “How big is each piece?” I asked. The both knew it was one-half. “So there you have it: 1 ÷ 2 = 1/2.”

But why stop there? I divided one of the halves in half again, and I asked, “How big is this piece?” They both knew it was one-quarter. “So that shows that 1/2 ÷ 2 = 1/4.”

They saw that if we continued in this manner, we would get 1/8, 1/16, 1/32, and so on, a pattern they called the *upside-down tooth numbers*, because the numbers in the sequence are the reciprocals of the powers of two. (For them, *tooth* = 2^{th}.)

Looking at the pieces of paper on the table, I asked a more advance question. “What do you think we’d get if we added 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …, and we kept adding half as much each time?”

Eli thought for a few seconds. “I think it would be one whole,” he offered.

I was surprised. “Why?” I asked.

“Well,” he said, “I’m thinking about a circle.”

I looked at the rectangular pieces of paper on the table, which had nothing to do with a circle. *Sure, he got the right answer, but his reasoning is way off base*, I thought.

He explained further. “If you fill half a circle, then a quarter more, then an eighth, and keep going, you’ll eventually fill the whole thing. That’s why I think it’s one.”

Visually, Eli’s argument would look something like this…

*Holy sh*t*, I thought. *That’s pretty good for a five-year-old.*

I don’t even care that he doesn’t understand this joke:

An infinite number of mathematicians walk into a bar. The first asks for half a beer. The second asks for a quarter of a beer. The third asks for an eighth of a beer. The bartender interrupts, pours one beer for them, and says, “You guys don’t know your limits.”

I think I was so impressed with Eli’s solution because infinite series can be difficult, even for mathy professionals:

A mathematician will call an infinite series convergent if its terms go to zero. An engineer will call it convergent if the first term is finite.

And let us not forget the Eilenburg swindle, which proves that 1 = 0 since:

1 − 1 + 1 − 1 + … = 1 + (−1 + 1) + (−1 + 1) + … =

1

and

1 − 1 + 1 − 1 + … = (1 − 1) + (1 − 1) + … =

0

Related to all this, there is a perhaps apocyphal story in which it is rumored that a student once posed the following problem to John von Neumann:

Two trains begin a mile apart and head towards each other at 60 miles an hour. A fly on one train flies at 120 mph to the other train, and when it touches the other train, it immediately turns around and flies back to the first train, and so on, flying back and forth between the two trains until it gets squashed in the middle. How far does the fly travel?

von Neumann thought about it a moment and said, “One mile.” The student said to von Neumann that most people don’t realize that the problem can be figured out easily: the trains meet in 30 seconds, and the fly can travel one mile in half a minute; yet most people think they have to add up the infinite series to figure out how far the fly travels. As the story goes, von Neumann replied, “But that’s how I did it.”

### Math on Monday Night Football

Tonight’s Eagles-Panthers game had several mathematical incidents.

**A Counting Problem**

During pre-game warm-ups, a voice-over quoted one of the player’s thusly:

You want me to describe playing on Monday night in one word?

Prime time.

Computer scientists start counting at 0. Apparently pro football players start counting at 2.

Speaking of counting… did you notice that the sentence above had three hyphenated words? That’s just crazy.

**Flippin’ Out**

During the opening kick-off, Mike Tirico mentioned that the Carolina Panthers lost the coin toss. That alone is not exceptional, but it was the *eleventh straight game* that they had lost the flip. The odds of a team being that unlucky? How about 2,047 to 1?

Perhaps they can blame bad luck for their eight losses this season, too.

**Numerically Interesting Milestone**

Wide receiver Steve Smith of the Panthers caught a pass in the first half that took him to 745 career receptions for a total of 11,011 yards. What a cool number! First, it’s a palindrome. Second, 11011_{2} = 27_{10}, and he currently ranks 27th among wide receivers in career receiving yards. That’s a pretty fun coincidence.

### Throw a Few Logs on the Fire

When teaching apportionment and the Balinski-Young Impossibility Theorem to a group of gifted middle school students, it was necessary to discuss the geometric mean. Once the door was open, this was an opportunity to discuss logarithms, too. In particular, the geometric mean of two numbers is equal to the arithmetic mean of their logarithms.

As a trivial example of this property, the geometric mean of 10^{1} and 10^{3} is 10^{2}, and the arithmetic mean of 1 and 3 is 2. But you probably saw that coming. It’s a bit more powerful to use less nice numbers. For instance, the geometric mean of 10^{1} and 10^{2} is approximately 31.62, and the arithmetic mean of 1 and 2 is 1.5. So what, right? Well, the kids in my class gasped audibly when their calculators revealed that 10^{1.5} = 31.62.

One of the students even whispered, “Oh, man, that is SO cool.”

Though that wasn’t my best lesson, that was certainly the best reaction I ever received from a classroom of students. Quite honestly, it surprised me. But recently, cognitive scientists at MIT have theorized that it may be more natural for humans to think logarithmically than linearly, so perhaps there’s a genetic reason that kids think it’s cool. Apparently, both young kids and people from some tradditional societies will say that 3 is the number halfway between 1 and 9.

Of course, logrithms are great fodder for math jokes…

Lumberjacks make good mathematicians because of their logarithms.

Although some people think that a logarithm is a birth control method for foresters.

This, of course, is the natural log: ln(*x*). And this is an unnatural log…

But as we all know, a logarithm is nothing more than a misspelled algorithm.

### Gobble Up Some Math Fun

How’s this for a change? I’m actually going to start this post with a joke…

What did the mathematician say after finishing Thanksgiving dinner?

(I overate).

The following turkey was made entirely from pattern blocks:

For some wholesome family fun, try to construct a pattern block turkey with the following:

- three hexagons
- three trapezoids
- four triangles
- five fat rhombi
- five skinny rhombi

If you don’t have a bucket of pattern blocks at your disposal, download the following template from the MJ4MF website, copy it onto cardstock, and cut out the 20 shapes you’ll need:

If that puzzle doesn’t give you enough to think about, here are a few quotes that might:

If you want to make an apple pie from scratch,

you must first invent the universe.

~ Carl SaganThanksgiving dinners take 18 hours to prepare. They are consumed in 12 minutes. Halftimes take 12 minutes. This is not coincidence.

~ Erma Bombeck

Happy Turkey Day!

### Math Haiku and Limericks

Haiku have 17 syllables, right? Nope. They actually have 17 *morae*. Don’t know what a mora is? Don’t worry; neither do most linguists.

I find the 5-7-5 structure of haiku too restrictive, and apparently Roger McGough does, too.

The only problem

with haiku is you just get

started and then

~ Roger McGough

And Daniel Mathews thinks the structure is problematic for writing math haiku.

Maths haikus are hard

All the words are much too big

Likehomeomorphic.

~ Daniel Mathews

Limericks are a little more forgiving. With five lines in an AABBA pattern, you have a little more time to develop a story. Or not.

There was a young man from Peru

Whose limericks stopped at line two.

If you’re at a cocktail party, and you want to deliver the following one-liner, you better set it up with the two-liner above.

There was a young man from Verdun.

“Then there’s the one about the Emperor Nero,” quipped poets Elliott Moreton and Carl Muckenhoupt.

Personally, I think it’s pretty fun to turn traditional poetry rules on their ear. Here is a tradition-busting limerick for you.

A poet through efforts concerted

Ignored all the rules

He learned in the schools

Tradition he oft times skirted

And wrote all his limericks inverted.

And lest haiku feel neglected as a poetic form, here’s an abomination of that type, too.

The last line goes here.

It’s still 5-7-5, but…

Haiku inverted.

### Binarily We Roll Along…

If you were born after November 11, 1911, and before the start of this millenium, then you’ve experienced 36 binary dates in your lifetime; there were 9 each in 2000, 2001, 2010, and 2011. I’ve collected several binary jokes, and I was going to save them until the next binary date… but since that’s not until January 1, 2100 — a date for which I most likely I won’t be around; and if I am, God help me if I’m still trolling out math jokes on this blog — I figure I better share these jokes now.

I don’t know how many jokes are contained in this post, but you can count them. Luckily, counting in binary is as easy as 1, 10, 11, …

Perhaps my favorite binary joke:

01000001 00100000 01101101 01100001 01101110 00100000

01110111 01100001 01101100 01101011 01110011 00100000

01101001 01101110 01110100 01101111 00100000 01100001

00100000 01100010 01110010 01100001 00101110

A *binary* is a two-headed canary.

When Paul asked Saul how much he’d pay for the following book, Saul answered, “Two bits!”

The above silliness is a follow-up to the Gauss^{®} jeans ad that I created and the Leibniz jeans ad that was posted at Rhapsody in Numbers.

### Math Coinky-Dinks

I’m a math guy, so I know that most coincidences are nothing more than people making a big deal out of something that, in fact, is quite likely. I’m not impressed when two people at a cocktail party have the same birthday or when nearly 30% of the people at that same party have a street address that begins with the digit 1.

Nor was I impressed when the Oregon newspaper *The* *Colombian* printed a winning number for the state lottery *in advance*. The probability that the number they accidentally printed on June 27, 2000, which was 6-8-5-5, would actually win the Pick 4 game the following day was 1/10,000. Not likely, to be sure, but not out of the question.

But is it just a coincidence that Douglas Adams claimed that 42 is “the answer to life, the universe, and everything,” and that Oreo cookies can be obtained by pressing 42 on the vending machine in my office?

And is it just a coincidence that ELEVEN + TWO = TWELVE + ONE?

Well, yeah. Probably.

**But something happened yesterday that was so strange, it cannot be brushed aside as mere coincidence.**

My son Alex was home sick from school. Around two o’clock, he said, “Daddy, I smell blood.” I checked to make sure he wasn’t bleeding… then I checked to make sure that I wasn’t bleeding, either. There was no blood to be found. A couple of hours later, we went to pick up his twin brother Eli at school, and Miss Vanessa at after-school care told me that Eli had an accident.

“He fell and hurt his knee,” she said, “and there was blood everywhere.”

Blood? I asked what time that happened. “Around two o’clock,” she said.

*Freaky.*

With twin boys, I suspect that there will be similar coincidences in the future. For instance, I suspect that I will one day receive a call saying that both boys were caught in a co-ed dorm after curfew. How weird would that be?

But I’m not phased. Coincidences are very common in my family. For example, my mother and father got married on the same day!

To check out some truly random statistical coincidences, click on over to www.coincidenceithinknot.com.

The following joke is based on a fun math coincidence.

Saul: It’s -40 outside.

Paul: Fahrenheit or Celsius?

Saul: When it’s that cold, it’s impossible to tell the difference.

It’s just a coincidence that -40° F = -40° C.

Or is it?