Calculating Fahrenheit

The post about the math mistake in temperature conversation reminded me of a formula that a friend told me about (thanks DSD!).  She was traveling abroad, and the guide she was with said that to convert Celsius to Fahrenheit people should use the formula:

Double the Celsius, and subtract from it the amount obtained by moving the decimal place one unit to the left.  Then add 32 to get the corresponding Fahrenheit.

For example, with a temperature like 50°C, you’d double 50 to get 100, then from that subtract 10.0 to get 90.  Finally, you’d add 32 to 90 to get 122°F.

This is equivalent to the formula
Temp in °F = (9/5) (Temp in °C) + 32.

In particular, if C is the temperature in Celsius, the description to double and the subtract that amount with the decimal place moved describes 2C – 0.1(2C), which is 1.8C, or 9/5C.

It does seem to me to be quicker to compute 9/5C by doubling C and subtracting a tenth of the result than to multiple by 9 and divide by 5 in some order.  The conversion isn’t as quick as “Double and add 30″*, perhaps, but unlike that estimation it has the advantage of being exact.

*a formula that always brings to mind the movie Strange Brew

The thermometer is by Bernard Gagnon – Own work, CC BY-SA 3.0. It has Centigrade rather than Celsius at the top, which I found interesting since I remember learning both terms in school.

Have a power-ful day

Today is 2/4/16 or 4/2/16, depending on where you live and how you write dates.  Either way, it’s a great day because 2⁴=16 and 4²=16.  There aren’t many days like that (although we are treated to two this year), so it’s worth taking a moment to celebrate.

Gene sharing math

The other day I found myself wondering what proportion of genes cousins would expect to share compared to biological siblings.  This took more time to figure out than I would have expected, in part because I knew that siblings who share two parents have 1/2 their genes in common on average, so I thought cousins sharing two grandparents might have a quarter.  They don’t, though – it’s half that.  In reasoning it out, it turned out to be easiest to think of moving up the biological family tree to a common ancestor, which led to one general formula and a few specific cases:

The Generalization:

Given two people A and B, find their closest common ancestor C.  If there are n generations from A to C and m generations from B to C, then the expected proportion of shared genes is (½)^n+m.  If there are two closest common ancestors (for example, both parents) then this number would double.

In the case of a parent and child, for example, there is 1 generation from the child to the parent (the common ancestor) and 0 from the parent to itself, so the proportion of shared genes would be (½)¹, or just ½.  Cousins would each be 2 generations from common grandparents, leading to (½)⁴, or 1/16, for cousins with one grandparent in common (sometimes called half cousins) and twice that for cousins with two grandparents in common (sometimes called full cousins).  Double cousins  — that is, people who are cousins on both sides of the family tree (for example, cousins whose mothers are sisters and whose fathers are brothers) — would still have grandparents as the closest common ancestor, but now it would be up to four common grandparents instead of just one or two: the expected proportion of shared genes between cousins with four common grandparents would be 4·(½)⁴, or just ¼.  Likewise, an aunt and nephew with two parents/grandparents in common would be 1 and 2 generations respectively from this pair of common ancestors, so the expected proportion of shared genes would be 2·(½)³, also ¼.

Special Case 1:  great-great-…-great grandparents
In this case the older relative is the common ancestor, so if “g” is the number of “great”s then the proportion of shared genes is (½)^g+2.  The additional 2 in the exponent is because the number of “great”s counts the generations after grandparents, who are already 2 generations away from their grandchildren.  This is the only case where the proportion is exact:  in all the others, it’s only an expected proportion because siblings could have anywhere from no overlap of genes to complete overlap of genes from each common parent.

Special Case 2:  great-great-…-great aunts and uncles
In this case the older relative’s parent(s) are the common ancestor.  With a great-uncle and great-niece, for example, the great-uncle’s parent(s) are the great-grandparent(s) of the great-nephew.  This means that there is 1 generation from the great-uncle to his parent(s), but 3 from the great-niece to that common ancestor, with each additional “great” adding another generation.  If “g” is the number of “great”s, then the expected proportion of shared genes would be (½)^g+3 if there is one parent in common, and (½)^g+2 if there are two.  (I personally find it interesting that you can expect to share the same proportion of genes with a sibling who shares both parents as you do with either of the individual parents, the same proportion with an aunt or uncle who shares both grandparents as you do with either of the individual grandparents, and the same proportion with a great-great-…-great aunt/uncle who shares both great-great-…-great grandparents as you do with either of those great-great-…-great grandparents themselves.)

One clarification:  great-aunt is the term I grew up with, but in looking around I just discovered that “grand-aunt” may be the technically correct term, since that person is in the same generation as a grandparent; likewise, the sister of a great-grandparent would be a great-grand-aunt.  This appeals to me aesthetically.   If you were to use these terms, then you’d have one fewer “great” in describing the relationship, and you’d need to add 1 to the exponent in the formulas above.

Special Case 3:  second cousins once removed (and the like)
Cousins share at least one grandparent, second cousins share at least one great-grandparent, and x^th cousins share at least one great^(x-1) grandparents.  This means that x^th cousins are each (x+1) generation removed from the common ancestor(s), and would expect to share (½)^2x+2 of their genes if there is one common relative and (½)^2x+1 if there are two. Each removal  refers to one of the people being one more generation removed from any common ancestors, and so increases the power of ½ by 1.  This means that x^th cousins who are y-times removed would expect to share (½)^2x+y+2 of their genes if there is one common relative and (½)^2x+y+1 if there are two. Second cousins once removed would share either (½)⁷ or (½)⁶ of their genes, while first cousins twice removed would share (½)⁶ or (½)⁵.

For those who like the visual, there is a handy little chart below, which appears to be in the public domain on Wikipedia. It does make some assumptions, however – namely, that siblings, cousins, aunts and nieces, etc. have exactly two closest relatives in common (both parents, two grandparents, etc.).

Wedding Cake WIN

A few days ago while in a yarn shop I ran into one of our alumni from a few years ago.  Seeing her was fantastic, but the icing on the cake (Sorry.  Sort of.) was that she had gotten married last summer to a biochem major, and they designed their own wedding cake.  She sent me a couple photos the next day of what is perhaps the coolest wedding cake I’ve ever seen:

Here’s a detail:

Hey, I can answer that question!  I’m not as sure about the next one, though.

Congratulations (and Happy Anniversary!) to Emily and Glen — I can think of no better way to start a marriage.

Props to the photographer, Hilary Argentieri, for taking such clear picture!

Oddly, this isn’t even our first math wedding post (see this mathy proposal) although it’s the very first one involving cake.

 

Edited 9/19 to add:  Apparently this came from xkcd — I think that makes me like it all the more.

How many before the end?

Let’s start with ultimate (yes, that seems a bit backwards – stay with me.), as in “last”.  The next-to-last, then, is penultimate, easily one of my favorite words.  But there’s more!  The next-to-next-to-last is the antepenultimate.  Need another?  The -last is the preantepenultimate, a completely real word that Chrome’s built-in dictionary has never seen (it suggests “prearrangement”).

Why am I telling you all this?  I mean, besides the sheer ridiculousness/awesomeness of a word for the fourth-to-last item in a list?  I needed an excuse to post this video:

Alpha’s Curious Filter

For no reason that I can think of, I decided to see how much Wolfram Alpha knew about probability, so I typed “probability of a full house” into the search box and got the following:

I thought that was pretty cool, especially since it includes the derivations, so I asked a few more questions, such as “probability of at least 2 red cards in a 5 card hand“:

Odd that it will count the numerator but not the (easier) denominator .  At this point, I thought I’d try a standard probability question (balls in an urn) that might be harder to parse because of the additional statements: “probability of drawing a blue ball from an urn contaiing 5 blue balls and 7 red balls“.  However, I missed the ‘n’ key when typing “containing” and got the following:

So, yeah, OK, Wolfram Alpha doesn’t provide “adult” content (why the quotes?), and I’m pretty sure I know what it’s reading as “adult”, but c’mon.  Note that fixing the typo doesn’t alleviate the problem, but it does cause Alpha to hiccup and request more computing time.  With variations on the wording, I’ve also had it return a picture of a blue ball along with the HTML code to generate it.  Nice.

Twelves

The number 1729 has a right to be proud :  it initially had only a small role on a taxicab in England but its super-power of being the sum of  two positive cubes in not one but two ways (1³+12³ and 9³+10³) led to a big break in a Feature Story starring GH Hardy and Srinivasa Ramanujan, with follow-up appearances for years to come on the likes of Futurama and Proof.  So, you know, yay 1729.

But lest this Hardy-Ramanjuan number get too boastful, it’s not the only sequin at the Oscars.  Its neighbor, that unassuming 1728, turns out to be an interesting character in its own right.

The origin of this is in the dozen.  Although ten is a pretty natural base to use, in the sense that a lot of cultures break numbers up by tens in some form, it’s not the only possibility.   We have not only a special word for 12 (dozen), but a special word for 12² (gross), which suggests that our language carries hints of a Base 12 system.  And that leads to the question:  is there a special name for 12³?

There is!  The official name is a Great Gross.   And while dozen and gross show up in egg cartons, it’s in measurement that the great gross really shines:  there are a dozen inches in a foot, a gross square inches in a square foot, and a great gross cubic inches in a cubic foot.

But while the great gross is helping out with set design, there’s a rumor (which we’re apparently happy to help spread) that 1728 actually has a stage name.  That’s because there’s a theorem about L-functions of elliptic curves called the Gross-Zagier Theorem, named after Benedict Gross and Don Zagier.  So the natural extension of a gross is…a Zagier!  Or at least that’s the name that 1728 goes by on the cocktail circuit according to Wikipedia, our local gossip rag.  Which makes us wonder where this down-to-earth yet whimsical number will show up next.

In an amusing turn of events, it turns out that Gross and Zagier won the Frank Nelson Cole prize in Number Theory in 1987 from the American Mathematical Society for their paper “Heegner points and derivatives of L-series” which contained the above theorem.  The other winner that year for a different paper was Dorian M. Goldfeld who, the following year, published a paper with M. Anshel entitled “Applications of the Hardy-Ramanujan partition theory to linear diophantine problems,” bringing it all back full-circles to the people who made 1729 famous.   It’s like one giant family reunion.

Greek Math (OK, just Greek)

TwoPi and I just got back from a long-anticipated trip across the Atlantic.  And, except for the fact that camels are about fifty times taller than they look and really really scary to ride (unless you’re ten.  Then, apparently, they’re totally cool.) the trip was amazing.  Especially because we found mathy things, and who doesn’t like mathy things???

This, by the way, might be my favorite photo:

Two of the days we spent in Greece, and everything was in Greek.   Which is obvious, but it made it seem like there was math everywhere.  Even on Sprite bottles.

Plus a lot of the signs were posted in both Greek and Latin alphabets, so I could try to sound out the Greek and then see if I was right.  [I spent a similar amount of time reading signs in Montreal, once, and then checking myself on the English subtitles].

It was like all these years I’ve spent learning and teaching math symbols paid off in a completely unexpected way.  (Even though, ummm, joining a sorority might have had the same effect.  But I digress.)

Unfortunately, with only a couple days, we didn’t do anything that had any actual math content during this portion of the trip.  But I did find this sign, which made me really happy.  (I blacked out the Latin part so that you could sound it out.)

 

Leading digits

If you want to go to the Vatican Museums (home of, among other things, the Sistine Chapel) you can buy your tickets in advance online. During the checkout process they list the number of Euros you owe, but they also include a few leading zeros. Having a computer that allows room for all the digits you might need is a good thing, as anyone who remembers the whole Y2K thing realizes. So kudos to Vatican City for making sure that no transaction will be too large for the checkout system to manage. But seriously, just how much money did they expect people to spend in those online transactions???

 

What transpired at the JMM

Last month we skipped the first week of classes and high-tailed it for the west coast, to the Joint Mathematics Meetings in San Francisco.  The entire city was abuzz with excitement over this event, as evidenced by the San Francisco International Airport which had an entire ceiling thing devoted to mathematical symbols:

And the meetings were fun, which they always are.  I even got my very own sketch of a Brown Sharpie scaring Godzilla, thanks to Courtney (in response to my lovely rendition of Godzilla threatening to eat said sharpie).

As it happened, the last day was the one that turned out to be the most full of surprises.  And the best one — better even than my last minute “Heylet’sgoseeWickedit’srightdownthestreet!”, which we did, but with muted enjoyment because we’d already checked out of the hotel and had all of our luggage which we had to cram into our seats under our feet and whoa! those theater seats are place pretty close together, with no consideration for people who might bring all their worldly belongings to the theater — happened that last afternoon of the meetings, after the exhibits had shut down and the message board area was starting to get that Morning After look.

What happened is that I went to the tables to hang out for a bit,  and saw my friend Karrolyne at a table.  She and I were roommates in Project NExT way back when, so I sat down and we chatted and the conversation turned to kids.  There was another woman at the table that neither of us knew and she joined in the conversation.   The three of us continued to chat about our different and common experiences, and at one point Mystery Mathematician commented about talking to people she’d never see again, and Karrolyne laughed and said to watch out, because you never know!  The world is small.

A little later, when we’d turned to other topics, I gave Mystery Mathematician my email address so she could send me some math info, and she exclaimed that she recognized the address.  Hmmmm.  Very strange — we didn’t remember meeting before.  Then Karrolyne had to leave and while I was saying goodbye to her, Mystery Mathematician figured out how we knew each other.  It was from HERE!  She jumped up smiling and said “I’m Math Mama Writes!”  Sure enough, this person I’d just been hanging out with was none other than Sue VanHattum, and in fact we’d exchanged some emails recently (hence the recognition of my address) but had never met in person.  Indeed, she didn’t even recognize my name, nor TwoPi [who stopped by the table briefly while we were talking], because we’re all mysterious with the alias’s here [I was “heather360” for about a day when Batman and I started the blog, and then got bored with that and switched to Ξ]. Of course, that raises the question of why I didn’t recognize hers, but I can only guess that it’s because subconsciously I figured she lived in my computer screen in New York.

All in all it was a fantastic surprise.  And though we didn’t have a lot longer to talk — as it was I made it to TwoPi’s talk with about 20 seconds to spare — it was a delight to know that she was just as fun to talk to in real life as I would expect from her blog.

Small world, indeed.

So what’s interesting about 2010?

It seems (perhaps only to me) like it ought to factor nicely, because 20 is twice 10.  But once you factor out that 10 and that 3 you’re just left with a prime, since 2010 is just 2·3·5·67. (Speaking of four primes, did you know that you can get four prime New York Steaks for $132.95 on Amazon?  I was relieved to see that they were not available for Super Saver Shipping.)

Wolfram Alpha points out that 2010 itself is a factor of 29⁶-1.  And  Number Gossip adds that it’s untouchable, which means that there aren’t any numbers whose proper divisors add up to 2010.

It can be written as 133122 in Base 4, which is kind of cool, and as 6, 3, 12 in Base 18; my favorite, however, is that it is 5, 10, 15 in Base 19.

Finally, it’s equal to:
669+670+671
400+401+402+403+404
127+128+…+141 and several others
[Hmmm…I can find a string for each of the 7 odd factors, but I’m not sure that exhausts all of the possibilities.]

While getting ready to post this, I noticed that MathNotations has a similar post from yesterday.   Whoops!

How would Michael Palin say “Quod erat demonstrandum”?

The letters QED (or the unabbreviated “quod erat demonstrandum” [“which is what was to be proved”]) are often used to indicate the end of a formal proof.  This longstanding tradition goes back to a style of proof writing, where the culminating sentence of the argument gives a recapitulation of the statement of the theorem; the QED places a stamp of finality on the discussion.

In modern typesetting, the QED has been largely replaced with typographic symbols; typically a solid or hollow rectangle or square is used to demark the end of the proof.  (The cynic in me wonders if these just serve as flags for when the reader should take up reading carefully again.)

How does one indicate the end of a proof in a classroom setting?  Often I will scribble out a square (or whatever symbol our textbook uses); sometimes I’ve written out “Q.E.D.”.  Often I’ll pause, then solicit questions and comments.

But apparently I much more frequently channel Michael Palin.

Today I gave a final exam in Real Analysis II.  This group of students has gotten to work with me on proofs for a full year, so they know my quirks and foibles better than most.

On the last page of the final, most of the students ended their last proof with the phrase:

“And there was much rejoicing.”

This isn’t a phrase I’ve consciously chosen to use in class, but it rings true enough as something I’m sure I *have* said on occasion.  But if it made this much of an impression on the students, I wonder if I use it all the time, not just occasionally.  Hmmm.

Come to think of it, it isn’t such a bad replacement for Quod erat demonstrandum.  Captures the feeling of a good solid proof, well-understood.  And it is much more evocative than a box.

Two things I don’t know

These are not the only two things I don’t know, mind you, but they’re two things I want to know, that I’ve tried to find out, but which are failing to succumb to the magic of the Internet.

The first is a spinoff of yesterday’s post, in which I quoted a paraphrase of Nixon’s from the Fall of 1972 in which he said that the rate of increase of inflation was decreasing [which, since inflation measures the change in prices, amounts to saying that a third derivative is negative].   Although this was just a postscript, it got me curious as to exactly how he’d phrased it.  So I looked, figuring that Google would turn up something and…it did, but nothing about that speech.    There is something sort of close in this speech from February 1, 1971 in the Annual Message to the Congress: The Economic Report of the President:

Fiscal policy should do its share in promoting economic expansion, and our proposed budget would do that. But fiscal policy cannot undertake the responsibility of doing by itself everything needed for economic expansion in the near future. To try to do that would drive taxes and expenditures off the course that is needed for the longer run. The task of economic stabilization must be accomplished by a concert of economic policies. The combined use of these policies, starting near the beginning of 1969, finally checked the accelerating inflation that had kept the economy overheated for years. [bold added]

See, it uses the word “acceleration”!  Which is the main word that stood out, because I’m afraid that trying to sort through speeches by Nixon for references to inflation is a wee bit mind-numbing.

This 1971 speech occurred more than a year before the one referenced in the quote paraphrase paraphrased quote; the timing, however, seems to match the economic data (well, sort of).  From Inflationdata, the average inflation rate was:
2.79% in 1967
4.27% in 1968
5.46% in 1969
5.84% in 1970  [And somehow I bet that fact that the increase was decreasing wasn’t so reassuring at this point]
But in 1971 it started to go back down, so that in the Fall of 1972 it was back to 2-3% levels.  That doesn’t quite match the claim that the increase was decreasing then — you could just say that inflation was decreasing.  But then again, it doesn’t look like inflation was accelerating per se either — it was mostly decelerating prior to his 1971 speech.

So the end result is that I have no idea which speech it was, nor am I sure that it was right in any case.

That’s the first thing.  The second is a minor point.  I was just reading a student paper about the secret society Bourbaki [the paper came with a short film she made re-enacting the start of Bourbaki!] and there was a reference to a curvy Z-like symbol that Bourbaki used to use, to signify “dangerous bends” in the road where it woudl be easy to get lost.  I was curious as to what it would look like, but the closest I could find was the adaptation that Donald Knuth used: Which is all well and good, except that I’m curious as to whether the black curvy part is identical to Bourbaki, or merely inspired by “him”.  Maybe a search through online books is the next way to go for that search.

DIY Beanbags, or Tiling a Sphere

As an avid juggler, I have a rather large supply of props to juggle, most of which are balls or beanbags.  (Also on the shelves: clubs, rings, scarves, devil sticks, and a diabolo.)  As anyone who’s ever bought juggling equipment can tell you, this stuff isn’t cheap: decent beanbags can run $10-15 apiece, and rings and clubs are much more expensive.  So when I discovered these instructions to make your own beanbags, I was understandably excited.  (Of course, I’ll have to ask Batwoman to sew the pieces together for me.  Needles have a tendency to end up stuck in me instead of the fabric.)

Then I started clicking around the IJDb, and I found these.  Marylis Ramos has clearly spent a lot of time thinking about tiling a sphere.  Certainly any Platonic or Archimedean solid can be adapted to a sewing pattern to approximate a sphere, but a great deal of experimentation among jugglers and sewers has led to only a few becoming popular: the tetrahedron, cube, dodecahedron, icosahedron (one of the best, but really hard to sew), truncated tetrahedron, cuboctahedron, and the “lemon” (with 3, 4, 5, 6, or 8 panels).

The pictures are, of course, idealized beanbags with perfect 1-dimensional seams that cannot be achieved by terrestrial sewing machines (at least not the Singer in my basement).  Or maybe they’re just from Wikipedia’s spherical polyhedron page.

I’ll be making some 4-panel lemons, and I’ll post a follow-up to discuss their sphericity (or lack thereof).

Detangling the Math

You know how some math problems start off simple and then, the more you look at them, the more complex they appear, with hidden depths and meanings?

This isn’t one of those.  This one needs a sign with “No diving allowed.”  Nonetheless, it was surprising to me in its shallowness (like the joke about the smallest not-interesting whole number:  hey, isn’t that interesting?).

I wore my hair down yesterday, but I had to braid it at night because I knew it it would get tangled up something fierce if I slept with it down.  That got me wondering about the mathematical reason for the number of tangles.  In particular, how does the number of possible tangles decrease as you partition hair into three segments for braiding, assuming that tangles could still occur within a segment but not between the three segments?

Now with braids, a big reason that there isn’t much tangling is because of the added tension on the strands of hair.  So let’s switch to something else instead:  spaghetti.

Suppose you put strands of spaghetti into 3 pots, and counted the possible pairs that could tangle.  There would be pairs in each pot, for a total of pairs overall.  On the other hand, if you’d dumped all pieces into a single pot, there would be pairs.

If you look at the ratio:

you get a lot of canceling, and it simplifies to

which, since spaghetti and hair both have relatively large values of , is essentially 3.  There’s nothing special about 3 either:  if you split the spaghetti into pots first, you’d end up with as many potential tangles.

Why isn’t there a square root?  There should be, I thought, because the number of pairs increases like a quadratic as you add more strands.  But as TwoPi pointed out, when I interrupted his reading to discuss this problem, each pot or group of strands has roughly as many distinct pairs as the total, but there are also pots, and that’s where the comes from.  Or, as I thought about it, each individual strand can only tangle with as many strands if you split them up and that carries over into the totals.

So in the end, since I end up with far fewer than 1/3 as many tangles by braiding, it turns out that it has very little to do with the mathematics and pretty much everything to do with the tension in the braid.  I’m afraid that in the great Math-Physics Match, this round goes to Physics.

Isn’t that a great picture from Harper’s Bazaar?  It’s from 1868.   The spaghetti is published under GNU-FDL by Tim ‘Avatar’ Bartel.