Monday Morning Math: the First Digit Law

October 3, 2022 by

You can catch criminals with math!  You might expect that if you were to record a bunch of numbers, the first digit would be equally likely to be 1, 2, 3, 4, 5, 6, 7, 8, or 9, but it turns out that for many numbers (costs for a company, time spent working on something) the first digit is usually small:  1 is the first digit about 30% of the time, while 9 is the first digit less than 5% of the time! Here’s a picture of the distribution:

from Gknor 

This rule is generally known as the First Digit Law, although it is also called Benford’s Law after Frank Benford (who himself called it “The Law of Anomalous Numbers” in a 1938 paper) or the Newcomb-Benford Law in recognition that  Simon Newcomb had noted it more than 50 years earlier, in 1881, in “Note on the Frequency of Use of the Different Digits in Natural Numbers”.

There are also some restrictions on what kind of numbers follow the First Digit Law:  According to Statistics How To:

Benford’s law doesn’t apply to every set of numbers, but it usually applies to large sets of naturally occurring numbers with some connection like:

  • Companies’ stock market values,
  • Data found in texts — like the Reader’s Digest, or a copy of Newsweek.
  • Demographic data, including state and city populations,
  • Income tax data,
  • Mathematical tables, like logarithms,
  • River drainage rates,
  • Scientific data.

The law usually doesn’t apply to data sets that have a stated minimum and maximum, like interest rates or hourly wages. If numbers are assigned, rather than naturally occurring, they will also not follow the law. Examples of assigned numbers include: zip codes, telephone numbers and Social Security numbers.

(TwoPi, in a discussion about this, mentioned that books of logarithm tables tend to be dirtier in the beginning than at the end, in a visual application of the law.) According to J. Carlton Collins in the Journal of Accounting the data set should be somewhat large, at least 500 entries ideally.  Still, it’s a pretty impressive rule, and one that doesn’t quite make intuitive sense to me.

So about catching criminals?  Forensic accountants use this rule to catch people who falsify invoices, because falsified data doesn’t usually follow this expected pattern.  Go math!

Monday Morning Math: How many rationals?

September 26, 2022 by

Today’s Monday Morning Math was suggested by Justin, one of our Math Club Officers!  He requested a MMM about the fact that there are the same number of rational numbers and integers, because that’s that is really awesome.  Great idea Justin!

The idea behind this is that when we talk about infinite sets, we say that they are the same size if there is a bijection between them (a function that is one-to-one and onto – that is, it exactly matches each object in the first set with an object in the second set).  This leads to weird things, for example, like that there are the same number of integers as even integers.  This is because the function:

f(n)=2n

exactly matches each integer to a unique even integer, so the two sets (integers and even integers) must have the same size.  Which is weird, because half the integers are even, so you’d expect there to be twice as many integers as even integers.  Infinity is weird.

But finding a map between the integers and the rational numbers is not trivial.  One way to think about it is to put the numbers in each set in some sort of order, where if you count them, and you have a lot of time on your hands, you know you’ll reach each number in the set.  For the integers it’s not too bad – you could count:

 0, 1, -1, 2, -2, 3, -3, …..

but then the rationals are sneaky.  How do you even put fractions in order?  One way is to look first at the positive fractions and find a way, and there’s a pretty picture here of how to do that

By Cronholm144 (CC license)

So we could list the numbers as:

1/1, 2/1, 1/2, 1/3, (then skip over 2/2 because that’s the same as 1/1), 3/1, etc.  If you look at the way the picture weaves back and forth, you will eventually list every positive rational number. 

That gets us the positive rational numbers, and to get the rest we could alternative positive and negative, the way we did with the integers!  So our listing would be:

0, 1/1, -1/1, 2/1, -2/1, 1/2, -1/2, etc.

Then, since you can list all the integers in an order, and you can list all of the rational numbers in an order, then you can match the first numbers in each list, the second in each list, etc. and see that there are the same number of integers as rational numbers!

Fun fact – we posted about infinities on our unofficial department blog back in 2008.  I notice that a lot of those links are broken, but I’ll quote part that isn’t.  Happy infinity!

[Digression: for a cool description of the countability of the rationals, read Recounting the Rationals, part I and Recounting the Rationals, part II (fractions grow on trees!) at The Math Less Traveled, which is an exposition of the paper Recounting the Rationals by Neil Calkin and Herbert Wilf.)

Monday Morning Math: quipus

September 19, 2022 by

It’s the return of Monday Morning Math, a post that goes up each Monday(ish) morning(ish) during the semester with a mathematical tidbit.

Today’s topic is quipu.  Here’s a picture of one from the Museo Machu Picchu, Casa Concha, Cusco in Peru .

photo by Pi3.124, CC license

Quipu (also written as khipu) were used by the Inca people are a way of recording information, like a ledger, that could then be carried or stored. Numbers could be represented with different knots using a Base 10 system (so 2 3 2 would be  232).  Meanings could also be represented by different colored cords, and by having substrings off of strings. There could be hundreds of strings on a single quipu.

While relatively few quipu survived, there are examples that are about a thousand years old (!).  I first read about them in the book Code of the quipu: a study in media, mathematics, and culture by Marcia Ascher and Robert Ascher but my impression is that in the past few decades there has been a lot more research: (The Khipu Field Guide, for example, appears to have a lot of information.) Wikipedia also has an extensive list of sources.

Bonus Monday Morning Math to make up for not starting last Monday as originally planned!  During Hispanic Heritage Month (from September 15 to October 15) the website Lathism.org features a new mathematician each day!  Today’s mathematician is Ramón Emilio (Emilio) Fernández, a mathematician at Pace University in New York City who earned a PhD in Engineering Education, Management, and Policy with doctoral thesis in Mathematics Education.

Happy mathing!

Monday Morning Math: Origami

May 2, 2022 by

Good morning!  This is the first Monday in May and the trees and flowers are in bloom.  It’s also finals week at Naz, which means it’s the last Monday Morning Math of the school year.  I hope you’ve had fun – we’ll pick it back up in September, probably starting the Monday after Labor Day!  Same bat time, same bat channel.

For this final MMM we’re using a reader suggestion – Origami (thanks Phyllis!)  Origami is a pretty neat subject [one that I think we’ll come back to next year too] because you can use it to make things like a flapping bird:

or a modular pinwheel:

Want something more geometrical?  Here’s a whole page that shows you how to make all the Platonic Solids (symmetric three dimensional polyhedron made out of a single regular polygon) AND all the Archimedean Solids (same as Platonic, but using more than one shape) AND some stars.

Want something more practical?  How about a giant space telescope made out of origami: 

Too big? What about medical implants?

Or do you just want to see some pictures of origami?

Here’s a fish:

Here’s some more fish, each made from a dollar bill:

Here’s a lilac spider, in honor of the upcoming Lilac Festival:

And, finally, here’s a pretty figure in case you didn’t want to end with a spider:

Have a good summer/winter everyone [depending of course on which hemisphere you’re in]!  My own plans include making even more origami – I got this book by Thomas Hull but haven’t had a chance to make things from it yet.  I hope your own plans are equally fun!

Monday Morning Math: Illusions

April 25, 2022 by

Good morning!  Today’s post is about illusions – specifically, the kind of illusions you can verify with MATH!  

I kind of think calling them “Illusions” might be giving some answers away, but even suspecting the answers it can take some convincing, possibly by holding up paper to your phone or computer.

Which center circle is bigger?

Would lines m and n meet if they were extended?

Which segment is longer, AB or CD?

Will line m, when extended, meet point A, point B, point C, or none of these points?

We’ve posted about illusions before (e.g. here and here) and if you want to see more you can go to the contest for the Illusions of the Year! They include photos and videos of the best ones, with some hints of how they work:

Enjoy!

Monday Morning Math: Omar Khayyam

April 18, 2022 by

Good morning!  Our mathematician today is Omar Khayyam.

Omar/Umar Khayyam was born in Nishapur, Persia, (modern day Iran) in 1048. Not much is known about his mother, but his father was a doctor who hired tutors to teach Omar.   Omar Khayyam is known for his mathematics, including writing down the laws of algebra that we know today.  He was able to make progress toward finding a general formula for ax^3+bx^2+cx+d=0 similar to the quadratic formula:  Greek mathematicians had come up with solutions to the quadratic formula that used a straightedge and compass, but Khayyam conjectured that it was not possible to solve the cubic equation with just those tools, and so developed other means of finding the solutions geometrically, using a parabola.  (It would be more accurate to say solutions to cubic equations: although we write it as a single equation, at that time the quadratic and cubic equations were written as several different cases depending on whether the coefficients were positive or negative.)  It was 500 years before anyone found a more general solution than his.

Omar Khayyam was one of the earliest people to describe the Arithmetic triangle (which is sometimes called Pascal’s triangle, although this was 500 years before Blaise Pascal).  He also contributed to the fields of non-Euclidean geometry and number theory.

In addition to mathematics, Khayyam wrote about astronomy, geography, and music.  He is largely remembered for his poetry, especially the rubaiyat (aka  Rubā‘iyyāt, or quatrains)

The Moving Finger writes, and, having writ,
Moves on: nor all thy Piety nor Wit
Shall lure it back to cancel half a Line,
Nor all thy Tears wash out a Word of it.

(Translated into English by Edward Fitzgerald)

Khayamm passed away in Nishapur in 1131, and a mausoleum stands over his tomb

Sources:

Monday Morning Math: Dorothy Lewis Bernstein

April 11, 2022 by

Happy birthday Dorothy!  Today is the 108th birthday of the first woman to be president of the Mathematical Association of America.

Dorothy Lewis Bernstein was born on April 11, 1916, in Chicago, Illinois.  Her mother, Tillie Loyev (changed to Lewis upon immigration to the US), was born in Ukraine and her father, Jacob Louis Bernstein, in Russia.  Her parents married in Milwaukee (where her mother’s family lived) in 1912 and then moved to Chicago where her father was a dairy farmer.  Dorothy was the oldest of six children and grew up in Chicago and Wisconsin.  

Dorothy earned her BA and MA in mathematics from the University of Wisconsin Madison [the alma mater of yours truly.  Go Badgers!] and her PhD from Brown University.  She worked at several universities, including Berkeley where work was being done to support the army during World War II.  She then spent 14 years just down the street from Naz at the University of Rochester(!!!).

While at Rochester she was asked by C. B. Tompkins, who was working at Engineering Research Associates on a contract from the Office of Naval Research, to undertake a study of the current state of knowledge of existence theorems in partial differential equations. As she explained in her 1978 AWM talk, “some of the proofs could be used as basis for the computational solutions of non-linear problems that were just being tackled by high-speed digital computers.” Her 1950 book with Princeton University Press was the result of this undertaking. 

Supplementary Material for Pioneering Women in American Mathematics: The Pre-1940 PhDs

In 1959 she began working at Goucher College in Maryland, where she stayed 21 years.  Dr. Bernstein was a prominent researcher, and was active in several organizations both before and after her retirement in 1979.  She was the vice president for the Mathematician Association of America in 1972-73 and president (the first woman elected to that position) 1978-80.  She passed away in 1988 at the age of 73.

Sources:

Monday Morning Math: Angles in a Triangle

April 4, 2022 by

Good morning!  I’m teaching Geometry this semester, so triangles are on my mind.  And here are some facts about triangles that, even though I know them, still blow me away.

I was taught that the angles in a triangle add up to 180 degrees.  This is true, at least in Euclidean Geometry, which is geometry done on a flat surface like a plane or piece of paper..(That link was to an illustration using folded paper but there are many proofs too.)

But what if you’re not on a flat surface?  What if you’re on a sphere?  Then things get weird.  “Lines” on a sphere are defined by the shortest path between two points (formally called geodesics), and it turns out that if you draw a line between two points and keep going, it will cut the sphere in half, like the equator of a globe, or a line of longitude.  And if you put three of those together you get a triangle…but the angles don’t add to 180 degrees.  In fact, you can have have a triangle with not one but two right angles, as shown below:

You can even have a triangle with three right angles!  Or three obtuse angles!  And overall, there isn’t a fixed amount that the angles add up to: it can be anything from just over 180 degrees all the way up to just under 540 degrees (which would be a triangle with three really big angles, covering a significant amount of the sphere).   So weird.

That’s not the only thing that’s weird.  If, instead of being on a sphere (which bows out), you are on a hyperbolic paraboloid, which bows in and looks like a saddle, then everything is opposite and the angles of a triangle are smaller than they would be on a flat surface.  You can still have triangles with one right angle or one obtuse angle, but the other two angles will be a bit smaller, and some triangles will just have three very small angles.  As in spherical geometry, the angles of a triangle don’t add up to any one fixed amount, but can be any positive number less than 180 degrees. 

If you like exploring, the (free) program Geogebra [that mimics drawing with a straightedge and compass]  has tools that let you draw in spherical geometry and hyperbolic geometry. Enjoy! 

Monday Morning Math:1+2+3+…

March 28, 2022 by

Good morning!  It’s snowing today, it was sunny a few weeks ago, and who knows what will happen next.  In this spirit of surprise, today we’ll look at 1+2+3+….   Any idea what it is if you keep adding?  You might think you’d approach infinity, but actually….

….well, actually that makes sense.  But then this would be mighty short, so instead we’ll prove that the sum is -1/12.  This is called the Ramanujan Summation after the mathematician Srinivasa Ramanujan who was born in 1887 and who passed away in 1920.  

So, let’s get proving!  We’ll do this in parts:

Step 1:  Prove that 1-1+1-1+1-1+-… adds to 1/2.

We’ll call the sum of this sequence A, and do some fancy algebra:
Since A=1-1+1-1…. then if we subtract 1 (the first 1 on the right) we get:
A-1=-1+1-1+1…., which is the negative of what we started with.  That means  (A-1)=-A, so (2A-1)=0, and that means A=1/2.  All done!

This all assumes that we can treat infinite sums the same way as finite sums.  YMMV.

Step 2:  Prove that 1-2+3-4+5-6+…. adds to 1/4.

We’ll call the sum of this sequence B, and keep going with the fancy algebra.
Since B=1-2+3-4+5-6+….  let’s look at A-B
A-B=(1-1+1-1+1-1+…)-(1-2+3-4+5-6+…)   
Let’s reorder, putting the first terms together, the second terms, etc.
This is (1-1)+(-1+2)+(1-3)+(-1+4)+(1-5)+(-1+6)+…, which simplifies to
0+1-2+3-4+5.  And that’s just B!

So A-B=B, which means A=2B, so B is half of A, and therefore 1/4.

Step 3: .Prove that 1+2+3+4+5+6+… adds to -1/12.

We’ll call this sequence C.
Since C=1+2+3+4+5+6+…, let’s look at B-C
B-C=(1-2+3-4+5-6+…)-(1+2+3+4+5+6+)
Like we did in Step 2, we’ll reorder, putting the first terms together, the second terms, etc.
This is (1-1)+(-2-2)+(3-3)+(-4-4)+(5-5)+(-6-6)+…, which simplifies to:
0-4+0-8+0-12+…., which is -4-8-12-…
You can factor out a -4, and get -4(1+2+3+…), and that’s -4C!

So B-C=-4C, giving B=-3C, so C is (-1/3) of B, or (-1/3) of (1/4) and that, my friends, is -1/12!

What do you think?  If you think it makes sense, you’re in luck – there are some deep results in physics that use these ideas (although they are proved using something called the Riemann zeta function).   On the other hand, if you think there was some mathematical sleight of hand, well, you’re right also.  Treating infinite series like they are finite makes sense until it doesn’t, like adding up a bunch of positive integers and getting -1/12.

This subject was inspired by a reference in The Art of Logic in an Illogical World by Eugenia Cheng, and is also on a Numberphile video.  I used a post on Cantor’s Paradise for the notation, and Scientific American for additional background.

Monday Morning Math: The L’Hôpitals

March 21, 2022 by

Good morning! The math tidbit for today is a two-for-one special: the L’Hôpitals, who lived in France in the 1600s.  But we’ll start with the phrase that my brain jumps to when I see L’Hôpital, which is L’Hôpital’s Rule (also called L’Hospital’s Rule).  It’s about limits, so it shows up in Calculus.  Essentially, if you have a limit as x\to a of a fraction where both the numerator and denominator individually are approaching 0 or where both the numerator and denominator individually are approaching \pm\infty, then:

\lim_{x\to a} \frac{f(x)}{g(x)}=\lim_{x\to a} \frac{f'(x)}{g'(x)}

provided that second limit actually exists.  So, for example,

\lim_{x\to 0} \frac{\sin(x)}{x}=\lim_{x\to 0} \frac{\cos(x)}{1}=1.

Huzzah!

This rule was named after Guillaume François Antoine de l’Hôpital, where that last name was spelled different ways even in his lifetime, even by him.  He learned calculus by correspondence with Johann Bernoulli, who was busy figuring out calculus himself at that time, since the subject was only a few decades old.  In 1696 l’Hôpital published what is considered to be the first Calculus book: Analyse des infiniment petits pour l’intelligence des lignes courbes.  He thanked several people in the introduction, including Johann Bernoulli, but it wasn’t clear at that time how much of the book was really after l’Hôpital’s own work (none?) and how much was based on Bernoulli’s notes (all?).  Apparently Bernoulli was fine with l’Hôpital publishing the book, possibly because of the money l’Hôpital paid him, possibly because he was happy just to have these still-new ideas disseminated. L’Hôpital died in 1704 when he was about 43 years old.

A lot of that information  comes from the MacTutor biography, which also states, “L’Hôpital married Marie-Charlotte de Romilley de La Chesnelaye; they had one son and three daughters.”   The English Wikipedia page adds that his wife was “also a mathematician and a member of the nobility, and inheritor of large estates in Brittany” with a link to a page for her – in French – from a French biography that indicates that she worked in Geometry and Algebra and lived from 1671-1737.  The only other site online that mentions her is this dictionary, also in French, which indicates that she helped with the printing of the aforementioned Calculus book and impressed another math professor, Monsieur de la Montre, with her knowledge of Euclid.  So was she too involved in the creation of that first Calculus book but not mentioned?  It sounds like it, though the extent of her involvement is unclear.  

And we’ll leave on that uncertain note.  Any new information would of course be welcome.

References:

Monday Morning Math: Pi Day

March 14, 2022 by

Happy Pi Day everyone!  It’s Spring Break here, but it’s Pi Day, so worthy of celebration!  Here’s 7 math jokes about pi, because π ≈ 22/7

#7
What did pi say to its sweetheart?
You look radian today.

#6
Why did pi fail its driving test?
Because it didn’t know when to stop.

#5
What do you get when you divide the circumference of the moon by its diameter?
Pi in the sky

#4
How many bakers does it take to make a pie?
3.14

#3
What is the most mathematical kind of snake?
A pi-thon

#2
What do you need for dessert seafood?
Octo-pi

#1

Monday Morning Math: Maryam Mirzakhani

March 7, 2022 by

It’s Women’s History Month, and we’ll celebrate with the first (and so far only) woman to receive the Fields Medal in Mathematics – an award given every four years in recognition of outstanding mathematics. 

Maryam Mirzakhani was born in Tehran, Iran, in 1977.  She loved reading novels and writing as a child, and became interested in mathematics when her brother would come home from school and tell her what he learned – her earliest memory of math specifically was when he told her the story of a young Carl Friedrich Gauss being asked to add the numbers from 1 to 100 and surprising his teacher by seeing a pattern that allowed him to do the computation in moments.

Maryam Mirzakhani went on to participate in (and win) the Math Olympiads as a teen; she then went to Sharif University of Technology in Iran and then Harvard University in the United States, where she earned her doctorate.  

In 2014 Dr. Mirzakhani was awarded the Fields Medal. Her research is sometimes described as a complicated version of billiards, where a ball bounces along the edges of a table.  “You want to see the trajectory of the ball,” Dr. Mirzakhani explained in a video produced by the Simons Foundation and the International Mathematical Union. “Would it cover all your billiard table? Can you find closed billiards paths? And interestingly enough, this is an open question in general.” You can see a short (under 3 minute) video of Dr. Mirzakhani explaining her work here:

Mirzakahni passed away in 2017 from breast cancer.  She was 40 years old.

The 2022 Fields Medals will be given out at the  International Congress of Mathematicians this summer.  The conference was originally scheduled to be held in St. Petersburg, Russia, but due to the Russian invasion of Ukraine the organizers have announced that the conference will be held virtually, without any contributions from the Russian Government, and will be free for everyone who would like to attend.  

Sources:

Monday Morning Math: Voronoi Diagrams and Georg Voronoy

February 28, 2022 by

Good morning!  Today’s math is Voronoi Diagrams, but I’ll lead into it with some Geometry.  Suppose you have two points A and B, and wonder which points on the plane are closer to A and which are closer to B.  Maybe A and B are schools, and this is for figuring out districts, or maybe they are pizza places and you’re wondering where to order from.  It turns out that the border between the two regions is on the perpendicular bisector of A and B.

But what if there are three points? It’s a little more complicated, but the boundaries are still made up from the perpendicular bisectors of the different pairs of points.  And this idea continues even when there are more points, as this picture shows.

Diagrams like these are called Voronoi Diagrams named after Ukrainian mathematician Georgy Feodosevich Voronoy (also written as Georgii Voronoi).

Wikimedia Commons

He was born in the village of Zhuravka in the north central part of Ukraine on April 28, 1868, and while he was still in the equivalent of high school he solved and published the result of a problem in algebra.  He then went to the University of St. Petersburg in Russia, first as an undergraduate but eventually as a doctoral student under Andrey Markov (himself well known because of something called Markov Chains, which are ways of calculating probabilities).   Both his Master’s thesis and his doctoral thesis were awarded the Bunyakovsky prize for outstanding work in mathematics by the St Petersburg Academy of Sciences.

Voronoy became a professor at the University of Warsaw in Poland, where he continued to do research and also supervise students: one of his students was Wacław Sierpiński, for whom Sierpinski triangles are named.   When he was only 40 years old he developed severe gallstones, and passed away on November 20, 1908.

Sources:

Monday Morning Math: Presidents’ Day edition

February 21, 2022 by

Today is the day we observe Washington’s Birthday, popularly known as Presidents’ Day, and that makes it a good day to look at the Pythagorean Theorem.  Specifically, a proof of the Pythagorean Theorem.  Specifically, the proof created by James A. Garfield, the 20th president of the United States.

The Pythagorean Theorem, often written as a^2+b^2=c^2, says that squares on the legs of a right triangle, added together, have the same area as a square on the hypotenuse,   Garfield’s proof, published 5 years before he assumed the presidency, used a trapezoid.  Here’s a photo from the New-England Journal of Education on April 1, 1876

(from Mathematical Treasures)

Garfield’s proof compares the area of the trapezoid (the height times the average of the parallel sides) with the area of the three triangles that make up the trapezoid.  Garfield wrote out the details for you above, but feel free to try it using the common a b and c notation.

Monday Morning Math: Valentine’s Day

February 14, 2022 by

Happy Valentine’s Day everyone!  I thought I’d give you a valentine.  Maybe a cardioid like this:

By AtomicShoelace – Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=39636709

Or, if you don’t want to have to turn your head sideways, here’s a heart made out of Sierpinski Triangles:

image.png

Here’s how to make it

And here are interlocking Mobius Valentines!

image.png

And here’s how to make these!

Have a happy day!