Monday Morning Math: Mathematically Gifted & Black

February 6, 2023 by

This post is a complication, about several mathematicians!  Every day during the month of February the site Mathematically Gifted & Black honors a mathematician.  For example,

February 1 featured Ruthmae Sears, an Associate Professor of Mathematics Education and the Associate Director for the Coalition of Science Literacy at the University of South Florida.  An article from the University of South Florida contains additional information about her:

As a mathematics educator, Ruthmae Sears has a true flair for problem solving. Her work extends far beyond using formulas and finding solutions to abstract mathematical problems. Using mathematical reasoning to examine social disparities such as poverty, literacy and structural racism, Sears develops community-centric solutions. Her work emphasizes inclusivity in all spaces, stemming from her belief that schools are microcosms of a community.

February 2 highlighted Clarence W Johnson, a Math Professor at Cuyahoga Community College in Ohio.  In that article he offers the following words of inspiration:

Never let other people convince you that you are incapable of succeeding. Draw strength from both the positive and negative actions of others. However, most of your success will be drawn from qualities within yourself.

February 3 recognizes Ayanna Perry the Associate Director for the Teaching Fellows Program at Knowles Teacher Initiativewith in New Jersey.  There is more information in this article, including a link to an article in Mathematics Teacher entitled “7 Features of Equitable Classroom Spaces”  that I just requested through InterLibrary Loan.

You can come back each day to learn about a new person, or browse previous years to learn even more, including the 2018 Nominee Clarence Francis Stephens, who for several years taught just down the road at SUNY Geneseo. 

Happy Black History Month!

Monday Morning Math: More Math and Music

January 30, 2023 by

Good morning! This week we have a bit more info about music (thanks TwoPi!):   Last week I mentioned a “just fifth”, which is when the frequencies of two notes are in a 3:2 ratio: A is 440 Hz (on a piano at least) and the note E above that is at 660Hz.  These two notes are a perfect fifth, and the ratio 660:440 is a 3:2 ratio.

The frequencies in an octave, on the other hand, are in a 2:1 ratio:  The A above the one with 440 Hz has 880 Hz.  So the work that Jing Fang did, showing that 53 fifths was almost exactly 31 octaves, amounted to showing that \left(\frac32\right)^{53} (which is approximately 2,151,972,563.22) is very close to  2^{31} (which is 2,147,483,648).  If you want to avoid fractions, this means that 3^{53} is almost the same as 2^{84}.  They won’t be exactly the same, but it’s pretty close.

But let’s go a little more in depth!  The savvy reader (that would be you) might realize that we can’t ever have a power of 3 exactly equal to a power of 2, because they are different prime factors.  This means we’ll never have a power of 3/2 equal to a power of 2.  This is a fundamental problem with piano tuning:  if you prioritize fifths then the octaves don’t exactly match up, and if you prioritize the octaves then the fifths don’t exactly match up.  Piano tuning today prioritizes the octaves, and uses the fact that 12 fifths is approximately the same as 7 octaves.  That is, \left(\frac32\right)^{12} (which is approximately 129.7) is very close to 2^7, which is 128.  

Isn’t that cool?  If you want to hear some of it, here’s a 3-minute video of mathematician and musician Eugenia Cheng giving more detail and connecting it to Bach and to Category Theory:

And if you want even more music, here’s some music played on a Mobius strip, just for fun:

  (And there are more videos on the AMS page on Math and Music.)

Monday Morning Math: Happy New Year!

January 23, 2023 by

Good morning everyone! Happy 2023!    Today’s post is in honor of two new years.

The first is the new calendar year: 2023.  If you want to do something fun (which of course you do) then you can see if you can use exactly the digits 2, 0, 2 and 3  and different math configurations to write the numbers 1-100.  For example: 1=\frac22+0\cdot3.  Or  1=2\cdot0+3-2.  (There are examples all over the internet, so a quick search reveals many solutions should you wish.)

The other is the Lunar New Year, which began yesterday.  This is also known as the Spring Festival, and is observed around the world, including China, Indonesia, Japan, Malaysia, the Philippines, Singapore, South Korea, Taiwan, Thailand, and Vietnam: we are now in the Year of the Water Rabbit in many countries, and the Year of the Cat in Vietnam.

In honor of the New Year we’ll talk about the mathematician Jing Fang  京房.  He was born in China 2100 years ago (78 BCE, during the Han Dynasty).  He was a mathematician and, appropriately enough for this post, he described astronomy – the solar and lunar eclipses.

But these weren’t his only accomplishments.  He was also very good at making predictions using the Yijing, or I Ching. There are 64 hexagrams, each made up of 6 rows that have one long or two short marks in each row.  While this isn’t mathematics, it does lead to the math question to ponder: namely, can you explain why there are exactly 64 configurations?

Picture from Wikimedia showing a diagram that Gottfried Wilhelm Leibniz owned.

And his math-connection doesn’t end there either – he used mathematics to describe music theory, particularly that 53 fifths (technically “just fifths”, which may or may not be the same as a perfect fifth – but you can here one here) was almost exactly 31 octaves.  It took more than 1600 years for anyone (said anyone being Nicholas Mercator) to caculate the difference between the two more exactly than Jing Fang. 

Math Moons and Music – a good way to start the year.

Sources: Wikipedia and…just that, because the few other sources I found had the same information.

Monday Morning Math: the Winter Solstice

December 26, 2022 by

December 12 was going to be the final Monday Morning Math of the semester, but finals had started two days earlier and with one thing and another (well, really just one thing – the aforementioned final exams), it didn’t happen.  

😬

It felt a bit odd to take a break without announcement, however, so here is one final MMM for 2022.

😀

(Arriving on a Monday at least, even though it’s not quite morning anymore.) And the timing is perhaps good for a math-adjacent topic: the Winter Solstice, which happened on Wednesday, December 21.  This is the shortest day of the year in the northern hemisphere, with just under 9 hours of daylight (technically 8 hours, 59 minutes, and 10 seconds) here in Rochester.  But there are two things about the solstice that I find interesting mathematically.

The first is that if you google “When is the Winter Solstice?” you get not just a day but a time: 4:47pm here.  This feels a little weird to me if I think about it being a day, but it has to do with northern hemisphere being tilted as far away as possible from the rays of the sun, as in this tweet from NASA below:

Or, if you want to envision the Earth with its axis vertical, it’s moving along a plane that is not horizontal, as explained on NASA’s blog

This is the image that makes the idea of a moment for the solstice make sense to me: it’s at the very peak of the ellipse, and that happens at one particular moment rather than a full day.

The second math adjacent thing is about subtraction.  You might think that the shortest day must have the latest sunrise and the earliest sunset, but actually neither is true: the lastest sunrise doesn’t happen until January 3, 2023, at 7:42am, which is about 3 minutes later than it rose on Dec 21.  So that’s kind of a bummer, in terms of how dark the mornings are.  But that’s compensated by the fact that the earliest sunset happened several weeks ago, back on December 9.  The sun set at 4:35pm that day, about 3 minutes earlier than it set on the solstice.  The sunrises and sunsets don’t quite change symmetrically, and that’s why the shortest day is about halfway in between. 

Happy Solstice and Happy Holidays to everyone! We’ll start again in about a month – see y’all in 2023!

Monday Morning Math: Whose sequence?

December 5, 2022 by

Today’s post is a follow-up to last week’s about Leonardo Pisano, also known as Fibonacci.  And it is specifically about the sequence that bears his name: 0, 1, 1, 2, 3, 5, 8, … where each number is the sum of the two before it.  This appears as the answer to a puzzles that he proposed in his book Liber Abaci:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

The sequence was given the name “the Fibonacci Sequence” by Edouard Lucas (1842-1891), a French mathematician for whom a similar sequence: 1, 3, 4, 7, 11, … is named.  Blog commenter S shared that leading up to that time “it was rarely known and when used was called by many names including “the sequence of Lamé” (as Gabriel Lamé had used the sequence for analysis of the Euclidean algorithm)” (Thanks S!)  

I recently got the book Finding Fibonacci by Keith Devlin, and while I haven’t read the whole book yet (something for the break!), he does talk briefly about the history of these numbers, which poetically were used in prosody (the study of rhythm and sound/stress in poetry):

They first appeared, it seems, in the Chandahshastra (The Art of Prosody) written by the Sanskrit grammarian Pingala sometime between 450 and 200 BCE.  Prosody was important in ancient Indian ritual.  In the sixth century, the Indian mathematician Virahanka showed how the sequence arises in the analysis of meters with long and short syllables.  Subsequently, the Jain philosopher Hemachandra (ca. 1150) composed a text about them.

S, too, linked to some photos of the sequence appearing in these sources.  I am wondering if the long and short analysis is similar to some problems we’ve given in our classes, which essentially boil down to the number of ways that you can add to 1, 2, 3, 4, 5, 6, etc. using just the numbers 1 and/or 2, where order matters: for 1 there is only 1, for 2 it can be 2 or 1+1, for 3 it can be 2+1, 1+2, or 1+1+1, and for 4 it can be 2+2, 2+1+1, 1+2+1, 1+1+2, or 1+1+1+1.

The idea that the sequence comes up in poetry makes me happy.  The idea of continuing to call it the Fibonacci sequence doesn’t, however, since despite what may have been Lucas’s good intentions, it ignores these earlier contributions.  I’ve seen reference to calling it the Fibonacci-Hemachandra numbers, but even that would bypass Pingala and Virahanka, as well as some other mathematicians, as noted in the abstract of “The so-called fibonacci numbers in ancient and medieval India” by Parmanand Singh

What are generally referred to as the Fibonacci numbers and the method for their formation were given by Virahṅka (between a.d. 600 and 800), Gopla (prior to a.d. 1135) and Hemacandra (c. a.d. 1150), all prior to L. Fibonacci (c. a.d. 1202). Nryana Paita (a.d. 1356) established a relation between his smasika-paṅkti, which contains Fibonacci numbers as a particular case, and “the multinomial coefficients.”

I will admit to still feeling like I don’t have a handle on the history of the sequence – just enough to be uncertain how to call it.  I’ll close with a photo of a bust of Acharya Hemachandra at Hemchandracharya North Gujarat University (CC license)

Monday Morning Math: Leonardo Pisano (Fibonacci)

November 28, 2022 by

Good morning!  I hope you all enjoyed your Thanksgiving Holidays. Both boys came home for the weekend, which was a treat. =)

On Wednesday I got a text wishing me a Happy Fibonacci Day.  I had to think a minute: Nov 23, so 1123, from the sequence 1, 1, 2, 3, 5, 8, 13,….  And this inspired this week’s post about Fibonacci.  

Leonardo Pisano was born around 1170 in Italy, probably Pisa — hence the “Pisano” part of his name.  He was born to the Bonacci family — hence the  “filuis Bonacci” (abbreviated to “Fibonacci”) part of his name.  His father was a diplomat, and as a result of his father’s post Leonardo was educated in North Africa and traveled widely, which meant he was exposed to different number systems, including the base ten number system that we use today.  Indeed, it is likely that Leonardo himself is the reason we use it: he found it to be much better for calculation than the Roman number system (which would have used XXIII for a number like 23).  He returned to Pisa around 1200 and wrote several books that illustrated this system, the most famous of which is Liber Abaci (Book of Calculation – abaci is related to abacus).  Here’s a statue, by Giovanni Paganucci, of Fibonacci holding a book (CC license).

Although Fibonacci’s most significant mathematical contributions are related to his books sharing the decimal number system and methods of calculation with western Europe, he has become most famous because of a single problem that was in the book:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

This problem leads to the number system 1, 1, 2, 3, 5, 8, …, where each number is the sum of the previous two, which now bears the name the Fibonacci sequence in his honor.  Although maybe it shouldn’t – the sequence was known in India well before Fibonacci. By whom, you might wonder?  I started to write a brief summary, but realized I didn’t know enough about the history myself to do it justice so that will have to wait for next week…

Sources: 

Happy Thanksgiving!

November 21, 2022 by

thanksgiving

(This was originally published on November 27, 2008, but I ran across it in looking up Thanksgiving Math and thought it might be good to update the links and repost!)

While you’re baking your turkey and pondering why the temperature continues to rise even after the bird comes out of the oven or thinking about how Game Theory showed that tryptophan affects trust and cooperation, you might also be wondering, “What kind of math did the folk at the first Thanksgiving study?”

The answer is…I don’t know. So I looked around at what textbooks would have been available. I couldn’t find any math texts of the Wampanoags from around 1621, and while I was enticed by a reference to “our earliest native American arithmetic, the Greenwood book of 1729” from David Smith’s History of Mathematics Vol.2, in looking further it seems that the phrase “native American” was used only in contrast to having been originally published in England, Spain, or other non-American country. So as far as the Wampanoags are concerned, I’ll admit to remaining in the dark about what specific kinds of math they would have been learning (formally or informally).

Then I looked into the Pilgrims. As they hailed from England, it seems likely that they’d be using a an English text. And for this they had few choices: there was a 1537 book entitled An Introduction for to : Lerne to Reckon with the Pen and with the Counters after the True Cast of Arsmetyke or Awgrym [that last word is Algorithm], which was a translation of a book in Latin by Luca Pacioli. Wikipedia mentions that another textbook was published in 1539, but I couldn’t find reference to the title or author (though I suppose I could add one myself).

A more likely candidate would be Arithmetick, or, The ground of arts by Robert Recorde, first published in 1543 and listed in several places as “The first really popular arithmetic in English”. The title is short for The grou[n]d of artes: teachyng the worke and practise of arithmetike, moch necessary for all states of men. After a more easyer [et] exacter sorte, then any lyke hath hytherto ben set forth: with dyuers newe additions. Here’s a woodcut from the 1543 version

robert_recorde-1543

and here’s the title page from the 1658 edition, with its fancy modern spelling:

recorde-title(Heh heh — I just ran that page through Adobe’s Optical Character Recognition, and it pretty much didn’t recognize a thing.)

I couldn’t track down any copies of what exactly was in the book, but this article about medicine in English texts quotes quite a bit from Recorde’s book. For example:

In 1552, Record amplified his arithmetical text with, among other material, a “second part touchyng fractions”. This new section included additional chapters on various rules of proportion, including “The Rule of Alligation”, so-called “for that by it there are divers parcels of sundry pieces, and sundry quantities alligate, bounde, or myxed togyther”.

The rule of alligation, we are told, “hath great use in composition of medicines, and also in myxtures of mettalles, and some use it hath in myxtures of wynes, but I wyshe it were lesse used therein than it is now a daies”. Despite Record’s regret about the adulteration of wine, the first problem the Master uses to exemplify his discussion of alligation in fact deals with mixing wines; the second involves a merchant mixing spices; and the remainder involve the mixing of metals. None of the examples involve medicine, although the merchant’s spices are once called “drugges”.

A third part was added in 1582 by John Mellis. Here’s a page (from Newcastle University) showing one of the pictures in the margin (possibly from that year, or possibly a later edition; I couldn’t tell for certain).

Records Arithmetic

Robert Recorde is actually better known for a later book: The Whetstone of Witte, which was published in 1557 and so is another candidate for An Original Thanksgiving Math Text.  It was called because, according to the poem on the front, “Here, if you list your wittes to whette, Moche sharpenesse therby shall you gette,” and appears (using Google books) to have been well-known enough to be referenced by Shakespeare in Act 1 Scene 2 of As You Like It (written around 1600) when Celia says, “…for always the dulness of the fool is the whetstone of the wits. How now, Witte! whither wander you?”

The Whetstone is best known for its introduction of the equal sign =, which Recorde explains below:

whetstone-equal-1

Which reads as

Nowbeit, for easy alteration of equations.  I will pro(?provide?) a few examples, because the extraction of their roots, may the more aptly be wrought.  And to avoid the tedious repetition of these words “is equal to”, I will set as I do often in work use a pair of parallels or Gemowe lines of one length, thus: ======, because no two things can be more equal….

So there you have it: math that some of the first Thanksgiving folk could possibly have studied.

thanksgiving

Monday Morning Math: Goldbach’s Conjecture

November 14, 2022 by

Good morning!  Today we’ll talk about something we don’t know.  Or, rather, something we don’t know *for sure*, because it hasn’t been proven (yet?)

It’s possible to write numbers (bigger than 1)  as a sum of prime numbers. For example, 14 is equal to 3+11. The question is, what’s the smallest number of prime numbers that you need?  Well, 280 years ago the German mathematician Christian Goldbach wrote to the Swiss mathematician Leonard Euler about this very thing:

He wrote in this letter – specifically, in the margin sideways – that he thought you needed at most three primes.  Euler replied and mentioned an earlier conversation where Goldbach had thought that the even numbers needed only two primes, which would imply what Goldbach mentioned in this letter. 

There’s a little bit of squirreliness in terms of what a prime number is, since Goldbach and Euler considered 1 to be prime (which is why there are all those 1s in the letter), but with a little bit of modification the two questions are:

  • Can every odd integer [>=9] be written as the sum of three odd primes?
  • Can every even integer   [>=6] be written as the sum of two odd primes?

The answer to the first question was proved to be YES less than ten years ago, by the Peruvian mathematician Harald Andrés Helfgott.  Yay!  But the answer to the second question remains “Probably”, so we will have to wait for another breakthrough.

Sources: 

Monday Morning Math: Code Talkers

November 7, 2022 by

Good morning!  November is Native American History Month: the website https://indigenousmathematicians.org/ highlights many mathematicians and was updated in 2021.  

Today’s math topic is codes – the kind where messages are encoded for secrecy.  During World War I, 19 Choctaw soldiers used the Choctaw language as a code for sending military messages in secret.  As described in the website for the Choctaw Nation of Oklahoma:

During the first world war, with the tapping of the American Army’s phone lines, the Germans were able to learn the location of where the Allied Forces were stationed, as well as where supplies were kept. When the Choctaw men were put on the phones and talked in their Native speech, the Germans couldn’t effectively spy on the transmissions.

Native Americans did not receive nationwide citizenship until 1924, yet the Choctaws were both patriotic and valiant, with a desire to serve in the war effort. Many Choctaw men volunteered in WWI to fight for our country. Choctaw Code Talkers of WWI were instrumental in ending war. Members of Choctaw and other Tribal Nations also served with distinction using Native languages in World War II, Korea and Vietnam.

The Choctaw Code Talkers

In World War II the Marine Corp recruited 29  Diné (Navajo) men to develop a much more complicated code based on their language:

The code primarily used word association by assigning a Navajo word to key phrases and military tactics. This system enabled the Code Talkers to translate three lines of English in 20 seconds, not 30 minutes as was common with existing code-breaking machines. The Code Talkers participated in every major Marine operation in the Pacific theater, giving the Marines a critical advantage throughout the war. During the nearly month-long battle for Iwo Jima, for example, six Navajo Code Talker Marines successfully transmitted more than 800 messages without error. Marine leadership noted after the battle that the Code Talkers were critical to the victory at Iwo Jima. At the end of the war, the Navajo Code remained unbroken. (from intelligence.gov)

Some of the Diné Code Talkers

In 2000 the Code Talkers were awarded with Congressional Gold Medals.

Sources:

Monday Morning Math: the witch of Agnesi

October 31, 2022 by

Happy Halloween!  For this day, it seems appropriate to talk about the Witch of Agnesi. Oooohhhhh!  Spooky! 

🎃

But this witch is a curve.  It’s described with the algebraic formula:   y = \frac{64}{x^2+16} (or, more generally, as y = \frac{8a^3}{x^2+4a^2} – in the previous equation I used a=2).

The original construction is described geometrically, starting with a circle of radius a.  Here’s a short video of how it is constructed

So how did this curve get its name?   Maria Gaetana Agnesi was a wicked smart woman who was born in Milan, Italy, in 1718, about 15 years after this curve was first studied by Pierre de Fermat and Guido Grandi.  By the time she entered her teens she spoke 7 languages, and by the time she entered her twenties she was also accomplished at philosophy and mathematics, which she discussed with her father’s visitors, part of an intellectual salon.  After her mother died Maria Agnesi took over running the house and wrote a calculus book for some of her 20 younger siblings.  This book was published when she was 30, when the study of Calculus itself was only decades old.  It was over one thousand pages, and Agnesi was granted an honorary appointment at the University of Bologna.  Agnesi spent most of her adult life focused on theology and serving others, particularly people who were poor or sick.

In 1901, John Colson translated it into English, and here is where a significant mistranslation occurred.  Agnesi had called the above curve averisera (related to the word for “turning”), but the word for “witch” is similar, avversiera, and that’s what Colson used. 

(Public domain from Wikimedia)

Sources:

Monday Morning Math: Happy Divali!

October 24, 2022 by

Good morning!  Today is Diwali: for those unfamiliar with the day, it is a celebration of the victory of light over darkness that originated as a Hindu religious festival and is now observed by people in many religious traditions. (More details can be found in, for example, the Times of India.)  Many years ago I ran across an article discussing mathematics and Diwali and saved the link.  Alas, when I looked for it now the original article appears to be gone, but in doing a new search I found a few more recent posts:

First grade teachers Michelle L and Michelle M talked about patterns and symmetry with their students, using designs on rangoli.  Rangoli are intricate patterns made on the floor out of flowers, rice, sand, or other materials:

CC Subhashish Panigrahi
CC by Rajeev Nair
CC by Dinesh Korgaokar

One thing I find interesting about this is that the symmetry is different in each case – it’s not always 8 or 12 for example.  

And for those of you craving something a little more Calculus-oriented, take a look at this video:  

Enjoy the week!

Monday Morning Math: Tying a Tie

October 17, 2022 by

Good morning everyone!  Today we’ll learn about tying a tie.  This is a Windsor knot:

But there are many other ways!  How many?  That’s the question that two physicists, Thomas Fink and Yong Mao, asked back in 1999.  They used math to prove that if you have a few rules, like that any tucks (folding one end of the tie under the rest) occur at the end of tying, then there are exactly 85 ways to tie a tie.  Not all of these are pretty, though: they thought only 13 of the ways really looked good.  Still, that’s a lot of neat ways, and they wrote a book called The 85 Ways to Tie a Tie: The Science and Aesthetics of Tie Knots.

But the story doesn’t end there!  Fifteen years later mathematician Mikael Vejdemo-Johansson saw a fancy tie on the Merovingian in The Matrix Reloaded that wasn’t in the list of 85 possibilities.  He loosened some of the rules that Fink and Mao had used and redid the math: with more options, he discovered a whopping 177,147 ways!  Here’s a knot called, appropriately, the Merovingian: 

Using this new collection, if you tried a new way every hour, it would take you just over 20 years to get through all the possibilities.

More detail about the discovery, including a video of how to tie your tie, can be found in this 2014 article by Rose Eveleth in Smithsonian Magazine.

Monday Morning Math: Star Quilts

October 10, 2022 by

Happy Monday!  In honor of Indigenous Peoples’ Day our topic today is Star Quilts.  Here are some examples:

Here is the  Blue Day Star Quilt, probably of the A’aninin (Gros Ventre) in Montana, made in 1990-91.

Here’s the Fall Time Star (Bright Fall Day) Quilt by Marie Kinzel of the Lakota (Teton/Western Sioux) in South Dakota, made between 1968 and 1988. 

Created by Marie Kinzel (Marie Strong Heart Kinzel)

Below is the Earth and Sky Quilt created by Chantelle Blue Arm of Cheyenne River Lakota Sioux in South Dakota, made in 2014.  

Created by Chantelle Blue Arm

You can see many more examples at the National Museum of the American Indian.  I first read about them in the January 2021 issue of Crux Mathematicorum, published by the Canadian Mathematical Society. The article “Explorations in Indigenous Mathematics: No. 1” by Edward Doolittle begins:

The starblanket design is popular among the Indigenous peoples of the Plains region, particularly in quilted blanket designs, but also in other crafts. In the Plains Cree language, the word for star is atāhk and the word for blanket is akohp, so starblanket is atāhkakohp. Chief Ahtahkakoop, so named because “the stars blanketed the sky, more numerous and brighter than usual” the night he was born, was one of the first signatories of Treaty Six. Ahtahkakoop Cree Nation is named after the chief.

As explained in more depth in that article, these beautiful quilts lend themselves to math explorations. For example, and this will be familiar to anyone who had Problem Solving with me back in 2021:  How many tiles are there in a Star Quilt? (I added that the answer should have n in it, although each person needed to specify what n represented.) I remember enjoying reading these solutions, and I think people liked coming up with them, too.

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.)