Monday Morning Math:1+2+3+…

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

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 of a fraction where both the numerator and denominator individually are approaching 0 or where both the numerator and denominator individually are approaching , then:

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

.

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:

• Wikipedia (English and French)
• MacTutor
• this dictionary
• some pictures of the book

Monday Morning Math: Pi Day

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

CC

Monday Morning Math: Maryam Mirzakhani

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. 

Photo by Gert-Martin Greuel, Creative Commons

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:

• The Guardian
• Scientific American
• Stanford News