What is Infinity?

To read more by Shunsuke: https://msurjblog.com/author/shunsukekatayama/

“I love you to infinity”, “I love you to infinity plus one”, “I love you to infinity plus infinity”, …

Let’s look at this mathematically, together, as you were just about to do on your own.

Mathematicians are to mathematics as geologists are to rocks.

Not many people gain pleasure from observing an ordinary rock, just like nobody ever gets excited when seeing an integral. Rocks have been in our hands as tools for a very long time, and we have developed many great things from them. Similarly, mathematics has been the flashlight we’ve used to discover new things: from the Pythagorean theorem to Einstein’s theory of general relativity. But it’s not mathematics that’s fascinating, but the results that we find beauty in. While having said that, geologists look very carefully at rocks and listen to the stories that they have to tell.

So let’s listen carefully to the sweet sounds of whisper of mathematics – to infinity and beyond.

Yes, infinity – the idea of infinitude is as beautiful as it is vast. Let’s put aside the “you’re infinitely beautiful,” “no, you’re infinity plus infinity times more beautiful,” “no no, you’re infinity multiplied by infinity times more beautiful” he-said-she-said – neither is complimenting the other any more than what they just said. As heartbreaking as it was when Andy had to leave for college in Toy Story 3, there are so far only two (sadly not infinite) types of infinities; namely, countable and uncountable infinities.

Countable infinity is a set of numbers that if you lived to be infinity years old, you can count all the numbers in the set. For example, the set of odd numbers {1, 3, 5, …}. If you count an odd number everyday for infinite number of days, you can count all odd the numbers. Let’s take a closer look at this after we define “uncountable infinity”.
Uncountable infinity is a set of numbers that even if you lived to be infinity years old, there’s no way that you can count all the numbers in the set. An example is the set of real numbers which is all the numbers that can be represented by decimals. You can spend an infinite amount of time counting all the numbers between 0 and 1 but you’ll still have the numbers between 1 and 2, 2 and 3, and infinitely many remaining intervals.

So, back to countable infinity. Here is a question: which set of numbers do you think is bigger, the set of prime numbers {1, 2, 3, 5, 7, 11, …} or natural numbers {1, 2, 3, 4, 5, …)? Euclid proved that there are an infinite number of prime numbers, which is a cool little fact, but would you believe me if I told you that there are exactly the same number of prime numbers as natural numbers?

Let’s say that there is a set of boys each representing a natural number and, likewise, a set of girls each representing a prime number.  We are going to form couples to have them go on a blind date. If there are more boys than girls – in other words, more natural numbers than prime numbers – we’ll know that by seeing that at one point we’ve run out of all the girls and we’ll have a (fairly sad) situation with only boys. But that’s not the case, since even when 20 couples or 100 couples or a million “couples” are matched, there will still be an infinite number of boys and girls that can be happily matched.
This is called the one-to-one correspondence: we will never find a boy or a girl – a natural number or a prime number – who cannot “find a date”. That may sound unrealistically perfect for the real world, of course,  but the world of mathematics is as perfect as it gets.
So, to recap: infinity = infinity + 1 = infinity + infinity = infinity × infinity. Those lovebirds were just repeating the same thing to each other [1].

Another fun fact is that the number of rational numbers – the set of all the numbers which can be written as fraction (e.g. 1/2, 2/3, 11/4) – is countable infinity as well. This means that there is the same number of rational numbers as prime numbers or even numbers.

Now that we have a little taste of what infinity is, do you think there is another infinity that we don’t know of? Maybe something that’s in between uncountable and countable infinity? I have no idea what that infinity might look like, but if you have an idea, perhaps this branch of mathematics – called number theory – might be for you.


[1] Assuming that these infinities being referred to are countable infinity.


Breaking memories: one data point at a time

Breaking Memories_2By Andrea Weckman | Illustration by Wanying Zhang

I began my first undergraduate research experience in the summer after my first year with a glorified impression of what it was going to be like. As I walked into the lab on the first day, I said to myself:

Andrea, this is your ticket to the Nobel Prize.

I think you’ll find that several things are a little off about this sequence of events. First, and most notably, I was talking to myself. (But every research scientist does, right? It’s normal, right?)

Secondly, at age 18, I was telling myself I would win a Nobel Prize: the gold medal of lifetime achievement, the ultimate symbol of scientific prestige. It didn’t even stop there — I’ve played with the idea of being the next Marie Curie, tucking two Nobel Prizes under my belt. Was I overestimating my role as a first-year undergraduate researcher in a small lab in Waterloo, Ontario?  Maybe. But a girl can dream.

This past summer, as I walked into St. Michael’s Hospital in Toronto to start another unpaid, 9-5 summer research experience, I said to myself:


Andrea, mark this day on your calendar as your first step towards saving the world. 

Again, I was glorifying my role – telling myself I would save the world. (Side note: when I say this to myself, I envision myself as a scientific Superman). But as I progressed through my first experience and then moved on to my second and third, rather than having my notions of grandeur squashed by reality, I came to realize that this delusion is necessary for maintaining your sanity in the world of undergraduate research. As you dig through piles and piles of data and run the same tests countless times, you need that sense of a greater purpose for your research. Your biggest motivation is the possibility of your efforts leaving a mark.

Now, at my current research position, as I make what seems like millions of histograms and residual plots and run different SPSS statistical analyses on what seems like billions of data points, the possibility of making a difference is what drives me.

I’ve met many people in my current research experience, all suffering from posttraumatic stress disorder (PTSD). I’ve encountered people whose memories keep them from sleeping at night, who startle when somebody speaks to them, and who are thrown into an all-consuming panic by a trigger that would seem inconsequential to anybody else. I’ve heard stories that make my life seem like a cakewalk. So when I start feeling sorry for myself after I’ve sat in front of a computer all day making graphs and exploring data, I think:

Andrea, this is your chance to make a difference. This is your chance to help. 

Since beginning work on a project that directly involves patients, my perspective has changed. It has shifted from a focus on achieving great things for myself (i.e. double Nobel laureate) to achieving things that will ultimately help others. If I can have a hand in finding a drug that will help these people get back on their feet and dampen debilitating traumatic memories, I will achieve my own little version of saving the world.


Andrea is a U3 Neuroscience student, and will be posting more in her Breaking Memories series next semester. 

Wanying is a U2 Chemistry student. 

The Chemical Conundrum: How to work efficiently in and out of the lab

By Isabella Liu

Like I said in my first entry, I work in the lab two days a week due to my schedule. Sometimes, when my course-load becomes heavy, I’m unable to come in and run my reactions. To compensate for this, I do a lot of literature searches and ask the necessary questions before coming to the lab to run my reactions; when I do come in during my two days, I make sure that no time is being wasted.

To be honest, after I completed my honours project, there was a period where I felt like I deserved a break. I would go into the lab, re-do some of the reactions I worked on during the summer, and make suChemicalConundrom_Booksre that I got replicable results. People in the lab encouraged that; they all said that I should be thankful that I have this opportunity and enjoy my time in the lab. What they said was partially true, what they didn’t include is the hard work that comes after “enjoying.” October came along and my graduate student was going to graduate. That was when it dawned on me: I was going to have to work independently, solve questions by myself (or ask my Professor Li), all without the help of my graduate student. Her departure gave me a little nudge – it was time for me to start doing my research.

On the topic of literature searches, people often fall into the trap of getting distracted – you’re on your computer, so you automatically get the feeling that whatever you do on your computer is personal. That’s when the personal and work space blur. To counteract this, I make sure that my workspace is as foreign to me as possible (i.e. I don’t put up photographs or have snacks laying around). Basically, I’m invisible.  My desk is also clean – the mess I make on other days is swept away when I write my interim report.

Once your workspace is prepared, finding and organizing relevant articles efficiently is the next challenge. There are several sites that I go for literature searches:

  • Google Scholar (GS) – it’s awesome for recent articles, but if you want to look for specific articles, such as reviews, go to
  • SciFinder (SF) – this was introduced to me by one of my lab courses. You can filter your searches according by authors, the number of cited articles, etc… The only downside is that it is extremely slow, and logs you out quite frequently.  Moreover, it opens up a new window every time you click on the article link.

To balance the advantages and disadvantages of each system, I use GS in conjunction with SF. Once SF gives me the article titles I’m interested in, I go to GS and type in the title. In a split-second (literally, 0.01 second), the article is found and its citation exported.

For exporting citations, GS is awesome. Log into your Google account and head to GS. Click on “settings” in the right-hand corner.

Scroll down to the lowest option – you can choose to export your citations directly according to the type of citation software that you use.

I use BibTex, so every article I search in GS will give me the option of “Import to BibTex.” All I have to do then is copy/paste the BibTex information in a text document. Give it a try – it will change your world!


Look out for more Chemical Conundrum posts from Isabella in January! 


(Photo : By Tom Morris (Own work) [CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], via Wikimedia Commons)

Breaking memories

Breakingmemories_1This is the first post in a twice-monthly series. 

Hello fellow scientists!

Have you ever wondered what the life of an undergraduate researcher is like? Well I’m here to share my perspective on what it’s like to work in a lab as a student. My name is Andrea Weckman, and I’m a fourth year neuroscience major doing a full-year research project in psychiatry.

The typical image that people envisage when they think of a lab involves test tubes, petri dishes, or rats. I spent my summer working in such a stereotypical lab — but my current research is completely different.

I stumbled upon the name of my supervisor while browsing Minerva for courses to take. I read his research blurb, which led to reading abstracts of his published papers, which led to me emailing him about the possibility of working for him during the school year. If there’s one thing I have learned during my four years as a student, and especially a student hunting for research positions, is that perseverance is key! So naturally, when my potential supervisor didn’t reply after one week, I emailed him again. And again. Until finally, a reply! My apparent desperation for this research position was based on two things: my supervisor is in the department of psychiatry, a field that I am very interested in as a potential career path, and, his research doesn’t involve test tubes or rats, it involves people! Real-life, living, breathing people! An extremely refreshing and welcome change from my previous research position analyzing histological specimens all day, every day.

I work at the lab, or from home on lab related stuff, for approximately 9 hours each week. I am working on a clinical project involving patients with posttraumatic stress disorder (PTSD). There are two phases to the trial, but the first phase is the one that I will focus on. It is a double-blind, placebo-controlled trial, the objective of which is to test whether the drug propranolol, when given before evocation of a traumatic memory, is capable of reducing subsequent physiological trauma-related responses such as heart rate, skin conductance and EMG recordings, as well as self-reported PTSD symptoms. The trial involves one experimental group, three control groups, and an overwhelming amount of data to deal with! The hypothesis, of course, is that the experimental group will have a greater reduction of these responses and symptoms than the other groups. Since the project has been underway for two years already, my main role so far as been to sift through mountains of data to get a statistical idea of what the results will eventually look like.

The monotony of exploratory data analysis, however, is broken up with exciting data collection sessions involving real patients, that make it all worth it! The ultimate goal of this research experience, as it is with most research experiences, is the production of a coveted publishable report.

Will I make it? Stay tuned to find out!

The Chemical Conundrum


The Abstract will feature every-other-weekly posts from three undergraduate students who are doing research at McGill. This is the first post in the first of those series. Stay tuned for introductions from our other regular bloggers next week!

Hello fellow Abstract readers!

I’m Isabella Liu, a fourth year Chemistry student. I completed my honours project last summer and am currently continuing said project. I am working in the green chemistry laboratory in the Otto Maass building – basically, look up to the fourth floor,  and that’s where you’ll find me working two days a week under the supervision of Professor C.-J. Li. (In the photo above, Dr. Li is on the very left. I’m the one in pink!).

In my lab, we search for greener and more atom-efficient methodologies to replace the reactions that you’re learning in your organic chemistry courses. What I do is search for green methods of C–C bond formation, an essential task in constructing organic molecules.

Historically, C–C bond coupling has only been possible if functional groups were attached to the coupling carbons (pathway 1). An example of that would be a Grignard reaction – something that should be very familiar to people who’ve taken Organic Chemistry 2!   However, reactions that require functional groups have low atom economy (which means not 100% of your reactants are converted into your product, a.k.a. you have undesired products in your reaction). This type of C–C bond coupling requires the pre-functionalization of C–H bonds, which is time consuming. Since the use of functional groups  is not green, we use metal catalysts to activate the C—H bonds. This technique is called Cross-Dehydrogenative Coupling (CDC), and was actually developed by my supervisor! CDC reactions allow chemists to skip the pre-functionalization step (pathway 2).  The advantages of this type of reactions include: (i) fewer synthetic steps; (ii) higher atom economy; (iii) less toxic waste; and (iv)the ability to use safer and cheaper starting materials.


What I did over the summer was an optimization process for the coupling of isochroman and dimethylmalonate. This reaction produces an enantiomer, which means the products are mirror images of each other (like your hand!). We are interested in enantioselective coupling reactions due to their potential pharmaceutical applications (many drugs only work properly if they are a specific enantiomer, – if the drug is “left-handed”, so to speak, instead of “right-handed”, it won’t work). Over the summer, we’ve optimized it using a chiral ligand, L*82, catalysing the reaction with Cu (I), which gave us a pretty good enantiomeric excess value (measures how much of a particular enantiomer we can get). Over the course of the semester, I’ll update you on my project as well as the little anecdotes of what it’s like working in an organic chemistry lab.

It was nice meeting you all, hope you enjoy the blog!


(Photo credit: Ping Wang, a visiting scientist)