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Why do some people hate music?

Have you ever come across someone in your life, who hated music? Did you force them to listen to your favorite song and expected them to get excited but you got a poker face instead? Did  you fully give up and just assumed that they were a weirdo? Well to be honest they aren’t weird, its just that they are unique in their own way! And most likely they might be experiencing a condition called as  Specific Musical Anhedonia!

Source: Neurocurean.com

Anhedonia is a condition wherein a person is unable to derive pleasure from any activity, in simpler terms it could also be referred to as depression. Musical anhedonia is a specific form of anhedonia, wherein a person is able to experience pleasure in every other activity however when it came to music, they couldn’t care less!

Although it is not a serious medical condition, did you know this condition is experienced by 3-5% of the global population? Well you could say that’s a small percentage of people, but why should you care? Read more to find out!

According to a recent study published in the journal of PNAS, researchers were able to establish the neurological basis of this condition. The study provides a direct evidence that sensing pleasure from music is mostly likely linked to the reward processing center of the brain. Well what exactly does this reward processing center do? Let me break it down in simpler terms, did you ever win money while gambling, had the first sip of hot chocolate you were craving for the entire day! what was the first thing you felt after doing these above activities, you must have gained some sense of pleasure right!. So,the reward processing center in the brain mostly helps in experiencing pleasure from most activities like good food, monetary gains, sex etc..

Based on the observations conducted on people with different spectrums of interest towards music, the researchers through brain imaging analysis (fMRI) were able to decipher that music anhedonia is most likely caused due to less connectivity or wiring between the auditory cortex (associated with hearing) and the reward processing center in the brain.

Source: S.K Wang et al (2016) American journal in Physiology

What do you think would be the significance of the study? The findings of the study conducted could potentially be used to develop new reward based therapies for people experiencing depression, autism and gain further insight about other addiction based disorders. This could probably help us understand  reward mechanisms in connection with different regions of the brain. Also, one could probably be able to look into the evolutionary aspect of how music integrated itself as a pleasurable activity in  the lives of us humans!

Source: Billboard.com

So now that I provided you with some extremely rewarding information , I hope you are amazed with how differently each of our brains are wired and how different each of them function and help us experience things!

Fun fact: One ongoing research on music anhedonia is actually funded by the Grammy museum!

Published on 18th November 2019

-Harshitha Nagesh

Is a Lack of Sleep Causing you to Age Quicker?

Are you pulling all-nighters to catch up on assignments and study for exams?

A new study  has found evidence that a lack of sleep can lead to many affects that could be detrimental to your health, such as cardiovascular disease, and most surprising of all, premature aging.

The study, which used fit-bits and other wearable sleep-quality assessing devices to track the average hours slept and and quality of sleep people got per night found that people who slept less on average per night had shorter telomeres than similarly aged peers who slept a healthy amount, which may prove that they’re aging quicker. The researchers considered above 7 hours to be a healthy amount for an adult to sleep each night.

Telomeres are part of your DNA, at the very end of the strands. Their purpose is to act as a prevention from important DNA code from being cut short. This is important because DNA actually gets shorter every time it is used, thus the shorter your telomeres are, the closer you are to losing important parts of your genetic code.

DNA in it’s signature “coiled ladder” formation.

Because telomeres often lose length in a predictable way, shorter telomeres are a very good indication of aging, and are thus often used as biological markers for age. Furthermore, telomere shortening has been linked to all causes of death, as well as quicker onset of age-related diseases. Due to this, shorter telomeres in sleep deprived people is evidence that their poor sleep may cause them to age quicker, and ultimately die younger.

Beyond premature aging and cardiovascular disease, which I mentioned earlier, a lack of sleep has also been linked to many other health complications such as high blood pressure and obesity.

Often times, we students glorify our lack of sleep, and accept it as a natural consequence of the heavy course load that we chose. Some people may even claim that lack of sleep is impossible to avoid as a University student while using caffeine to fight exhaustion. A recent study found that up to 60% of University students are not getting enough sleep. Research such as these studies show that there are serious consequences to a poor sleep schedule, and that getting a good sleep should be a priority.

Where do we go from here? The researchers believe that this important insight into the health effects of poor sleep should inspire public policy changes. For example, later starting times for schools and work places could be a good place to start, so that people can fit a healthy amount of sleep into their schedules. 

And as for you, the person reading this, remember: get your seven hours – you’ll live longer.

-Gurkaran Bhandal

Google AI Dominates 99.8% of Population – At Least In StarCraft II

AlphaStar, DeepMind’s StarCraft II playing AI, has achieved Grandmaster level in StarCraft II and is now ranked higher than 99.8% of the active StarCraft II player base.

StarCraft II Logo – pngimg

StarCraft II is one of eSports most beloved games. The strategy game features fast-paced real-time strategy and battle between players who take command of one of three fictional in-game races.

However, players faced a new opponent this summer – AlphaStar, Google AI firm DeepMind’s foray into the world of eSports.

 

StarCraft II‘ s Extraordinary Complexity

StarCraft II  was chosen for its incredible complexity. StarCraft players control hundreds of soldiers, vehicles, and warships simultaneously and in real-time. Furthermore, unlike common board games like chess and go, StarCraft II players cannot see what the opponent is doing unless the player physically sends a unit to gather information. This meant that AlphaStar had to choose the best of the 1026 possible actions that it can make based on the information that it had gathered.

Gameplay of StarCraft II – Vicente Alfonso, Flickr

 

AlphaStar’s Journey and Its Next Steps

AlphaStar used a fully-automated system to learn how to play StarCraft II. It used many general-purpose learning techniques, such as self-play and imitation learning, where the AI plays against itself and imitates others respectively, to teach itself.

Depiction of some learning methods and challenges involved – DeepMind

AlphaStar was first pitted against two professional level players in December of last year, however many felt as though it wasn’t a fair fight.  Therefore, DeepMind adjusted and limited AlphaStar’s reflex speed to even up the playing-field before unleashing AlphaStar onto the European StarCraft II servers in July.

In order to keep the experiment blind, DeepMind used a myriad of techniques, such as playing on multiple different accounts and constantly changing between these accounts.

Although AlphaStar wasn’t able to beat all opponents it met, many of which were some of the best in the world, it was still able to achieve a very high ranking and can be considered a huge step in progress for artificial intelligence.

Although DeepMind considers AlphaStar a success, StarCraft II is far from being beaten. AlphaStar does not quite play at the world champion level and you should expect to see more from AlphaStar and other artificial intelligence in the future.

 

Written by Tim Chan

How we recycle isn’t helping the environment

(Credit: Vasiliy Ptitsyn/Shutterstock)

The effects of recycling seems to be over exaggerated due to the fact that not many are aware that improper recycling harms the environment than it does good. Tragically, improper recycling is done by many, whether intentionally due to situational circumstances or because they are unaware of proper practices. For your understanding, improper recycling is defined to be recycling strategies that makes it harder for recyclable materials to be recycled, or harms the environment more than intended. This article will go over some of the major improper recycling methods that we need to be aware of, so that we can assure that we take full responsibility for our future – a responsibility that everyone must hold.

 

One thing that we do that represents improper recycling the most is how we dispose of our plastic bags. Interestingly, it has been found that many people put plastic bags in the plastic section, or one of the recycle bins. However, these plastic bags, even if they are stated to be biodegradable, cannot be recycled in this manner. This is because, firstly, biodegradable plastic bags need to be in a condition where they can be broken down by bacteria – hence “bio”-degradable. This condition is only met if the plastic bags are in landfill, but the bags will never be able to reach landfill if it is put in the ‘plastic’ section.

 

Ironically even if they are thrown away properly, green consumer advocate Debra Lynn Dadd states that contemporary landfill cannot degrade materials “because they are compacted so tightly, [they] do not let much air in”.  The severity of the issue can be further implied as she was able to find “25-year old hotdogs”, as well as “50-year-old newspapers” in the landfill that were still readable. This is what we define as situational improper recycling, where we are not able to recycle efficiently even when putting in our efforts to do so, because the circumstances does not support it. This calls for attention by the government, or any other institutes to devise a new method into reinforcing biodegradation in landfills. However, the general public is not demanding, or aware of this fact because they are negligent towards it.

 

Furthermore, Items that are in the plastic section is moved to a recycling center, where workers sort items accordingly. One would assume that machinery would be involved, as the daily trash production globally is immensely high – and yes, machinery are involved. However, when plastic bags are in the recycle bin, the “soft plastics clog up the machinery”, meaning that you could break the recycling center. When plastic bags enter the recycling center, Mike Tunney, area recycling director for Waste Management states that it “contaminates their recycling streams, [slows down] their systems, drives up [the] cost [of the process], and hurts the quality of the materials”. Specifically, Tunney informs us that the plastic would plug the gaps of the machine, and that more than five hours are spent removing these materials from the machine. Plastic bags, therefore, should either be “reused”, returned to retailers, and most importantly, thrown into “regular trash”.

  • Figure 1. News that outlines how improper recycling practices of plastic bags do more harm than good to the environment

 

Another issue that needs to be outlined is that people are simply, not recycling properly. Mark Badger, the executive vice-president of Canada Fibers state that “almost one in three pounds of what goes in a blue box shouldn’t be there”. This, contaminates the existing recyclable materials from being recyclable to non-recyclable, as it is stated that even a few spoonful of peanut butter can contaminate a tonne of paper. Furthermore, China, the world’s biggest importer of recyclable material, started banning imports of paper with more than 0.5 percent contamination, making contamination an even more significant problem. This meant that these paper, had to be in almost perfect condition. Any small deformities, such as a stain on paper has now made it become utter trash. To add insult to injury, it has been found that “North American cities are struggling to meet” the standard of the 0.5 percent contamination, serving as a further implication of how poorly we are recycling.

  • Figure 2. This outlines the contamination rate of recyclable material that is caused by improper recycling practices all across Canada. Statistics are shown to be from April 2018. Further implies, that most of North America isn’t passing the threshold to pass paper material over to China.

 

So, what can we do? First, it is clearly evident that we must be careful of how we dispose of our plastic bags, or any other items. Never assume that something should go in a blue bin, even if it has the recycling symbol. This is because depending on your region, it may not be recyclable, like how a “coffee cup lid [is] not recyclable in Toronto”. Search up what is recyclable to be able to identify what materials go in what, and plastic bags should go into the regular trashcan. There are many more things that you can do to properly recycle – and there are many more improper recycling practices that are continued today, that you may not have known. There are simply too much to cover in one article. So, do the research – bear the responsibilities that we must take in order to protect our world, for today – and for tomorrow.

 

Written by – Isaiah Youm

Air Pollution in New Delhi

Air quality in New Delhi has worsened in the last decade. According to a WHO survey, Delhi has the worst air quality as compared to other major cities around the world. The poisonous particulate matter of air in New Delhi is 440 pm which is 12 times worse than the recommended level of 35 pm. Air pollution is responsible for killing millions of people across India, from lung cancer, ischaemic heart disease, chronic obstructive pulmonary disease, acute lower respiratory infections, and diabetes. Air pollution in New Delhi is expected to reduce the life expectancy of its residents by 20 months and responsible for irreversible damage to the lungs of 50% of all children in the state. 

Reason for Pollution 

  •  Air quality worsens in the capital around this time of the year due to stubble burning when crops are being harvested. 
  • Around 60% of the population in India still use fossil and solid fuels to cook food, thus increasing the carbon dioxide level in the air. 
  • Firecrackers burnt on Diwali, known as the festival of lights also add to the increasing smog levels. 
  • New Delhi has a population of around 20 million people and carbon dioxide emissions from 8.8 million registered motor vehicles.  

Current Situation    

On 3rd Nov 2019, the air quality level climbed to a hazardous level and 37 flights were diverted from the New Delhi Airport due to poor visibility. New Delhi’s Chief Minister had to shut down schools due to the current hazardous air quality and announced that 5 million masks would be distributed to schoolchildren across the state. The government has given out temporary bans on construction activities and industrial plants to control the pollution level. Further, the government has implemented an odd-even scheme for cars under which vehicles with an odd number will not be allowed to drive on even dates of the month.

Next Steps to reduce air pollution

  • Burning waste – Heavy fines should be imposed on people who burn garbage, tires, leaves and other wastes.
  • Solar energy – Installation of solar panels at home should be encouraged and subsidies should be provided to residents.
  • Carpooling- Congestion free lanes can be made for cars with three or four people to encourage people to carpool to work.
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Group A- Simarbir Singh Dhaliwal

Random math

Consider the direction of a leaf floats as it falls from the tree, is the direction random? Is a coin flip random? Is the following sequence random?

5359408128 4811174502 8410270193 8521105559 6446229489 5493038196

What is randomness

Before revealing the answer to those questions, I want to introduce what is randomness. It is very hard to define randomness. Take a minute and try to think of a way to define randomness in your own words. You might think of flip a coin, roll a die, or draw a card from a deck of cards. There are many definitions of randomness in different branches of mathematics. However the basic intuition of randomness is the same; randomness is events happen by equal chance, and there isn’t a pattern can help people to recreate the sequence, like 10101010… just “10” repeats itself.

Randomness and Chaos

Is there a difference between randomness and chaos? Let’s first consider a dice. Is dice rolling random? In a paper in 2012, researchers from the University of Lodz developed a three-dimensional model of the dice throwing to predict the outcome of the result. With some initial conditions like the viscosity of the air, the friction of the table and so on, the model can provide an accurate prediction. This means starting from 2012, at least for scientists, throwing dices is no longer random. Chaos system, in mathematical terms, is a system that has extreme sensitivity to initial conditions, but with precise initial conditions, the results can be calculated.  A good example of the chaos system is the weather system. With the perfect knowledge of the weather, like the real temperature, humidity, cloud and etc. in every unit square of the earth, with support of a good model, we can calculate the exact weather data for any location. However, for a random system, by definition, the output is independent with the input, which means even with the complete information,  people still cannot predict the output.

So the answers to the questions at the start of the blog are all false. With perfect initial condition and a proper model, the direction of a leaf floats as it falls from the tree can be calculated. And this is true for the coin toss as well. And in fact, there are many studies about coin tossing, and I linked a few interesting ones in the further readings section of the blog. The sequence of the numbers is from 141th to 200th digits behind the decimal point of the famous irrational number pi. With this information, we can easily reconstruct the sequences.

Application to randomness

Randomness seems like a very theoretical concept, however, it has many applications. Randomness is used in computer science, information theory, physics, statistics, gaming, music, art, and so many other areas. I will quickly go through a few important ones.

In computer science, pseudo-random number generators (PRNGs) are used very often. The idea of a PRNG is using a seed(integer), and a set of rules to generate more integers with a given range with the probability of each number within the range to be the same. True random number generators (TRNGs) is another random number generator used in computer science. A TRNG generates the random number from a physical process and directly measure it with an external device. By comparing the two pictures below, the first picture has no clear pattern; however, the second picture has strip patterns.

Generated by a true random number generator,from https://www.random.org/analysis/

Generated by a true random number generator, from https://www.random.org/analysis/

Generated by a pseudo-random number generator, from https://www.random.org/analysis/

Generated by a pseudo-random number generator, from https://www.random.org/analysis/

 

In many scientific fields, computer simulation is an important tool. Computer simulation relies on a predetermined model and either pseudo-random number generators or true random number generators.

An online pseudo-random number generators:

https://www.mathgoodies.com/calculators/random_no_custom

A true online random number generators:

https://www.random.org/

Further readings/watching material

An interesting video about pseudo-random number generators and more

 

the butterfly effect in chaos system:

https://en.wikipedia.org/wiki/Butterfly_effect

Comparison between real random number generators and pseudorandom number generators:

https://www.random.org/randomness/

Research paper of the three-dimensional model of dice throwing: https://aip.scitation.org/doi/10.1063/1.4746038

Research paper of coin tossing:

https://arxiv.org/pdf/1008.4559.pdf

-(Sylvester) Shunkai Li

Video

Animal testing: Where do we stand?

A sketch depicting mutual dependency in human-animal relationships. Source: The Telegraph UK

Human dependency on animals has a long history spanning over a million years. Historically humans have partnered with animals for a variety of purposes required for their mutual consumption and existence. From domesticating animals for meat, agricultural practices, hunting activities, transport etc.., to  employing them for developing and testing therapeutics, it would not be completely wrong if I were to say that we humans became a little selfish along the way. For more than a decade there has been an increase in this awareness but there is still an ongoing debate with the pros and cons of employing animals for research. But how far have we come along in developing effective medications at the expense of animal welfare? Read more to find out! (P.S Some images have graphic content!)

Rodents are most commonly used laboratory organisms. Source: Flikr

First of all, if we look at the historical origin of using animal models for  research, the idea was born 2400 years ago when researchers realized the functional and anatomical similarities between humans and animals. Prominent thinkers such as Aristotle documented their findings based on observational and experimental studies conducted on animals and this was further spread across Europe and other parts of the world. With technological advancement in the previous decade such as sequencing of mouse and rat genomes and the development of the first knockout mice/rat , animal models have become an indispensable tool in biological research. Advantages such as the ability to recapitulate life processes, understand mechanisms under normal and diseased conditions and manipulate these mechanisms to develop novel effective therapies have favored their increased use in research. Significant breakthroughs have been made using animal models in the past decade in fields such as aging, neurodegenerative disorders, different types of cancers, gene editing etc…

Bovine fetuses are subjected to cardiac puncture during FBS extraction.Source: Trancend.org

One could argue that animals models promote animal welfare by developing therapeutics for animal diseases however majority of experimentation conducted on animals are extremely inhumane. To give you an image of how badly animals are exploited, let me state an example. Foetal bovine serum (FBS) is a constituent derived from young calves that is used for culturing human/animal cells outside the body. It is widely used by labs across the world but not many researchers are aware of its ethical concerns. FBS is harvested from bovine fetuses by puncturing the heart using a syringe and drawing blood. All this is done without anesthesia! And the reason this is being done in live organisms is because blood drawn from dead fetuses have the tendency to clot which is unfavorable for research purposes.

 

Source: Flikr

For experimental purposes animals are subjected to various activities that involves restraining, collection of  blood samples from various parts of the body, performing surgeries with or without anesthesia, overdosing, behavioral tests, inoculation of tumor producing cells, sacrificing and harvesting body organs for analysis etc.. which causes severe discomfort to animals. Even though its extremely difficult to eliminate animals from being used in research, steps can still be taken to reduce suffering as much as possible.

So where do we currently stand and what can we do to promote animal safety ?There has been a major debate in the past decade and  a lot of steps have been adopted in hope to promote animal welfare. Recently, 3D cell culture systems, micro arrays, artificial organs, organs on an electronic chip, computer based simulations have been developed, it is better to reduce animal experimentation as much as possible or replace them completely using these or other alternative strategies. Adaptation of the 3 R’s that aims to replace, reduce and refine the use of animal experiments needs to be strictly employed by research investigators to ensure animal welfare. Scientific personnel must be well trained in animal handling and certified prior to conducting experiments. Furthermore, the concept of ethical applications have been adopted by various countries that evaluates project plans prior to conducting experiments. These evaluation committees are composed of panelists from various backgrounds that assess  and approve whether a certain study ensures animal safety or not. Most cosmetic industries are also adopting to cruelty free methods of testing their products, which are increasingly popular among their consumers, this should be encouraged further.

Source: Flikr

In conclusion, it is possible to adopt alternative strategies but currently it is not feasible to completely abandon the use of animals in research. Some would argue that this is still beneficial for  a mass population when compared to the sacrifice of a small percentage of animals. That small percentage could grow rapidly and possibly lead to ecological imbalances and suffering in the future so this issue still needs to be taken seriously. With rising technological advancements and inventions such as organ on a chip  (video by Wyss institute included below) there is a small ray of hope that animals will be completely spared from being used in scientific experiments.

Written by Harshitha Nagesh

 

 

How Do Wombats Make Cube Shaped Poo?

How do wombats make cubed poo? That was what Patricia Yang and her fellow researchers at the Georgia Institute of Technology wanted to know. Their research, which was first presented at the 71st Annual Meeting of the APS Division of Fluid Dynamics and which has since gone on to win an Ig Nobel Prize earlier this month, sheds light on the processes involved in the creation of the distinctively shaped feces.

Wombat – Wikipedia Commons

 

Why Cubes and Why Is It Interesting?

In the modern world, most cubes or rectangular prisms are formed by one of two methods, extrusion and/or injection molding.

Extrusion is the same process that occurs when you squeeze a tube of toothpaste out, producing a long cylindrical piece. In essence, extrusion involves pushing a material through a die shaped like the cross-section of the final product (in the case of a cube, the cross-section would be a square). The resulting extrudate (the material that has been extruded) is then cut to size.

Injection molding is takes a material in a heated and liquid state, and injecting it into a mold. As the  material cools, it takes on the shape of the mold cavity. The mold is then separated and the resulting injection molded piece is then cleaned.

Cubes are a shape rarely found in nature and therefore, the wombat’s cubic scat has always been a fascinating mystery.

Lego, Injection Molded Plastic Toy Bricks – Semevent, Pixabay with Editorial License

 

As it turns out, wombats use piles of their poop to mark territory, much like how dogs use urine to do the same. Like their plastic brick counterparts, cubed poos are simply better at stacking than conventionally shaped fecal matter.

Wombat Excrement, Showing Distinctive Cube Shape – Wikipedia Commons

 

So How Do Wombats Make Cubed Poo?

When Dr. Yang and her fellow researchers were given two corpses of wombats euthanized after vehicle accidents, they began to dissect them immediately. The researchers began by inflating a long balloon into the animal’s intestines. They measured the elasticity of the intestinal walls and found that the elasticity varied at different points of the wall. The researchers theorize that the stretchier sections were responsible for the curved corners where as the more rigid sections were responsible for the straight and flat edges of the poop. They also found that only the last 8% of the intestines were actually responsible for the shaping of the excrement. Some also think that wombat diet has much to do with the formation and stability of the cube shape. The dryness of wombat diet helps maintain the rigid shape.

Why Is This Important To Us?

As mentioned previously, we normally use extrusion or injection molding to form cubic shapes. The revelation that we can use soft elastic tissue to form cuboids may provide new techniques for manufacturing.

 

Written by Tim Chan

One Atom Thick “Supermaterial”: Graphene

What is graphene?

Essentially a 1 atom thick sheet of pencil lead, graphene is the first “2D” material we’ve ever come across and holds unique properties that boast a wide berth of potential applications. Like all other purely carbon materials, it is in the arrangement of the carbon atoms that give the material its specific properties. The difference between soft, flaky graphite and extremely tough diamonds are in how their carbon atoms sort themselves, and evidently, in how many atom layers they do so in. The hexagonal arrangement of carbon in graphite boasts strong bonds between the atoms, playing a part to graphene’s immense strength once isolated. University of Manchester researcher Andre Geim and his team were able to perform this isolation by using tape on graphite repeatedly, until only graphene remained, earning Geim a Nobel Prize in 2010 for his findings.

Alex LMX / Shutterstock

What’s special about graphene?

The unique properties the carbon sheet takes once taken from graphite into a single atomic layer are what earn graphene the title of a supermaterial. First and foremost, its incredible strength is over 40x stronger than that of a diamond. Prof. Hone at Columbia University puts this into perspective in his words, “it would take an elephant, balanced on a pencil, to break through a sheet of graphene the thickness of Saran Wrap.”. Beyond that, graphene is also a great conductor of both heat and electricity, moving electrons 200x faster than silicon. Graphene also absorbs very little light, making it translucent. With all these amazing properties in one material, the applications are countless and exciting, some currently in practice include glasses, and sporting equipment. While the current logistics of incorporating the material into transistors and other high-scale applications are yet to be feasible, further research and investment into graphene is in progress and marks an exciting development in the future.

 

-By: Tae woo Kim

infinitely interesting

What is infinity? I heard many different answers. A math student told me that infinity is the number with the greatest value, since infinity has larger value than any finite value, and it is great! In this post, I will give an introduction to infinity and present some of the interesting paradoxes of infinity.

Most people know infinity from the high school math courses. Infinity is just a result of a number divided by 0. The statements high school teachers taught are correct, but that is not the end for “infinity.” Infinity is a state or a concept. In the Oxford dictionary, infinity is defined as “the state of having no end or limit.” We know that infinity is greater than all other numbers, but what if we compare infinity with infinity? If there are two sets, each of which has infinitely many elements, which one is bigger and which one is smaller?

To answer those questions, let us start with how mathematicians compare the size of the set. To compare sets, mathematicians usually pair up elements, one from the first set and one from the second set. If each component can be paired with a component from the other set, and there is no element left after the pairing, they can conclude the size are the same. Consider the following scenario, we want to give candy to children, and we do not know how many candies are there. We can give one candy to each child. If we still have candy left after we give each child one candy, we know the number of candy we have is more than the number of children. If we run out of the candy before each child gets one candy, and some of them start to cry, you will know there is less candy than you expect (hope that will not happen).

Consider the following question, which set is larger, the set of odd numbers or set of even numbers? I think most of you will believe they have the same size. If you think this way, congratulations, you are correct! And consider this question, which set is greater, the set of natural numbers of the set of even numbers. I believe that most of you will believe the set of natural numbers is larger than the set of even numbers. Since the set of natural numbers are the set of odd numbers combining with the set of even numbers. However, they have the same size. Let’s verify the method we described above. We can try to match the even number with the natural numbers see if there are any numbers left after we match them. We can map two from the even number to the one in natural numbers, 4 to 2, 6 to 3,… In this way we can use the formula the even number 2n can map to natural number n to infinity, with no left elements from each set. The picture below also shows the idea.

mapping from natural numbers to even numbers

The picture shows the mapping from natural numbers to even numbers. From the article “Infinity and Trying To Do Maths With It”.

If you want to see some mind-boggling example of what happens if infinity happens in real life, see this video about Hilbert’s paradox of the infinite Hotel down below.

All the above examples I gave are only one kind of infinity. Mathematicians call it countably infinite. Countably infinite sets are the sets that we can pair each element in the set with a natural number, like even numbers, odd numbers, prime numbers and so on. But is that all kind of infinity? Does there exist other sizes of infinity? The answer is yes for both questions!

Picture from Wikipedia Cantor’s diagonal argument page

Consider a set of infinitely many strings with infinite length. For each pivot of the string, it is either 0 or 1. Then is this set countably infinite?  Actually, the set is not countably infinite. This is proven by famous mathematician Georg Cantor in 1891, and his proof is known as the diagonalization argument. The idea is as follows: construct the following string, take the first element of the first string and flip it (if the original digit is 0, then take 1; if the original digit is 1 take 0) as the first element, and take the second element of the second string and flip it as the second element, and continue (show in the picture below). The constructed string is different from any string since it is at least one fliped digit that is different from any other string. So we created a string that we cannot assign natural numbers to. So the set is not countably infinite. A direct consequence is that the real number is uncountably infinite. This is a link to a formal proof of why real numbers are uncountably infinite using diagonalization argument.

http://mathonline.wikidot.com/the-set-of-real-numbers-is-uncountable

In fact, there is a more important consequence of the diagonalization argument. The power set of a set has greater size, even the set is infinite. The power set is a set of all the subsets. By this consequence, there are infinitely many uncountably infinite sets.

Although there isn’t much application to the size of infinity, they are the fundamental part of set theory, number theory, real analysis, and more. There are so many branches of mathematics that are built upon these abstract and fascinating ideas. Other than it’s importance in mathematics, it is also something really poetic about infinitely many sizes of infinity.

For a review of concepts of infinity, watch the video down below.

-(Sylvester) Shunkai Li