Not taking Add Math and not having the slightest bit of differentiation in IGCSE unlike the syllabus now, I was terrified of studying calculus as I came in like a clean slate without any prior knowledge. Of course, I’ve always heard about how arduous it was from my seniors and classmates and noticed how the atmosphere went cold at the word “calculus”, but calculus in my opinion at least, isn’t as difficult as what I had expected dare I say compared to cough complex numbers. I enjoy doing calculus now to some extent because the problem is stated just as it is, leaving little room for misinterpretation.

Riemann Sum

The video above delves into and explains the Riemann Sum pretty well hehe. Butttt I will give a summary of the main concept anyways.

So if you were lazy to watch that almost 30 minute video, you might still be wondering what Riemann Sum is.

If you were told to find the area of ACDB, what would you use? No regular shape would fit that shape to give us an exact formula to find this area. Instead we can approximate the area obtained by summing up the areas of rectangles that try to approximate the shape of this curve in that specific region, which is what Riemann Sum is all about.

Like in this figure for example, we can split this region into three rectangles above the curve and below the curve. Rectangles above the curve include rectangles ACFE, EPHG and GQIB which when added together gives the value of the upper sum. This is because it gives an area bigger than the actual area of ACDB as seen by part of the rectangle above the curve. Upper sum is also known as left Riemann sum, named so because the upper left corner of the rectangles used touches the curve.

Similarly, rectangles below the curve include rectangles AEPR, EGQS and GBDT which when added together gives the value of the lower sum. This is because it gives an area smaller than the actual area of ACDB as seen by part of the curve not covered by the rectangle. Lower sum is also known as right Riemann sum, named so because the upper right corner of the rectangles used touches the curve.

The actual area of ACDB is going to be in between this upper sum and lower sum and can be approximated further to be equal to the average of these two sums.

Note that the area you want to find has to be defined in the sense that it is bounded between for example two vertical lines, f(x) and the x-axis as in this case.

Finding the upper and lower sum was repeated again but with 6 rectangles each above and under the curve instead of 3. When this is done a slightly different value is obtained as seen in the video. Butttt, if you compare the average of the upper and lower sum using 6 rectangles to the one with 3 it will be closer to the actual value under the curve found with definite integral.

Now, what if we have 12 rectangles or 24 or even 100? We can generalize the method done to find the area with 3 and 6 rectangles in the following formula. Take note this specific formula is only applicable for decreasing functions and would need to be switched in the case of an increasing function.

To make things more clear, X sub 0 would be point A in Figure 3 and point B would be X sub 1 and so on.

Since Riemann sum is just an approximation, if we wanted to find the exact area under a curve we need to utilize the concept of limits. To do this, the width of each rectangle must approach zero, which in turn means that the number of rectangles n that make up the region would approach infinity. All of the above has now helped us derive, or better said define the Riemann Sum to find the area under the curve, which is:

This equation can be rewritten as what is globally known as the definite integral that gives us the area between the x-axis and the curve over a defined interval [a, b]:

You might have noticed that using a definite integral is essentially the same as following Riemann’s sum using an infinite amount of shapes (rectangles) and an infinitely small value for the width of each rectangle.

The Fundamental Theorem of Calculus

Differential and integral calculus are related to each other, although they might seem unrelated at first. This relationship is very important as seen from the self-explanatory title of this theorem. We will see how they are related in the following parts.

First Theorem

If f is continuous on [a,b], then the function defined by

is continuous on [a,b] and differentiable on (a,b), and S′(x)=f(x).

In simpler words, integration is the opposite of differentiation. This can be more clearly rewritten as below:

Second Theorem

If f is a continuous function on [a,b], then

where F is an anti-derivative of f, i.e. F′=f.

Thus, from these theorems we see that differentiation and integration are indeed interrelated with each other.

International Mindedness

Before formulas were invented to calculate the area under a function, this concept was explored by the ancient Greek astronomer Eudoxus using the method of exhaustion. He attempted to find the volumes and areas by splitting up the region into an infinite number of divisions. This method was further applied by Archimedes in the 3rd century BC to find the area of shapes like circles, ellipses etc.

Much later on, the relationship between calculus and integration was made by Liebniz and Newton independently during the late 1600s and early 1700s. Both of them have made equally important contributions to the mathematics field (Rosenthal, n.d.).

Real Life Applications

Calculus is used everywhere especially by engineers. There is a strong connection between physics and calculus.

1. Before cars can be sold, the center of mass needs to be checked using integration. This aspect is important to check as it determines the safety of a car on different types of roads and speeds (“Center of mass and gravity”, n.d.).

2. To model predator-prey populations can be done with the Lotka-Volterra equations which are differential equations (“Lotka-Volterra Equations — from Wolfram MathWorld”, n.d.).

3. To determine the rate of a chemical reaction using integration (Reyes, n.d.).

4. To study the spread of an infectious disease which relies immensely on calculus. It can even determine how far and fast a disease is spreading amongst other uses (“Using Calculus to Model Epidemics”, n.d.). This is very important especially in today’s world where Covid-19 is still rampant.

IB Learner Profiles

Communicators – Throughout this whole investigation all the way to the finishing touches leading up to our presentation, communication was an important aspect in completing our work. This learner profile allowed us to delegate tasks and correct each other’s mistakes.

Balanced – We made sure to split up the work in a fair way. More than that, doing the toolkit we made sure to not do all the work the night before and did it in small increments which ensured that we were able to get a good nights rest and have some personal time.

Risk-taker – Even though one of our teammates wasn’t able to present abruptly, we thought of a way to overcome this. Although it was a challenge to present with 3 people rather than 4, we found a way to overcome this issue when we decided that prerecord the voice.

Thinkers – One part of the toolkit which required of me to come up with a generalized formula made me think deeply. It took me a while to get the solution but once I did it it felt like an aha moment.

Open-minded – There were probably times in this toolkit where each one of us made mistakes, at least I know that I made some. Instead of looking down on these mistakes, I took people’s thoughts on what I did wrong and made it into a learning experience to grow from.

Sources

Center of mass and gravity. Retrieved 1 November 2020, from http://ruina.tam.cornell.edu/Book/COMRuinaPratap.pdf

Fundamental Theorem of Calculus | Brilliant Math & Science Wiki. Retrieved 1 November 2020, from https://brilliant.org/wiki/fundamental-theorem-of-calculus/

Lotka-Volterra Equations — from Wolfram MathWorld. Retrieved 1 November 2020, from https://mathworld.wolfram.com/Lotka-VolterraEquations.html

Reyes. Differential and Integrated Rate Laws. Retrieved 1 November 2020, from https://laney.edu/abraham-reyes/wp-content/uploads/sites/229/2015/08/Differential-and-Integrated-Rate-Laws1.pdf

Rosenthal, A. The History of Calculus. Retrieved 1 November 2020, from http://people.math.harvard.edu/~knill/teaching/summer2014/exhibits/lagrange/history_calculus_rosenthal.pdf

Using Calculus to Model Epidemics. Retrieved 1 November 2020, from https://homepage.divms.uiowa.edu/~stroyan/CTLC3rdEd/3rdCTLCText/Chapters/Ch2.pdf

Journey in MAAHL

Entering IB, I decided to take MAAHL because I wanted to pursue my aspirations of becoming a chemical engineer – which one of the university requirements usually entailing a 5-7 in MAAHL with Imperial College London requiring a 7 (jokes on me who won’t make the requirements lol). I didn’t expect to be taking IB and so I didn’t take Additional Mathematics in IGCSE. At that time, I was thinking that if AS Level Mathematics is like Additional Math, I will just learn it later on :D. Plans changed and I was stuck doing IB, but little did I know what I was getting into.

I still remember the first few days of math class where I felt so lost (I lowkey still feel lost now but not as much), everyone around me had already learnt log, binomial theorem, basic calculus, permutation and combination. I still remember how I got a 1 in one of my Amazing Tests, and 3’s on a Unit test and its retake respectively. The retake was only 1% from being a 4 which is the passing score :((. Along with the failing, I have become worried before every math test with butterflies in my stomach and feeling sick. I almost had a taste of passing but it was not until Semester 1 Term 2 did I pass. Taking MAAHL has allowed me to experience failing, but what’s more important is how you bounce back. I had to be putting in more effort to cover the Additional Mathematics topics in addition to MAAHL.

Now, currently in Semester 2 Term 2 having stayed the whole term because of COVID-19, I can look back at all this and say that this was an unpredictable ride. I would have never thought a day would come where I would help make a video (Video 1 below) for math class or the Mathletes ceremony or helping out during SISMO (check out my other post on this ;)) happened which was an interesting and fun project to handle that I probably will not have the opportunity to help out again with in the future :(.

In a way, not much has changed, this ride is still unpredictable, I am not sure how I will do in my math future as it is still the subject I am struggling most with. Although, I feel how well you do in a subject slightly correlates to how much you like a subject (as my love for math has definitely decreased exponentially :))) I sometimes do look forward to math class and certainly miss it right now, when online class can’t do justice to real life learning. I hope that I can work harder and stop being lazy so that I can improve my math grade and not feel math anxiety anymore (as I’m writing this I’m dreading how I would need to learn Add Math calculus in summer ugh).

TOK in Math

Is mathematics invented or discovered?

Math is also known as the language of the universe and for a good reason too. In physics, we learn that math shares no boundaries and is the common foundation that everyone can understand, hence its nickname as ‘The Language of the Universe’ that can explain the smallest of objects from the fundamental particles to the universe which is made of these particles. Mathematicians and scientists have been debating over whether this language is made up by humans or is just lying there waiting for someone to discover it, ever since Albert Einstein said his famous quote, ‘How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?’. This got people to really ponder whether or not math is truly as Einstein stated ‘made up by humans’.

Everyone has their own personal opinions and I hold the non-Platonist (not to be confused with platonic hehe) view. This view is held by those who believe that math is nothing more but a construct made from the minds of humans. This would explain why math can describe our world and universe so well, as it was invented to do so. As history tumbles along its path, so does math as it develops into a bigger and bigger hub of information, growing in size to accommodate and satisfy our purposes. I believe that if humans were to disappear one day, math would as well unlike the Platonist view which believes that math exists independently of humans much like electrons and protons that continue to exist whether or not humans are around.

International Mindedness

Which one: The Bourbaki Group Analytical Approach or the Mandelbrot Visual Approach?

As math evolved to the math we know today, there have been many different approaches to math. Two of the approaches which are often compared with one another is the Bourbaki and Mandelbrot approach. The Bourbaki group analytical approach was founded by a group of mainly French mathematicians. This approach to math is based on a more abstract foundation, and they are slightly against the use of images in their presentation. This has come to be as a result of the Bourbaki being Puritans, and Puritans have strong opinions against pictorial representations of truths of what they believe in. They were under criticism due to the extreme view the Bourbaki had as they have even reduced geometry as a whole into abstract algebra and soft analysis instead of using visual methods such as pictures.

On the other hand, the Mandelbrot – named after Polish-born French and American mathematician Benoit Mendelbrot – visual approach relies more on pictures as implied by ‘visual’. For example, entrancing images of the Mandelbrot fractal set exhibit an elaborate and infinitely complicated image that keeps on continuing into even more detail when magnified. Due to the beauty of the Mandelbrot set, it has become well-known outside of the mathematics field and has become a good example of visualizing math and is an excellent representation of the beauty that math can bring.

In conclusion, I believe that the Mandelbrot visual approach is more superior because it uses a lot of pictures to convey math which can aid understanding. Furthermore, as it is aesthetically pleasing, people are more likely to get amazed by it and as a result it can become the push one needs to be interested in studying math, especially for visual learners.

IB Learner Profiles

Reflective

As mentioned in the above paragraphs on my journey in MAAHL, I experienced failure during the first term (and perhaps during this SA as well :”), but I had to see what I did wrong and evaluate my mistakes in order to grow from them and prevent making the same fault again. Moreover, I discovered the optimal way for me to learn is by writing coherent notes and by going through the steps out loud, as if you’re teaching it to someone else, which I managed to discovered by trying out different methods.

Balanced

Although I felt the need to practice math everyday so I can improve, which it probably would have if I did do that, but I also had to acknowledge that I am human and it’s not worth it to burn out from overworking myself by doing extra math problems everyday amidst the work and projects that I am handling. At first, I struggled to find intellectual, physical and emotional balance as I always felt like one had to be sacrificed in order to achieve the other two due to time restrictions. Even to this day, I still feel that when I achieve two of these, one of them seems to be missing, an issue which I am still working on but it has definitely improved. As a result, I am more productive yet nevertheless still in the process of achieving harmony between these three elements.

Knowledgeable

Over this past year, I managed to apply math to problems that I never thought I would do such as the IA I did which investigated the percentage of the urban population living in slums against the prevalence of contraceptives using any methods by women ages 15-49 in Indonesia, which was eye-opening as it was the first time I made a research paper on issues of local significance. My knowledge was further enhanced by writing answers to TOK questions which are globally applicable.

Inquirers

The longer I stay in MAAHL, the more I want to know more about math although it is scary, it is interesting at the same time (?). Especially during the IA, we had to learn things that were not in our syllabus which enabled me to have the skills to conduct a more thorough research than what would have been possible if I did not go out of the syllabus.

Communicators

We learnt to communicate information in a plethora of ways, from making that video to making an IA to writing on our blogs. Again, as I’ve mentioned in my last blog post, I learnt the importance of communicating when collaborating with others especially in an important event like SISMO.

Thinkers

The many tests and dreaded so-called amazing quizzes and tests have really pushed me to think outside of the box and think of how to solve problems without the question being so direct in either their instruction or the given values. I’ve had to develop critical and creative thinking especially in chapters like complex numbers or vectors where the concepts seem so abstract, especially after all we’ve been learning the past years of our life deals with real numbers and vectors in 2d.

Principled

One of the biggest moments to show this characteristic is the one mentioned in my SISMO blog about how we had to not leak the questions to others even though they were our friends to make the competition fair.

Caring

This has been shown through my voluntary desire to tutor people math specifically the Sec4’s a couple of times, but other than that I help a couple of my friends with their math questions.

Risk-takers

I guess I took a risk when I wrote in pencil in that very first amazing test which I am very much traumatized by since I got a 1. But for real, this learner profile has made me less scared to try out a problem that seems unfamiliar even if I get it wrong.

Open-minded

When working in with partners or in a group, as I often had to such as the very recent Towers of Hanoi math assignment we had. I had to be considerate and willingly listen to other people’s opinions so that it could enhance my work.

Sources

https://www.huffpost.com/entry/is-mathematics-invented-o_b_3895622

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

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

“Anything is possible when you have the right people there to support you.” – Misty Copeland

Preparation Day

Ever since EE week when I first found out that we would be holding a SIS Math Olympiad (SISMO), the whole MAAH class have been invested in preparing for this event. I played a minor (and useless like I am irl) role throughout this event but still tried to do as much as I could and helped whenever I had the resources to. Along with Samantha, I helped gain math teacher contacts (liaison) from schools that were in the JABODETABEK area, by contacting my limited friends who went to those schools. I helped out with small tasks such as providing printer and ink cartridges, as well as glueing the name tags to the pins which the candidates would be wearing and stapling papers. Furthermore, Samantha and I were in charge of the social media of mathletes (follow @siskgmathletes on ig hehe) and would post instagram stories of the progress.

I really enjoyed doing this as it was memorable – it’s not everyday that you will be helping out in preparing for a math olympiad. Although I was surprised that the olympiad would be taking place on February 2 as it seemed to soon, we managed to attract around 130 people from around 14 schools within two months which may seem like an impressive or even impossible feat for the first ever SISMO but with everyones perseverance we pushed through. At times the olympiad seemed like an unattainable goal but as time passed by everything worked out. I feel like I learned not to underestimate what working in a team of determined people can result in. Also, I was happy that unlike a lot of events that international schools hold, the event was even open to national schools which definitely makes this event especially more inclusive and breaks the borders. Next year, I hope to be of more use in running this event.

SISMO Day

Right before the actual day we had a group meeting of the rundown of the whole event. In this way we knew what our jobs were gonna be and what/where we were expected to do/be at different times. We jokingly agreed to have a late fee depending on the amount of minutes we were late by. Surprisingly I think this was an effective incentive for people as many came on time (like me who usually comes pas”an) or almost on time, although it was raining, and some who came late even paid hahah. On the actual day, February 2, I was in charge of the reregistration for the advanced group which includes grade 11 and 12. I just had to mark whether or not someone has arrived and give them their name tag.

When the event finally started and my job as a registrar ended my job became to help out with the icebreaker which is the game 24. We had a crisis as we did not have enough card decks, but somehow we managed to overcome that problem and did not cause a commotion. The winner from each group would get the chance to draw an angpao from a box with varying amounts of money.

Later on as the event proceeded, along with the rest of the SISMO committee, invigilated while the participants took their tests while looking very stressed with their foreheads resting on their hands EHEHEH. Right after that we checked the papers right after their respective session ended so we could mark it and figure out who earned the most number of points as quick as possible. During the second session and lightning round, I unexpectedly took over the job of being an MC as one MC was missing.

Looking back on the event, it went considerably smoothly especially considering that this was the first time we held such an event. Furthermore there was some emergencies that was handled calmly and correctly by the others. I honestly never expected to help out with MC-ing an event ever in my life, as I was always a quiet person (but people have been calling me crewed recently :(( ) but I was glad I had the experience as it was eye-opening and a step outside of my boundaries. It was a really fun event as unlike other math olympiads, there was an icebreaker which I feel like helped made the competition a little less intense which is good and also the lightning round and Kahoot (although someone interfered with this and put bots heheh) was interesting as it is seldom seen in olympiads.

International Mindedness

Golden ratio

During the lightning round for the advanced category, one of the answers for one of the questions included the golden ratio. I remembered this as I have read about the golden ratio as it is used to judge how aesthetically pleasing something is. We can judge the attractiveness of a monument (such as Parthenon) or even a face depending on how close one’s face is to this ratio and which can be rounded to 1.62. There is even a list of the most beautiful women according to Greek mathematicians. Isn’t it cool? This ratio is represented by the Greek symbol “Phi”, named after the Greek sculptor Phidias who discovered it.

Combination and Permutation

AHH one of my favorite parts of math although I’m bad at it ;-;. I find it really interesting, like who does not want to know the total number of combinations of something occurring. Ex: The number of ways you can arrange 3 out of 4 letters. Based on Quora (idk how trustworthy this is), this discovery was greatly developed by Blaise Pascal and Pierre de Fermat – hmm don’t those names sound familiar ;). Blaise Pascal was a child prodigy (ugh can’t relate) who also discovered Pascal’s triangle which is used in binomial theorem. The name may be especially familiar for those who take physics and can recall that the SI unit for pressure is Pa (Pascal). Fermat on the other hand is famous for “Fermat’s Last Theorem”.

IB Learner Profiles

Risk Taker

Ahh, the IB learner profile no one fails to forget. This learner profile definitely played a role throughout this process of preparation leading to the big day. As aforementioned, I felt like it was such a big risk to be holding a math olympiad which was due to happen in less than two months as the school wanted to push it so it happened on the weekend of STEM week. This is also why I put that quote on the top haha. Being a risk-taker has helped me understand when it is feasible and reasonable to take a risk, as taking risks has helped me grow as a person outside of my usual comfort zone.

Communicator

As throughout the whole preparation we had to express our ideas fluently to one another in order to get our ideas across, in hopes of delivering the best experience during the olympiad. Not only were we expressing our ideas, but we also had to listen to each other in order to collaborate well. Also, during the actual day of the event, we often had to communicate with each other be it with hand signals or whispers to get things done the right way.

Principled

During the last few weeks of preparation, the rest of the SISMO gang got to see the papers. We had to be morally ethical in not revealing the questions to the participants who we knew personally.

Concluding rites (OoooOO sounds so fAncY)

I would just like to express my gratitude towards everyone (MAAH students , JC2 people and math teachers and anyone else not mentioned) who helped out as everyone’s effort contributed to the success of the olympiad, amidst all the school work piling up. I especially want to mention Audrelia, Roberto and the math teachers of course such as Mr Kichan who had such a heavy task to execute but managed to do.

Resources

(n.d.). The Golden Ratio. Retrieved from http://www.geom.uiuc.edu/~demo5337/s97b/art.htm

Waran, R. L. (2018). Who discovered permutation?. Retrieved from https://www.quora.com/Who-discovered-permutation

To what extent do instinct and reason create knowledge? Do different geometries (Euclidean and non-Euclidean) refer to or describe different worlds?

Instinct is a natural or intuitive way of acting or thinking and reason refers to the power of the mind to think, understand, and form judgments by a process of logic. Knowledge is formed from a mixture of both these characteristics. In my opinion, instinct influences a significant amount of knowledge, as there is a natural tendency for your mind to gravitate towards pondering on the unknown and ask the unanswered questions as to why things are. This question in turn leads to more questions and to more discoveries, hence the creation of knowledge. Reason, I believe also leads to the creation of knowledge, as through reasoning (which is a vital component of the basic fundamentals of math) a plethora of discoveries have been made, not only in the mathematical department but also in mathematically-related field such as chemistry and physics, to name a few.

Non-euclidean geometry is any geometry that is different from euclidean geometry. In Euclidean geometry, given a point and a line, there is exactly one line through the point that lies on the same plane as the other line. Non-euclidean geometry is divided into spherical and hyperbolic geometry. In spherical geometry there are no such lines. In hyperbolic geometry there are at least two distinct lines that pass through the point and are parallel to (in the same plane as and do not intersect) the given line. I think Euclidean geometry and non-euclidean both refer to our worlds, but Euclidean geometry refers to a simplified version of our world where everything is in two dimensions while non-euclidean geometry refers to the actual world and is needed for real-life application of mathematics.

What are the Platonic solids and why are they an important part of the language of mathematics?

The term Platonic solids are used when talking about shapes in three dimension. This name is given to shapes which have a regular, convex polyhedron. These kinds of shapes are made up of polygonal faces which have 2 characteristics: regular (equal angles and sides) and congruent (identical in shape and size). Furthermore, these polygonal faces must all also have the same number of faces meeting at each vertex. There are 5 solids which meet this criteria:

Let me give you a little history on these shapes. Their history goes way beyond our time, back to the time of the ancient Greeks and this is why some mathematicians credit the discovery of these shapes to Pythagoras. But in fact, these shapes were named after the famous Athenian philosopher, Plato, instead. He wrote about these special shapes in a book in which he compared 4 of these 5 shapes to the 4 classical elements, that is water, air, earth and fire. For the fifth Platonic solid, Plato remarked that the Gods used it for arranging the constellations on the whole heaven.

Fast forward to the 16th century, Johannes Kepler, a renowned German astronomer attempted to relate the five currently known planets of our solar system to the five Platonic solids. In the end this idea had to be abandoned, but at least out of the extensive research he had conducted, he has managed to make a name out of himself and contribute to some of the most famous scientific discoveries to this day in astrophysics.

You might think, well what are the use of these Platonic solids? Why are they so special? Even when we don’t know always know how they are used in the language of mathematics, we see these shapes around us whether we acknowledge it or not. For example, these 5 dice in the photo below are pretty common. Of course, the cube is the most prevalent type of die but I’m pretty sure you’ve encountered the other three types of die one way or another. It’s not a coincidence that die are made up of these shapes, and it’s because these shapes make a dice fair which is a quality we require when playing board games.

They are an important part of the language of mathematics because platonic solids are found everywhere in nature. The tetrahedron, cube, and octahedron all occur naturally in crystal structures. This does not mean that crystals are limited to those shapes though. Even viruses, such as the herpes virus, have the shape of a regular icosahedron. In fields of chemistry, platonic hydrocarbons have been synthesized, including cubane and dodecahedrane.

Is it ethical that Pythagoras gave his name to a theorem that may not have been his own creation?

Babylonians have already discovered a formula for finding an estimation of circumference using the formula 3D, where pi is taken as 3 at that time and D is the diameter. Furthermore they also discovered the formula for the area for a circle as well. Other civilizations such as the ancient Egyptians discovered areas and volumes. Erasthostenes, a famous philosopher also discovered the Law of Erasthostenes as well as the circumference of the earth. But, it is still disputable to say with absolute certainty that one person discovered something, as although we can’t disprove that they did have knowledge of that, we also can’t prove that they were the first ones to have discovered it. Civilizations that came before them might have discovered it but may not have written it down, or their records might have been lost.

Thus with all this being said, in my opinion it is ethical. We will never know whether these famous scientists especially those who were born a long time ago during the ancient times actually discovered whatever they are given credit for inventing. This is because it is easy to pretend to be someone else due to the lack of evidence. Hence, there is a high probability for many other theorems to be named after the wrong person. As Isaac Newton once said, “If I have seen further than others, it is by standing upon the shoulders of giants. ” Referring back to this quote, it resonated with me when I read the question. This is because I feel like any theorem is not created solely by ones own effort, but by the combined effort of many people (the Babylonians, Egyptians etc. for example), and for as long as we cannot falsify for sure that he did not create the theorem, we also cannot say that it is unethical. For example Shakespeare is rumored as well to not have written his own plays, but would it be ethical for us to strip away his name from the books? No, not until we prove that he did not write his own plays, we cannot accuse him for doing so and remove his name.

Citations

“Platonic solid.” Wikipedia, Wikimedia Foundation, 14 Jan. 2020, en.wikipedia.org/wiki/Platonic_solid.

“INSTINCT: Meaning in the Cambridge English Dictionary.” Cambridge Dictionary, dictionary.cambridge.org/dictionary/english/instinct.

“REASON: Meaning in the Cambridge English Dictionary.” Cambridge Dictionary, dictionary.cambridge.org/dictionary/english/reason.

A function is a relationship or expression involving one or more variables.  It is a relation wherein a set of inputs to a set of possible outputs where each input is related to exactly one output. In other words a function is a one-to-one or a many-to-one function (shown below).

An easy way to deduce if a graph is a function is by the vertical line test. This test is simply that if a vertical line at any point of the graph intersects the equation at the two points it is not a function. Some examples of functions are y = 3 and y =3x + 5.

These functions can further be transformed into other functions which look similar to the original function but are for example shifted to the left/right, up/down, reflected and/or horizontally/vertically stretched/shrink according to the change in function (as seen in the image below).

g(x) = f3

h(x) = f7

j(x) = f5

k(x) = f1

p(x) = f4

r(x) = f3 & f8

s(x) = f9

t(x) = f6

v(x) = f2

I could match the function to the line based on the characteristic of the equation. Equations like g(x) & r(x) were straight horizontal lines as it was y = a. I could deduce what j(x), k(x) & s(x) was based on the domain given. Domain is a restriction on the range of possible x values a function can have. The remaining functions, h(x), p(x), t(x) and v(x) are the formulas for semi-circles so I could match them accordingly based on the negative sign and the values inside the function which I will explain shortly.

• 

I used the Desmos application to graph this. I changed some parts of the original functions to fit the range desired, and also based it on the structure of a real bicycle including the pedals, gear and spokes I graphed. The functions depicted in the photo above include linear, semi-circular and square root functions (which will be explained shortly).

Linear Equation:

y = mx + b; where m is the slope and b is the y-intercept

Semi-circular equation:

where b is the height of the lowest point in the semi-circle from the y-axis, r is the radius of the semi-circle, and a is the x-coordinate of the highest point of the semi-circle

Square root equation:

The image below describes some of the changes in the shape of the function based on the changes in the variables.

Real life application

1. Forensic scientists use functions to help determine the height of a person based on the length of the femur. The function h(f) = 2.47 + 54.10, ±3.72cm. Where 𝑓 is the length of the femur. I find this example really exciting because I have watched numerous forensic TV shows which use this technique but didn’t exactly know it was an application of functions.
2. To determine the position of an object the distance an object travels as a function of time is given by 𝑠(𝑡)=12𝑎𝑡2+𝑣0𝑡+𝑦0𝑎, where a is the acceleration due to gravity (−9.81232m/s^2), 𝑣0 is the initial velocity and 𝑦0 is the initial height. This formula is used by many people who take physics.
3. In determining the price of objects such as the price of a car, it depends on the cost of many variables or raw materials: metals, plastics, glass, salary, location of the company, taxes, quality of manufacturing, etc. There are functions which include these variables to help determine the selling price. Similarly, the climate of any region is a function of temperature and air pressure, density, ocean-atmosphere interaction, CO2 presence, altitude, etc.

In conclusion, we can see that functions like other areas of math are often used in our daily lives, whether we realize it or not. I hope that this blog post has sparked your curiosity in thinking about how other math topics you have learnt so far can be relevant in real life not just for learning at school to pass a test.

IB Learner Profile

1. The learner profile that resonates with me the most during the production of this ejournal is reflective as at first I lost all the functions I used when I quit the application. Thus, I had to reflect on what I had previously done and ponder on how to further improve the bicycle even though I was really disappointed that I had to restart my work again.
2. Another learner profile that I can relate to is thinkers, as some of these functions like the one for producing a semi-circle were new to me and I had to figure out how to manipulate the values into giving me the graph I wanted. I had to persevere in testing different values of variables to see the effect it had on the graph, and from that I was able to implement my new found knowledge.
3. The last learner portfolio is inquirers as during the process of making the bicycle, I wanted to make a sideways parabola for the gear. I had to research how to make one even though it was not required, but after making it I realized it was not a function as a vertical line would pass the graph twice. In the end for making the gear, I used two different functions to replace the one equation that was not a function

Sources:

https://mathinsight.org/definition/function

https://www.quora.com/What-are-the-examples-of-an-application-of-a-function-in-real-life

https://www.quora.com/What-are-the-applications-of-functions-in-real-life

COMPLEX NUMBERS

Although the name sounds intimidating, it really isn’t once you get used to it. Generally we can separate numbers into two broad categories: real numbers and imaginary numbers. Some of you may have never heard about imaginary numbers, but I’m sure most of you are familiar with real numbers as these are the numbers we deal with in our daily lives and use most of the time.

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An imaginary number is the square root of -1, suitably called so as in the real numbers system, inputting this into a calculator would have resulted in math error as it is not possible to compute that. Based on imaginary numbers we can create numerous more numbers such as 5i, -12i etc. These are all examples of pure imaginary numbers. Well, what about numbers such as -2 + 3i? This gives rise to complex number, which is a number that can be represented in the form a + bi, where a and b are real numbers and i is the square root of -1. It is a complex number as it is neither a real nor an imaginary number.

Another interesting thing to note is that all real numbers and imaginary numbers are complex numbers as well, but not all complex numbers are real numbers/pure imaginary numbers. This is because a real number such as 5 can be represented as 5 + 0i, and an imaginary number such as 3i can be represented as 0 + 3i.

APPLICATIONS IN OUR LIVES

Although we may not be able to think of the use of complex numbers in our daily lives, some people do encounter these numbers often and without complex numbers, many of the things we encounter or use everyday would not be possible. Complex numbers are used especially in jobs requiring the field of physics and math. Here are some of the applications:

1. Most of us – especially the ones who are able to read this blog – are lucky enough to have access to electricity and power. This electricity comes from generators in power stations, and these generators use the concept of electromagnetism and require some knowledge of electrical engineering.

Both of these fields require as in electricity, one of the units we use is the unit Voltage which is usually represented as V = V0 (cos wt + j sin wt), where j is complex to find the difference between real and reactive power in a system. Real Power is power which results in active work. In other words, active power is solely that work which is done for producing of real (useful) work in this case electricity. Whereas Reactive Power can be stated as “The power which is required to establish and maintain a magnetic field”, which helps produce the electricity, but is not the power which is directly turned into electricity.

2. Robots are also built and programmed with the help of complex numbers. Control systems – which use complex numbers – are used to make these robots by figuring out if the system is stable or not.

3. Another use of complex numbers is in signal processing. Signals are useful in cellular technology (the technology our phones use), radar and wireless technologies such as radios. Even more surprising is that they can be used in even biology when studying brain waves.

IB LEARNER PROFILE

The first IB Learner Profile which resonates with me while typing up this blog post is caring, because I have developed a sense of respect towards people whose occupation deals with complex numbers (as we can see that it is not an easy thing to comprehend). Doing complex numbers at an IB level makes me feel like the person in the meme below, so it’s almost unfathomable for me to know that there are people whose occupation relies heavily on this topic.

Another learner profile that I found myself relating to is knowledgeable as through this blog post, it really pushed myself to find and understand various uses of complex numbers as I originally could not imagine where it would be applicable. This has really widened my horizons and made me more aware of how the same math is used locally and globally. It amazes me that even though people live all over the world in different countries speaking in different languages, the same math is being used.

The last learner profile that kept popping up in my mind while writing this is reflective, as I had to think about where I might possibly encounter and experience the use of complex numbers. Even though I was unable to think of an application where I personally used complex numbers, I could find things I use on a day-to-day basis that relied on the concept of complex numbers. It made me ponder on my daily routine and realize that there are probably a lot more ways in which math is involved in my daily life, whether I know it or not.

REFERENCES

https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:complex-num/a/intro-to-complex-numbers

https://www.quora.com/What-are-some-real-life-applications-of-complex-numbers-in-engineering-and-practical-life

https://www.livescience.com/42748-imaginary-numbers.html

https://www.quora.com/What-is-real-reactive-and-apparent-power

Why it is possible to solve for the sum of an infinite geometric series?

First of all, let's start off by defining an infinite geometric series. An infinite geometric series is the sum of an infinite geometric sequence. There wouldn't be a last term in this series. 

 The formula a1+a1r+a1r^2+a1r^3+…, where a1 is the first term and r is the common ratio, is the general pattern of the infinite geometric series.

It is possible to find the sum of all finite geometric series. Although in the case of an infinite geometric series when the common ratio is more than one, the terms in the sequence will only become bigger and bigger and if you add the larger numbers, there will be no final answer. The only possible answer would be infinity. So, we can only find the sum of the series when the common ratio greater than one for an infinite geometric series.

But, it is possible to get an answer because it converges (has a finite sum even when n is infinitely large). The sum of an infinite geometric series can be calculated as the value that the finite sum formula approaches as the number of terms n approaches infinity.

Relating “convergent and divergent” notes to “Sequences and Series”

In both ways of thinking, whether using imagination or logic, we have to use both when solving problems related to sequences and series. Mainly, convergent thinking is used as we have to analyze the question before answering it so that we can apply the right formula, but we also have to be creative in thinking as that type of problem might be the first you’ve encountered so we have to approach it with many ideas on how to solve it.

Real life application of Sequences and Series

1. Tumour growth, the growth rate is exponential except when it becomes so large that it cannot get food to grow effectively. So it starts off like an exponential graph but then stops completely. A more precise statement is known as Gompertz Law of Mortality – “rate of decay falls exponentially with current size”.
2. The Taylor series was used by Albert Einstein in his 1905 paper on Brownian motion to deduce the motion was the accumulated effect of momentum transfer from many individual atoms, thus proving their existence.

IB Learner Profile

The IB Learner Profile which best describes this eJournal is ‘Inquirers’ because I had to be curious and ask questions to research.

References:

https://www.varsitytutors.com/hotmath/hotmath_help/topics/infinite-geometric-series

http://www.nabla.hr/CO-SequAndSeries4.htm

3 Modes Of Thinking: Lateral, Divergent & Convergent Thought

3 Modes Of Thinking: Lateral, Divergent & Convergent Thought

3 Modes Of Thinking: Lateral, Divergent & Convergent Thought

3 Modes Of Thinking: Lateral, Divergent & Convergent Thought

https://www.teachthought.com/critical-thinking/3-modes-of-thought-divergent-convergent-thinking/

https://www.quora.com/What-are-the-applications-of-sequences-and-series