Issuu

Plateau's Problem: The Double Bubble Theorem

Pg.24

Becoming The Next Zuckerberg: An Introduction to Graph Theory Pg.30

The Collatz Conjecture Pg. 18

The Golden Ratio in Architecture Pg. 42

Editor’s Note I am beyond excited to share the first edition of COSMOS with you all. After months of dedication, creativity, and collaboration, we have crafted something truly special. COSMOS is the first-ever stem-based publication in JIS and a student-led quarterly journal. The name COSMOS is derived from our goal, which is to make mathematics a universal language and interpret the universe around us through the lens of mathematics. To achieve this goal, our journal is divided into two sections: introductory mathematics and advanced and applied mathematics. In section 1, we introduce math concepts using easy explanations and interesting real-life examples. These articles are also translated into Bahasa, to be shared with public local schools in Jakarta, which aligns with our goal of making mathematics understandable to everyone. In section 2, students either dive deeper into mathematics and explore concepts outside the school curriculum or find a link between mathematics and other fields such as natural science, technology, and economics. Therefore, section 2 articles contain more difficult concepts and are targeted at those who have a strong passion for mathematics. As the editor-in-chief of COSMOS, it was a great pleasure to read and edit every article in this issue. I can guarantee you that all articles were written using each student's best abilities. I want to thank all staff writers for their hard work and for making this issue a contentful one. Lastly, I would like to thank the managing editors Jinwoo Yang and Justin Wu, who worked with me from the beginning, developing this project from a small idea into the school's official journal.

Dowon Her, Editor-in-Chief

Who Should Karim Hire? The Basic Concepts of Sequences and Series

By Brian Lee 72090@jisedu.or.id Introductory Mathematics Key Concept: Arithmetic and Geometric Sequences and Series

Introduction

Have you ever heard of sequences and series? Some might be familiar with this term, but some might not be. Regardless, sequences and series can be often found and applied in our everyday lives. Let’s take a look.

Arithmetic sequence

Let’s say Karim interviews A and B to work in his new luxurious cafe for the next 5 years. A: “I would like to receive a wage of 1,000 dollars per month.” B: “I would like to receive only 100 dollars in my first month of employment. However, I would like to receive 200 dollars in my second month, 300 dollars in my third, and 100 dollars more for each consecutive month.” Who should Karim hire for his own good? The math that answers these questions is sequences and series. Sequence is a set of numbers in a particular order or pattern. In the case of A, and B, the sequence of their monthly wages can be shown like this: 𝐴: 1000, 1000, 1000, 1000, ··· 𝐵: 100, 200, 300, 400, 500, ···

In mathematical terms, a certain term of a sequence is expressed as ‘u.’ ‘un’ would be the nth term of the sequence. Finding the nth term of a sequence is not too tricky, but could often lead to a choice of formula from two options: an arithmetic sequence and a geometric sequence. As for B, he said that he wants +100 dollars of the wage he received in the month before. In this case, Karim can use the arithmetic sequence, a sequence where the new term is the previous term added by a certain constant, and where all terms follow the same pattern.

𝑢𝑛 = 𝑢1(𝑛 − 1)𝑑 -

𝑢1 = first term of sequence

𝑑 = common difference (aka the value being constantly added)

Since A’s wage is fixed, d will be 0, and therefore 𝑢60 for A will be the same value as 𝑢1. 𝑢60 = 1000 + (60 − 1) × 0 = $1000 To find the amount of money B will receive in his last month of employment, Karim can simply find n, and input the value into the formula. 1 year is 12 months;

therefore, 5 years will be, 5 × 12 = 60 months. Since B demands an increase of wages of 100 dollars each month, d will be 100.

the common ratio, and where all terms follow the same pattern. The expression of Un as for geometric sequences is slightly different from an arithmetic sequence. 𝑛−1

𝑢60 = 100 + (60 − 1) × 700 = $6000

𝑢𝑛 = 𝑢1 × 𝑟

Comparing the wages of A and B in the final month, Karim is already a bit doubtful about hiring B over A. Now, Karim decides to make a new calculation: the total amount of money he has to give away to A and B. In mathematical terms, he aims to find the total sum of an arithmetic sequence. The total sum of terms in sequence ‘S’ is called a series. ‘Sn’ would be the sum of the sequence from the first term to the nth term. Again, the formula for a series depends on the type of the sequence. The sum of an arithmetic sequence can be expressed as:

𝑢1 = first term of the sequence

𝑑 = common ratio (aka the value being constantly multiplied) 59

𝑈60 = 1 × 2

= $576, 460, 752, 303, 423, 488

Now Karim is a bit doubtful about A too. In fact, he is very doubtful. He decides to find the new total amount of money he has to give away to A. The total sum of a geometric sequence can be expressed as:

𝑛

𝑆𝑛 = A: 𝑆60 =

60 2

𝑆𝑛 =

𝑛 (2𝑢1 + (𝑛 − 1)𝑑) 2

𝑢1(𝑟 −1) 𝑟−1

× (2 × 1000 + (60 − 1) × 0)

𝑆60 =

1×(2 −1) 2−1

= $1, 152, 921, 504, 606, 846, 976

= $60, 000

In this case, Karim would not hire A at all. B: 𝑆60 =

60 2

× (2 × 100 + (60 − 1) × 100)

= $183, 000

Based on his calculation, Karim finds out that B is playing nonsense mind games against him. In fact, B is asking for 3 times the wages of that of A. Without a sight of hesitation, Karim decides to employ A.

Geometric Sequence

But what if there’s a plot twist? What if A says that he wants to change the way he receives his wage? He says that he will receive only 1 dollar in his first month, but 2 dollars in his second, and double the amount of his previous wage in the month after. In this case, Karim can use the geometric sequence, a sequence where the new term is the previous term multiplied by a certain constant, called

Conclusion

As you can see, sequences and series are important mathematical concepts that can be often applied to our daily lives. Especially in money-related situations, such as calculating the compound interest of banks and deciding which bank you should invest in, arithmetic and geometric sequences are most used and are extremely helpful.

The Birthday Paradox: Why Outcomes Often Defy Expectations

Reyansh Kankhyan S 72090@jisedu.or.id 69249 Introductory Mathematics Key Concept: Probability

Introduction

The Birthday Paradox goes like this: How many people do you need in a room for there to be more than a 50% chance that at least two of the people share the same birthday? While it might seem like you need a large group - say, 183 people (half of 365, which is how many days there are in the year)—the surprising answer is only 23 people.

2 The Principle Behind the Birthday Paradox Because the question asks for the probability of ‘at least two people’, that means we would need to calculate the probability of 2,3,4… x amount of people. This would be very time consuming. In these kinds of situations, we can use complementary counting, where we calculate the probability of the condition not being met, then subtract that from 1. Instead of calculating the probability of at least two people sharing a birthday, it’s easier to calculate the probability that no one shares a birthday and subtract that from 1.

With 2 people, the probability that the second person doesn't share a birthday with the 365

364

first is 365 × 365 . With 3 people, the probability that the third person doesn’t share a birthday with the 365

364

363

first two is 365 × 365 × 365 . … and so on. Because these probabilities are codependent, ie, it depends on how many people are already in the group, we must multiply the probabilities for each person added to the group. For 23 people, the calculation looks like this: 365 365

364

363

343

× 365 × 365 × … × 365

Multiplying all these probabilities together gives a value of 0.4927, or a 49.27% chance that no one shares a birthday. Therefore, the probability that at least two people do share a birthday is: 1−0.4927 ≈ 0.5073 or 50.73%

With 1 person, there’s no one else to compare with, so the probability of no shared birthday is 100%.

The reason this works is because the number of comparisons grows rapidly - 23 people lead to 253 unique pairs of people that can be compared. To make this a bit clearer, I

graphed out the probabilities of at least two people sharing the same birthday for up to 100 people. As you can see, the probability rises exponentially up to a certain point, because the number of unique pairs rises exponentially. The number of pairs is how many comparisons we have to make, ie, the more the number of pairs, the more comparisons we make and hence the higher the probability of one of those pairs sharing a birthday. It starts slowing down after a certain point because probability can never go over 1, and it must asymptote at 1.

not the case, and switching gives you a much better chance of winning. How Does It Work? 1. Initial choice: When you first pick a door, you have a 1/3 chance of picking the car and a 2/3 chance of picking a goat. 2. Monty reveals a goat: Monty always opens a door with a goat behind it, which doesn’t affect your original 1/3 chance of having picked the car. 3. Why switching works: If your first choice was a goat (which happens 2/3 of the time), then switching will give you the car. If your first choice was the car (which happens 1/3 of the time), switching will make you lose. Since you were more likely to pick a goat in the beginning, you should switch because if you picked a goat and you switch, you win the car. The confusion comes from thinking that the host’s action makes the odds 50-50, but really, it just gives you a chance to increase your odds of winning by 33%.

The Monty Hall Problem

The Monty Hall Problem is based on a game show scenario. You are presented with three doors. Behind one door is a car (which you want to win), and behind the other two are goats (which you don’t want). You choose one door. Then, the host, who knows what’s behind the doors, opens one of the other two doors to reveal a goat. He then asks if you want to stick with your original choice or switch to the remaining door.

The surprising answer is that you should always switch, as it increases your chances of winning the car from 1/3 to 2/3.

This counterintuitive result stems from confusing test accuracy with actual probability. The test’s accuracy only tells you how well the test performs under ideal conditions (i.e., when people are sick or healthy). But to determine the probability that you actually have the disease after testing positive, you need to

Most people assume that once Monty opens a door, revealing a goat, there are two doors left, so the odds should be 50/50. That’s

Why medical testing accuracy isn’t useful Imagine there is a test for a disease that’s 90% accurate. If you test positive, what are the chances that you actually have the disease? Many people assume that if the test is 90% accurate, there’s a 90% chance you have the disease. But in reality, the chance can be much lower depending on how rare the disease is.

consider the chances of false positives and false negatives separately. Here’s an example using actual numbers: Suppose 1% of the population has the disease. This means for every 1000 people, 10 actually have it. The test is 90% accurate, meaning that 90% of those with the disease will test positive (true positives) and 90% of those without the disease will test negative (true negatives). Out of 1000 people, 10 have the disease. Since the test is 90% accurate, it will correctly identify 9 of those people as positive. Out of the 990 people who don’t have the disease, the test will also be 90% accurate, meaning it will correctly identify 891 people as negative. However, 99 people will still get a false positive result. So, in total, 108 people (9 true positives + 99 false positives) will test positive. But only 9 of them actually have the disease. Therefore, if you test positive, the probability that you actually have the disease is: 9/108 ≈ 0.0833 or 8.33% Even though the test is 90% accurate, the probability of actually having the disease after testing positive is just 8.33%—much lower than what you would think.

Probability and Its Applications By Vincent Tjoa 71255@jisedu.or.id

Introductory Mathematics Key Concept: Probability, Combination, Permutation

Introduction

What is probability, and why does it matter? An understanding of the concept would doubtlessly change the way you go about choosing day by day. You might be struck by how often you unknowingly apply probability in various random, everyday situations. In this article, the various elements of this concept are broken down. What is probability, why is it important, and how can you use it in real life? This article will explain all three questions, and introduce permutations and combinations—integral concepts within probability.

What is Probability?

Poobability is the measure of the likelihood of an event. For example, the probability of getting heads when one coin is tossed is 0.5 (50%), and that of getting tails is also 0.5 (50%). Probability is inclusive between 0 and 1 (0% and 100% respectively), where 0 suggests impossibility, meaning it cannot occur, whereas 1 suggests certainty, which means it will definitely happen.

Probability has great significance in our lives, yet most of us do not even pay attention to

it. Be it predictions concerning the weather, decisions pertaining to work or school, or the mechanisms of games, this branch of mathematics assists people in drawing conclusions about the likelihood of an occurrence.

Why is Probability Important?

Probability helps us draw valuable conclusions. For instance, if a weather application on your phone says there is a 70 percent probability of rain, it means it is more likely to rain than stay dry, so you will decide to carry an umbrella. Businesses use probability to predict consumer behavior, and insurance companies assess risks based on calculated probabilities to charge an appropriate premium.

Permutations and Combinations

There are many cases where you need to count how something can happen. That is where permutations and combinations come into play in computational mathematics to count the number of possible outcomes under various conditions. Whether you are calculating how many different ways a group of people can be arranged or how a set of items can be chosen in different ways, permutations and

combinations handle these calculations. Permutations deal with the number of ways something can be arranged when the order matters, while combinations refer to selections where the order does not matter. 1)Permutations: A permutation refers to the arrangement of items where the order matters. For example, if you have three colored balls (red, blue, and green), how many different ways can you arrange them? The answer is calculated using permutations.

𝑛𝑃𝑟 =

𝑛! (𝑛−𝑟)!

To solve this problem, we must first understand what it is asking for; does the question care whether the order of the books matters, or is it only asking how the objects can be placed? In this case, the order of the books matters so we must use the permutations formula to solve it. 10!

10!

P = (10−8)! = 2! =

10 8

3,628,800 2

1,814,400

Where: -

1) Dan has 10 books and wants to arrange them on his bookshelf which can only fit 8 books. How many ways can Dan arrange 8 books on the shelf?

‘n’ is the total number of items. ‘r’ is the number of items you want to arrange. "!" is factorial, which means multiplying all whole numbers from 1 to that number. (Ex: 5! = 5 x 4 x 3 x 2 x 1)

2)Combinations A combination is different from a permutation because the order does not matter. Imagine you are selecting 2 balls from the same set (red, blue, and green). The order in which you pick them doesn’t matter, so the formula for combinations is slightly different:

𝑛𝐶𝑟 =

𝑛! 𝑟!(𝑛−𝑟)!

Thus, Dan is able to arrange his books in 1,814,400 different ways. 2) Vincent has 10 different plants, but his garden can only fit 6 of them. He wants to choose 6 plants to display in his garden. How many different ways can Vincent select 6 plants from his 10? To solve this problem, does the order of the plants matter or not? Since he is only selecting which plants to display (not arranging them in a particular order), the order does not matter, so we must use the combinations formula to solve it. 10!

10!

C8 = 6!(10−6)! = 6!4! = 210

So, Vincent is able to select 6 plants in 210 different ways.

6 Applications of Probability, Permutations, and Combinations

5 Example Questions of Permutations & Combinations

1) Games of Chance: If you have ever played card games or rolled dice, you have already used probability. Games of chance often involve calculating the odds of drawing a particular card or rolling a specific number.

Understanding permutations and combinations can help you improve your strategy.

For example, in poker, let's say you are trying to complete a flush (5 cards of the same suit) and you already have 4 suited cards. Each suit has 13 cards in a deck, so there are 9 cards left that could complete your flush. With a 52-card deck and 5 cards already seen (your 2-hole cards and the 3-flop cards), that leaves 47 unknown cards.

hand, if you are selecting people to form committees or teams from a larger group, combinations can help you figure out how many different groups can be formed without worrying about the order in which people are chosen. For instance, if you want to know how many ways you can arrange 10 people in specific seats, you can use permutations. The formula would be: 10!

P = (10−10)! = 10! = 3, 628, 800 ways

10 8

To calculate the probability of completing your flush: -

Probability on the turn (4th card dealt): 9

P(Flush on Turn) = 47 ≈ 19.15% -

Probability on the river (5th card dealt, assuming you didn’t hit the flush on the turn): 9

P(Flush on Turn) = 46 ≈ 19.57% -

Total probability of hitting the flush by the river (either on the turn or river): 9

P(Flush by River) = 1 - (1- 47 )(1- 46

)

= 0.3261 or 32.61%

Thus, the total probability of completing your flush by the river is approximately 32.61%.

1) Event Planning: Imagine you are planning a group event, and part of your task is organizing the seating for a set number of guests. By using permutations, you can determine how many different ways the seating can be arranged, considering that the order in which guests sit matters. On the other

3) Genetics and Biology: In genetics, there are a few areas where biology depends upon probability. Biologists use a Punnett square to predict the probability that offspring will inherit certain genes from the parents. Suppose, for example, that flower color is determined by one gene with two alleles, R and r. Red flowers express the R allele, white flowers the r allele. A cross between a plant homozygous recessive and a plant heterozygous for flower color yields what probability that the offspring will be heterozygous for flower color?

Suppose one parent is contributed with one dominant gene for brown eyes, and the other parent has one recessive gene for blue eyes. You will be able to work out the possible combinations using a Punnett square: B

The genetic combination could then be: - BB homozygous brown: 25% - Bb (heterozygous brown): 50% - bb (homozygous blue): 25%

This will mean that 75% chances are for brown eyes, and 25% are for blue eyes. Knowing these probabilities, the scientist can predict genetic traits and variations in offspring.

Conclusion

Linked with permutations and combinations, probability provides some of the most powerful means whereby we can make sense of the uncertainty in our lives. Such notions enable us to measure and analyze possible outcomes to provide a basis for making wiser decisions—be it about games, events, or scientific predictions.

References:

Foto, Merah. “Resolusi Tinggi Dadu Baru Yang Bersih.” IStock, 26 Feb. 2019, www.istockphoto.com/id/foto/dadu-kasino-me rah-gm1132091114-299978666. Accessed 6 Oct. 2024. Gstatic.com, 2024, t3.gstatic.com/licensed-image?q=tbn:ANd9Gc TCAXA9g1R0S1l7jwbHVKMYYcvNH1oGb zbTMKsIlVaKbJTGQmWgk3D32DhiQGDC3 ft1. Accessed 6 Oct. 2024. GeeksforGeeks. “Difference between Permutations and Combinations | Permutations vs Combinations.” GeeksforGeeks, 27 Sept. 2021, www.geeksforgeeks.org/difference-between-pe rmutations-and-combinations/. Accessed 6 Oct. 2024. Quality Gurus. “Permutations and Combination.” Quality Gurus, 24 Dec. 2022, www.qualitygurus.com/permutations-and-com bination/. Accessed 6 Oct. 2024. The. “Punnett Squares - Basic Introduction.” YouTube, YouTube Video, 13 Nov. 2018, www.youtube.com/watch?v=agQpPPQ5IVQ. Accessed 6 Oct. 2024.

“What’s the Chance of an April Shower?” Official Blog of the Met Office News Team, 21 Apr.2016,blog.metoffice.gov.uk/2016/04/21/w hats-the-chance-of-an-april-shower

Mathematical Analysis on the Collatz Conjecture By Jovason Tiger Abishai Silaban 72378@jisedu.or.id Advanced Mathematics Key Concept: The Collatz Conjecture

Introduction

Think of a number, any number. 7? Good choice. Now multiply it by 3 and add 1. We get 22. It is even, so we cut it in half. Now we have 11. It’s odd so we multiply it by 3 and add 1. We have 34. It’s even so we cut it in half and repeat the process. 34 - 17 - 52 - 26 13 - 40 - 20 - 10 - 5 - 16 - 8 - 4 - 2 - 1. 1 is an odd number so we multiply it by 3 and add 1, now we have four. Divide by 2, we get 2. Divide that and we get 1. Now, we have this never-ending loop of 4-2-1. And according to the Collatz Conjecture, If you start with any positive integer, you will always end in this loop.

that explains how we get the numbers above and what causes the loop. A piecewise function is defined as a function that applies different rules depending on the model, domain, or range. In our case, let n be the value that we choose. If n is odd then f(n) = 3n + 1. If n is even, then f(n) = n/2. This notorious conjecture also has another name for the unique input for odd numbers, it is called the 3n+1 Conjecture. The function is shown clearly below:

To fully understand the mystery of the Collatz Conjecture, we will need a piecewise function

The Problem with the Conjecture

The numbers in the collatz conjecture are called “Hailstone numbers” because they go up and down like hailstone. You can view it like this: think of a mountain that you start at a certain point on journey to try and get down. The amount of “stops” is how many times you have to calculate 3x+1 into the numbers before you finally get down to the 4 - 2 - 1 loop. If you start at the number 26, you can think of it as meters. You need 10 stops before you get down to 1. Take the next number, 27, and it takes 111 steps to get to 1, reaching an altitude of 9232 meters. The variety is so wide that we are able to conclude that there is no pattern to

the Collatz conjecture but mere randomness, or so we think.

We can figure out the probability of the leading digit of every number in the conjecture. Mathematicians have measured 1 billion numbers and found that 1 is by far the most common leading digit with 30% of the numbers, 17% for number 2 and the percentage gets smaller as the value of the digit increases. There is the same number of even and odd numbers and we might think that the value for odd numbers is more than tripled while the even numbers only gets halved. Mathematically, the conjecture should be growing, not shrinking. Here’s the catch: every odd number will become even in the conjecture, and then we eventually have to divide that number by 2. Therefore, technically you don’t triple the number, because the maximum amount you can increase is by a factor of about 3/2. The +1 doesn’t really make a difference especially when dealing with large numbers.

Because for every odd number we get an even number according to the conjecture, the next

step will always be to divide that number by 2, and 50% of the time, we get an odd number, which means we apply the 3x+ 1 rule and ultimately divide the result by 2. So from the original number, ½ of the time it’s odd, and of that different values, ¼ of the time it’s odd. With this, you see that ¼ of the time you can divide by 4, and of the numbers that come out, the value of it will be ¾ of its original number. This goes on: ⅛th of the time you can divide by 8 before getting to the next odd number, and 1/16th of the time you can divide by 16 and son. And if you take the geometric mean of the pattern to get from one odd number to the next one, you multiply by ¾, which is less then 1. Statistically speaking, the Collatz Conjceture is more likely to shrink than grow, and if proven true, will ultimately end in the 4 - 2 - 1 loop.

The Solution

One of the ways we can visualize the path the numbers take is to show how each number connects to the sequence. Once we do that, we see a directed graph: a treelike pattern with streams.

only calculate such finite numbers. The only answer is absolute mathematical proof. 68

Now if the conjecture is proven to be true, every number should be connected to this graph. Up to infinity, the numbers should flow into the massive river of 4-2-1. What’s so beautiful about this conjecture is that if you rotate each number depending on its even or odd category, anti-clockwise for odd, and clockwise for even, and adjust its angle, you look at a beautiful organic structure.

Mathematicians have tested numbers up to 2 , and for us that is such a big number, but compared to all numbers, we have achieved so little.

Terrence Tao, one of the greatest mathematicians of this century made the most recent progress out of anyone that have tried to tackle this problem. He proved that the conjecture is almost true for almost “all numbers.” If we take a look ourselves, the conjecture gives this limit:

This structure proves the complexity but the beauty of this problem. The streamline that ends with 4-2-1, starting from any number possible is shown in this structure. When the degree of the branch is adjusted to a desired amount, that’s when you see this structure. There are 2 ways that we can prove that this conjecture is false. First is any single number that starts a sequence that keeps growing, up to infinity and don’t obey the 4-2-1 loop. Second is another loop. The only loop that we know so far and mathematicians have calculated is the 4-2-1 loop. But if there were another loop that doesn’t follow the 4-2-1, this conjecture would be false. One of the major problems with trying to a mathematical theory is that brute force is not the answer, you can’t test all the numbers in the world because it is infinite, and we can

Conclusion

It would take every number that exists to prove this conjecture, and that is not possible. But it takes only one to disprove it. If we take a look at the numbers of perfect squares from the range of 1 - 100, 10 of them are perfect squares, giving them a probability of 10%, and as we count the number of perfect squares with a wider range, this percentage gets lower, we only have 31 perfect squares out of 1000 numbers. We can conclude from this that almost all numbers are not perfect squares, and the amount that we have calculated so far is nothing compared to every number. So it begs the question, do we really know anything? Perhaps everything that we’ve proven so far in science is a miracle? As Paul Erdos once said, “Mathematics is not ripe enough for such problems.” Do we actually have the right to know the solutions to every problem? Maybe we don’t, and it’s better to leave some unsolved.

Euler’s Formula: The Bridge between Algebra, Geometry, and Trigonometry By Chaewon(Wendy) Yang 71446@jisedu.or.id Applied Mathematics Field of Application: Engineering and Computer Graphics

Introduction 𝑖𝑥

Euler's formula, 𝑒 = 𝑐𝑜𝑠(𝑥) + 𝑖 𝑠𝑖𝑛(𝑥), is arguably one of the most beautiful and deepest results in mathematics bringing together such distant branches of science as algebra, geometry, and trigonometry. This equation is not only a beautiful way of connecting the exponential functions and the trigonometric functions, but it is quite useful in many aspects of mathematics as well. In this article, we will explore the pure mathematical proof of Euler’s formula and its relationship with algebra, geometry, and trigonometry. We will then examine its practical applications in the real world, showcasing its relevance in technology and the sciences.

remembering that 𝑖 = and so on:

− 1, 𝑖

− 𝑖,

If we group real terms and imaginary terms, we will notice that the real expression corresponds to the Taylor series 𝑐𝑜𝑠(𝑥) and the imaginary terms correspond to the Taylor series of 𝑖 𝑠𝑖𝑛(𝑥):

Proof of Euler's Formula 𝑖𝑥

Euler’s formula, 𝑒 = 𝑐𝑜𝑠(𝑥) + 𝑖 𝑠𝑖𝑛(𝑥), can be proved using power series expansion. Let us think of extension of the Taylor series for 𝑥

𝑥

Now, let’s substitute 𝑖𝑥 into the series for 𝑒 ,

𝑒 , 𝑐𝑜𝑠(𝑥), and 𝑠𝑖𝑛(𝑥):

Thus, Euler's formula is proven!

3 Connecting Algebra, Geometry, and Trigonometry Algebra: 𝑖𝑥

Euler’s formula 𝑒 = 𝑐𝑜𝑠(𝑥) + 𝑖 𝑠𝑖𝑛(𝑥) creates a strong link between complex numbers and exponential functions. In algebra, this formula is an essential tool for working with complex numbers, which are important for solving equations that don’t have real solutions. For example, the solution

𝑥 + 1 = 0 involves the imaginary unit 𝑖. Euler’s formula makes it easier to work with complex exponentials. When doing exponentiation with complex numbers in polar form, their angles (arguments) are added, and their sizes (magnitudes) are

and cosine. This method is often used in calculus and solving differential equations, especially when dealing with wave-like motions. Expressing these in exponential form simplifies the process.

𝑖𝑥

multiplied—Euler’s formula, 𝑒 , makes this process much easier.

Geometry: In geometry, Euler’s formula can be visualized on the complex plane. The formula 𝑖𝑥

𝑒 = 𝑐𝑜𝑠(𝑥) + 𝑖 𝑠𝑖𝑛(𝑥) represents a point on the unit circle, with an angle 𝑥 measured counterclockwise from the positive real axis. The real part, 𝑐𝑜𝑠(𝑥), gives the x-coordinate, while the imaginary part, 𝑖 𝑠𝑖𝑛𝑥), gives the y-coordinate. This visual interpretation shows that Euler’s formula links complex numbers to rotations on the plane. When you rotate a point by an angle 𝑥, it’s the same as multiplying the 𝑖𝑥

complex number by 𝑒 . This is particularly helpful in fields like computer graphics and physics, where rotations need to be calculated efficiently.

Trigonometry: Euler’s formula also offers a simpler way to

4 Real-World Applications of Euler’s Formula In the applied world, Euler's formula plays a crucial role in various fields: - Electrical Engineering: In electrical engineering, Euler’s formula is very important for understanding alternating current AC circuits. AC signals, like those used in power systems, change over time and can be shown as sine or cosine waves. For example, engineers use Euler’s formula 𝑖𝑥

𝑒 = 𝑐𝑜𝑠(𝑥) + 𝑖 𝑠𝑖𝑛(𝑥) to represent voltages and currents as complex numbers. This makes it easier to figure out how these changing signals behave. By turning sine and cosine functions into exponential form, engineers can use methods like phasor analysis more easily. This helps them understand how voltage and current work together over time, ensuring that electrical systems run smoothly and safely.

𝑖𝑥

represent trigonometric functions. Since 𝑒 breaks down into cosine and sine, it helps make many trigonometric identities easier to work with. For example, adding angles is simple by multiplying their exponentials:

When we expand both sides using Euler’s formula, we get the well-known identity for adding angles: By matching up the real and imaginary parts, we can easily find the formulas for adding sine perform calculations and understand concepts like interference and superposition. For example, in the famous double-slit experiment, Euler’s

-Quantum Mechanics: In quantum mechanics, Euler’s formula is essential for explaining how very small particles, like atoms and electrons, behave. Scientists use wave functions, which are complex mathematical expressions to describe the state of particles. These wave functions contain information about the chance of finding a particle in a certain place or condition. Using Euler’s formula makes these wave functions simpler, helping scientists

formula helps explain the patterns formed when particles pass through two slits. These patterns are key to understanding wave-particle duality, which means particles can act like both waves and individual object.

- Computer Graphics: In computer graphics, Euler's formula helps with rotating and moving 3D objects. When animators and designers create scenes in movies or video games, they often need to rotate objects smoothly in 3D space. Using Euler’s formula, these rotations can be expressed as complex exponentials, which makes calculations easier. For example, when rotating an object around an axis, programmers can use Euler’s formula to figure out its new position after the rotation. This makes the math simpler and helps create more complex and realistic animations in video games and movies.

5 Conclusion Euler's formula is a remarkable connection between algebra, geometry, and trigonometry, all brought together in one powerful expression. Its impact goes beyond just theoretical math, as it plays a key role in many real-world applications like engineering, physics, and computer science. Whether it’s simplifying complex calculations or explaining how rotations work in geometry, Euler’s formula is a fundamental tool that helps us better understand both the mathematical world and practical problems in everyday life.

Plateau’s Problem: The Double Bubble Theorem By Jihoo Lee 70598@jisedu.or.id Advanced Mathematics Key Concept: The Double Bubble Theorem

Introduction Have you ever wondered about the appearance of soap bubbles? You may have seen bubbles in your daily life, even when you shower, wash the dishes, or conduct a chemical experiment! Bubbles have hollow spheres with an iridescent surface between them. But why so? Well, that’s what we are going to explore. Plateau’s Laws are a set of laws that describe the structure of soap films. Originating from the Belgian physicist Joseph Plateau in the 19th century, these laws help explore many patterns in nature, especially the formation of foams. According to the first law based on Plateau’s observations of soap bubbles, the dihedral angle must be 120 degrees. For example, beehives, which form perfect hexagons like bubbles, have their hexagonal walls meet at precisely 120 degrees to maintain an equilibrium state. Using its application, soap will meet in threes—which is also known as the ‘Plateau border’—at 120 degrees when they form walls to minimize surface area.

The second law is that four Plateau borders will meet at a tetrahedral angle of 109.5 degrees to form a vertex.

The renowned exemplification of Plateau’s Problem is the Double Bubble Theorem, a mathematical result based on Plateau’s Laws that states that a standard double bubble will have enclosed and separate two volumes of air with minimal area.

This article will explore the volume of enclosed surfaces and the geometry of a stable double bubble for a better comprehension of the Double Bubble theorem.

The Standard Double Bubble

To begin with, a standard double bubble follows two conditions: having dihedral angles of 120 degrees and having a constant mean curvature. But what is a constant mean curvature? A constant mean curvature measures the property of a surface when the mean curvature at every point on a surface is the same constant value.

long it takes for the smaller circle to the larger circle to turn a set angle. Then, how long will the smaller circle take to turn 90 degrees? It is just simple! As its surface is approximately a quarter-circle, the circumference equation of a circle can be used to quantify the curvature.

𝐶 = 2π𝑟 𝐶𝑠𝑚𝑎𝑙𝑙 𝑏𝑢𝑏𝑏𝑙𝑒 = 2π(1𝑐𝑚) = 2π𝑐𝑚 𝐶𝑏𝑖𝑔 𝑏𝑢𝑏𝑏𝑙𝑒 = 2π(2𝑐𝑚) = 4π𝑐𝑚 𝐶𝑠𝑚𝑎𝑙𝑙 𝑏𝑢𝑏𝑏𝑙𝑒 4 𝐶𝑏𝑖𝑔 𝑏𝑢𝑏𝑏𝑙𝑒 4

2π 4

= =

4π 4

= 2 𝑐𝑚 = π𝑐𝑚

A curvature simply refers to how quickly a line curves. Since the equation of 𝑎𝑛𝑔𝑙𝑒

curvature can be represented as k = 𝑙𝑒𝑛𝑔𝑡ℎ , the curvature of the bubbles can be found using the length that was previously calculated. 𝑘𝑠𝑚𝑎𝑙𝑙 𝑏𝑢𝑏𝑏𝑙𝑒 = Starting with two dimensions, imagine that there are two bubbles with radii of 1cm and 2cm, respectively. But how do we compare their curvatures?

𝑘𝑏𝑖𝑔 𝑏𝑢𝑏𝑏𝑙𝑒 =

90° π 2

180° π

90° π 1

𝑟𝑠𝑚𝑎𝑙𝑙 𝑏𝑢𝑏𝑏𝑙𝑒 = 2 𝑟𝑏𝑖𝑔 𝑏𝑢𝑏𝑏𝑙𝑒 𝑘𝑠𝑚𝑎𝑙𝑙 𝑏𝑢𝑏𝑏𝑙𝑒 = 2𝑘𝑏𝑖𝑔 𝑏𝑢𝑏𝑏𝑙𝑒 1

∴𝑘 = 𝑟

Hence, the curvature of the two-dimensional bubble can be shown as k = 1 . 𝑟𝑎𝑑𝑖𝑢𝑠

For the bubbles, k is constant at every

point - which means that bending is uniform along the entire curve. And therefore, the mean curvature is constant for a standard double bubble. As shown in the image above, when the bubbles are in contact, the smaller bubble curves faster than the larger bubble. To quantify this, it is important to determine how

Theorem 1 and Corollary 2

According to Theorem 1 of the Double Bubble Theorem, the double bubble uniquely minimizes the area among all 3

surfaces by 𝑅 - the three-dimensional

Euclidean space- enclosing two equal volumes. To prove this statement, isoperimetric inequalities are needed - a geometric inequality involving the perimeter of a set and its volume. Assume that A(S) is the surface area of the double bubble.

be rotated around a certain axis or plane without changing shape.

(To add, n is the number of dimensions of a space.)

Therefore, A(S) can never be less than the area of the double bubble, which proves the theorem. 3

For Corollary 2, any surface in 𝑅 enclosing two regions, each having volume V, 3

the area A satisﬁes 𝐴 ≥ 243π𝑉 with equality if and only if it is isomorphic to the standard symmetric double bubble enclosing 3

two regions of volume V by an isometry of 𝑅 .

Theorem 3 and Theorem 4 For Theorem 3, an area minimizing 3

enclosure of m volumes in 𝑅 , for m < n, is rotationally symmetric about an (m − 1) -dimensional plane. We need to enclose multiple distinct regions of specialized volumes. If we have given m volumes 𝑉1, 𝑉2.... 𝑉𝑚which neeed to enclose

For m=3, the symmetry will be around a 2dimensional plane. This is a two-dimensional Cartesian plane. A Cartesian plane is a two-dimensional coordinate plane formed by the intersection of two perpendicular lines. For m = 0, the symmetry is around a 0-dimensional point, so the shape will be a sphere. And lastly, for Theorem 4, if A(V1, V2) is the minimum area for surfaces in 𝑅 enclosing volumes 𝑉1 and 𝑉2 , then A is concave as a function of 𝑉1 and 𝑉2 .

during a surface in 𝑅 , what is the minimal surface that encloses these m volumes? Rotational symmetry about an (m-1) dimensional plane means that the surface can

This can be proved by inequality.

In mathematics and physics, the symbol λ (lambda) represents a parameter or variable, particularly in weighted averages and linear combinations.

Conclusion

The Double Bubble Theorem is significant since it is connected to nature and can be observed in everyday life. This theorem allows people to acquire a better understanding of surface behavior and find inspiration for mathematical problem-solving and architecture. To summarize, the Double Bubble Theorem adds to the field of geometry by providing insights for future research.

How long will it take for a country’s population to double? The Rule of 70

By Chaeyeong (Amy) Lee 71473@jisedu.or.id Applied Mathematics Field of Application: Demographics

Introduction

In a country, there are a total of 200 people. If the annual natural increase rate is 5%, how long will it take for the population to be doubled? This can be calculated in multiple ways. One is to solve an equation using exponential and logarithism. For the example above, the increase rate is 5%, so the population will be multiplied by 1.05 every

of the variable as A, the constant annual percentage growth rate as r%, and time as t. To find the doubling time, the outcome after t years will be 2A. 𝑡

𝐴 · (1 + 𝑟) = 2 · 𝐴 𝑡

(1 + 𝑟) = 2

𝑡

year. The equation will be 200 · 1. 05 = 400 , and t can be expressed as 𝑙𝑜𝑔 1.052, approximately 14.2.

This equation aims to find the time it takes A to double, so the logarithm can be used to make the process easier.

However, there is an easier way to calculate this: the formula 70/(growth rate). When 70 is divided by 5, it is 14 - similar to the answer derived from the equation. This formula is the Rule of 70.

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠 𝑡𝑜 𝑑𝑜𝑢𝑏𝑙𝑒 =

𝑡

𝑙𝑛((1 + 𝑟) ) = 𝑙𝑛2 𝑡 · 𝑙𝑛(1 + 𝑟) = 𝑙𝑛2 𝑡=

𝑙𝑛2 𝑙𝑛(1+𝑟)

𝑙𝑛2 and 𝑙𝑛(1 + 𝑟) can be approximated with tailor series.

70 𝑎𝑛𝑛𝑢𝑎𝑙 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑔𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑥

Derivation of the Rule of 70

The Rule of 70 assumes the growth rate is fixed and determines the time it takes for a variable to double(doubling time). Although it seems simple, the Rule of 70 contains complex mathematical information. There can be various approaches to derive the formula, but one is to use logarithm and tailor’s series, involving calculus. Let’s put the initial value

𝑥

𝑙𝑛(1 + 𝑥) = 𝑥 − 2 + 3 − 4 ··· In 𝑙𝑛(1 + 𝑟), r is less than one, so 2

− 𝑟 /2 + 𝑟 /3 − 𝑟 /4 ··· is a very small number that is disregardable. Therefore, ln(1+r) can be considered as r.

𝑙𝑛(1 + 𝑟) ≈ 𝑟 To calculate ln2, 1 can be substituted in x.

𝑙𝑛2 = 𝑙𝑛(1 + 1) 1

= 1 − 2 + 3 − 5 − 6 ··· There can by infinite numbers added in this series, but finite number of terms can be summed to get approximation. 1

1 − 2 = 0. 5 1

1 − 2 + 3 ≈ 0. 833 1

1 − 2 + 3 − 4 ≈ 0. 583 1

1 − 2 + 3 − 4 + 5 ≈ 0. 783 1

1 − 2 + 3 − 4 + 5 − 6 ≈ 0. 613 When the process is continued until only a small variation remains, the infinite series is converged to approximately 0.693147.

𝑙𝑛2 ≈ 0. 693147 ≈ 0. 693 When everything is put together, time can be expressed as 0.693/r. 𝑙𝑛2

𝑡 = 𝑙𝑛(1+𝑟) ≈

0.693 𝑟

For convenience, 100 is multiplied to the denominator and the nominator so that the r is converted to the growth rate as percentage instead of decimals. Then, we get the formula for the Rule of 70.

𝑡≈ ≈ 3

0.693·100 𝑟·100 70 𝑔𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒(%)

Variations of the formula

The formula has variations since the nominator can be rounded to different numbers. When the growth rate is high, bigger numbers tend to give a more accurate result. As an example, with low growth rates like 1% or 2%, numbers close to 69.3 such as 70 or 71 calculate a more precise doubling time. On the

other hand, if the growth rate is 16%, 74 will be a better choice. Among the variations, 69, 72, and 70 are used frequently. The Rule of 69 is a more approximate value from 69.3, so it gives a more precise measurement; it is often used when dealing with continuous compounding or situations that require mathematical rigor. The Rule of 72, on the other hand, gives a slightly more accurate calculation compared to the Rule of 70 when the growth rate is between 6% and 10%. For example, if the annual growth rate of a country’s GDP is 8%, the doubling time gained from precise calculation is 9.00647 years. Using the Rule of 70, it is 8.75 years, and using the Rule of 72, it is 9 years.

Application in various fields

The Rule of 70 gives quick calculation of doubling time, so it is utilized in various areas: finance, economics, and demographics. For example in finance, the Rule of 70 can be used to evaluate potential investments by assessing the time it takes for the investment to double. In economics, the formula is employed to estimate the doubling time of a country’s Gross Domestic Product(GDP). In demographics, it is a useful tool for calculating the doubling time of a country’s population.

Limitations

This formula, despite its utility, has some limitations. the Rule of 70 assumes that the growth rate is constant, which is unlikely in real life. Economic changes such as inflation may fluctuate the situation of investing, and the growth rate of a country’s GDP can shift according to the country’s economy, technology, or demography. Besides, the formula is an approximation, so when the growth rate is high, the estimate becomes less accurate. For example, if there is an investment growing at 25% annually, the doubling time will be 2.8 years using the Rule of 70. On the other hand, it will take 3.1 years

according to a precise calculation using the compound interest formula.

Conclusion

Doubling time is an important estimation for people, organizations, or countries to evaluate goals and make decisions. The Rule of 70 is a useful tool for this since it offers a clear and fast calculation of doubling time. However, it is crucial to note its limitations. The growth is not constant in real life, and there can be unexpected events.

Citation

Jeremy Finger. “Understanding the Rule of 70 for Investment Growth.” Riverbend Wealth Management, 20 July 2024, riverbendwealthmanagement.com/th e-rule-of-70/. Accessed 6 Oct. 2024. CFI. “Rule of 70.” Corporate Finance Institute, corporatefinanceinstitute.com/reso urces/wealth-management/rule-of-70/. Leavitt, Laura. “Rule of 72: What It Is and How to Use It.” Bankrate, 29 Aug. 2023, www.bankrate.com/investing/what-i s-the-rule-of-72/.

Becoming The Next Zuckerberg: An Introduction to Graph Theory

By Justin Tjitra 69631@jisedu.or.id Applied Mathematics Field of Application: Network Technology

Introduction

It’s a scene we’re all familiar with. A young Mark Zuckerberg runs up to his dorm room on a fall night in 2003. Grabbing a beer from his fridge, he takes a swig, sits down at his silver Macintosh, and begins typing out the first lines of code for FaceMash: a site where users were asked to compare the attractiveness of their fellow Harvard classmates by presenting pairs of student photos side by side. The university’s servers crashed at 4 PM the next day from the immense traffic coming from Kirkland House. Within a few days, the FaceMash was shut down, and Mark was put on probation. But this experience didn’t stop him from realizing the vast potential of social networking. Within a year, Facebook was launched, and the rest is history. Living in Jakarta, most of us are used to sitting idly in our seats, scrolling endlessly on Instagram Reels, and waiting for our cars to inch forward every few minutes. Oftentimes when I’m put into this situation, I open up Google Maps on my phone to find the shortest route home. But I’ve never really stopped to wonder how the app knew what routes there were in the city, let alone which one was the shortest path specifically back to my house. After all, Jakarta alone has over 6,000 km of commuter roads. To put that into perspective, the entire Indonesian archipelago extending

from East to West is 5000 km. Made simply, the app seemed to work like magic. Luckily, it doesn’t take a genius like Mark Zuckerberg to build a social network, or a lightning fast computer brain to find the shortest way back home. In fact, all it takes is a basic understanding of an important field in information science called graph theory. To get started on building the next social network, Google Maps, or any project that requires some form of connection, graph theory is the first essential step that you would need to know. 2

Basic Terminology You might initially think of a graph as a crisscross plane of lines with a y- and x-axis from math class. However, graph theory is an entirely different field often not taught in high

school, but usually in more advanced mathematics and computer science courses. The basic concepts and algorithms of graphs: A graph is a mathematical representation to represent the relation between two or more different objects or variables. All graphs consist of two basic components: nodes and edges. Think of a node as an object that could technically represent anything (person, cities, organizations, etc.), and an edge as some form of connection between them. Graphs could represent and model any sort of relationship between many different variables, such as social apps, cities, and even the world. For example, in Zuckerberg’s case, Facebook could be represented as a graph where each user represented an individual node, and an edge represented whether they were friends or not.

But to represent the different ways we can go from one node to another in a graph, we have two types of sequences: paths and cycles. A path is defined as a sequence of nodes and edges that are not repeated. A cycle, on the other hand, is a sequence of nodes and edges where only the first and last nodes are repeated. Hence, it’s a path that “cycles” back on itself, where its beginning is the same as its end.

But back to the case of modeling routes in Jakarta, how could we represent that as a graph? Sure, we could set each destination or intersection as a node, but different roads have different lengths and travel times. Some are one-way, and some cycle back to themselves. The solution is simple. Ever since the field's conception by Euler in the 1700s, mathematicians have found several ways to represent connections between different nodes. In other words, there are many different types of edges that we could use to show how one destination is connected to another. In some cases, one can only travel from one node to another in just one direction. Because of this, the edge that connects the two nodes is labeled as “directed”, and an arrow is drawn in the direction of travel.

In other cases, there can be different edges to travel between two points, each of which can have different weights on them. Weights could represent any sort of value, but in most cases, they represent the time or cost it takes to go from one node to another. To represent this, we simply draw multiple edges between the two nodes we want to connect.

Case Study:

Now that we’ve covered all the necessary terminologies, we can take a look at a specific case of graph theory’s real life applications. Earlier, I mentioned Google Maps’ use of finding the shortest path to any place while accounting for changes in traffic. So how exactly does the app know, at any one time, what’s the shortest path to travel to any destination? Graph theory gives us an answer through an algorithm named after the 20th Century computer scientist Edsger Dijkstra that efficiently finds the shortest path between any two destinations. Here’s how the algorithm works. We first have to set each node with a starting value that represents the shortest time or distance it takes to reach that node. Initially, the node we start from has a value of 0, and since all the values for the other nodes are currently unknown, we can set an arbitrarily large value to them, such as infinity. Because our algorithm involves individually traversing each different node, we

must also keep track of whether we’ve visited the node we’re currently on. It would be pointless to traverse a node we’ve already visited since we’re searching for the shortest path. We may set all nodes as unvisited except the one we start on, since we’re currently traversing it. Starting with our first node, we can traverse to each of its different neighbors that haven’t been visited. Let’s label the node we’re currently on as A, and the neighboring node we’re traveling to as B. If the value of A added with the weight of the edge connected to B is smaller than the current value of B, that means we’ve found a shortest path to B than we previously did. As a result, we can update the value of B to this new value. Otherwise, the distance we’ve found to B before is still the shortest, and we don’t need to update the node's value. Finally, we can set node A as visited. We repeat this traversal until all nodes have been visited, and return the value of the node of our destination. To use the algorithm to dynamically find the shortest path between two destinations in Jakarta, we can represent each specific destination on a map as a node, and the lanes that connect them as edges. In most cases, edges would be directed, as it’s uncommon to find a two-way lane in most areas of the city. Each edge is initially given a weight from the average time it takes to travel across it without any dynamic conditions, such as traffic. Based on live data collected from satellite images and user input, these weights can be readjusted in real time in order for the app to optimally find the shortest path.

Conclusion: From social networks to shortest paths, graph theory has proved to be an essential tool in representing connections and the data hidden in them. Its diverse Uses in several industries, ranging from social media, to transport infrastructure, to even politics, has far-reaching effects on how we see the world and the networks in them. With this newfound knowledge, no longer does analyzing and optimizing Complex networks have to appear like magic. Becoming the next Zuckerberg just became a little easier.

Three-Dimensional Representation of Euler’s Formula and its Applications

Dowon Her 67817@jisedu.or.id

Applied Mathematics Field of Application: Natural Sciences

Introduction

Euler’s identity is considered one of the most beautiful mathematical expressions. The mathematical expression links five of the most special numbers in mathematics, 0, 1, e, π, and i, with three math operations, exponentiation, multiplication, and addition, each occurring only once.

formula can be derived using the second method and how it can be represented in a three-dimensional plane as a helical structure. To understand how Euler’s formula is derived, it is important to review three mathematical concepts: the unit circle, the complex plane, 𝑥

and the exponential function 𝑒 .

𝑖π

𝑒 + 1= 0 Each of these numbers has a special identity. 0 is the additive identity of any real number, meaning that adding 0 to a number results in the same number. 1 is the multiplicative identity, meaning that any number multiplied by 1 is the same number. e 𝑥

is the base of the exponential function 𝑒 , which is the only function that has the same derivative function as itself. π represents the ratio between the circumference and diameter of any circle. Therefore, the beauty in this identity lies in how constants of such unique properties are connected only with a few mathematical operations.

The Unit Circle

Imagine a circle with its center in the origin with a radius of 1. This is called the unit circle. At angle θ, since 𝑠𝑖𝑛(θ) is equal to the length of the opposite side over the hypothenuse, which in this case is 1, the height of the triangle will be 𝑠𝑖𝑛(θ). Similarly, since 𝑐𝑜𝑠(θ) is equal to the length of the adjacent side over the hypothenuse, the base length of the triangle will be 𝑐𝑜𝑠(θ). Therefore, point A can be represented by the coordinate (𝑐𝑜𝑠(θ), 𝑠𝑖𝑛(θ)) in the cartesian plane.

Euler’s identity is a special case of Euler’s formula when the value of 𝑥 is π. 𝑖𝑥

𝑒 = 𝑐𝑜𝑠𝑥 + 𝑖𝑠𝑖𝑛𝑥 There are two popular ways in how Euler’s formula is derived. One is through the 𝑥

use of the taylor series expansion of 𝑒 , 𝑠𝑖𝑛𝑥, and 𝑐𝑜𝑠𝑥. A more interesting approach is through the geometric interpretation of the formula. This article will explain how Euler’s

The Complex Plane

A complex number can be represented by its real part, a, and its imaginary part, b, where a and b are both real numbers.

𝑍 = 𝑎 + 𝑏𝑖 Due to its imaginary part, a complex number can not be represented in the real number line. Thus, a complex plane, where the horizontal axis represents the real number part and the vertical axis represents the imaginary part, is needed.

Exponential function, eix The defining property of the 𝑥

exponential function 𝑒 is that its derivative is identical to itself. Using this property, let us 𝑥

If a complex number, 𝑎 + 𝑏𝑖, is multiplied by 𝑖, it produces − 𝑏 + 𝑎𝑖. Here, the real part changed to the negative value of the imaginary part and the imaginary part changed to the real part. In a complex plane, this means that the point is rotated 90° counter-clockwise along the origin.

take a physics approach. If we consider 𝑒 as a function for its position vector, its velocity vector will equal to its position vector because the velocity vector is the derivative of the position vector with respect to time. 𝑥 𝑑 𝑥 𝑒 =𝑒 , 𝑑𝑥

𝑥 𝑑 𝑎𝑥 𝑒 = 𝑎𝑒 𝑑𝑥

If the exponent 𝑥 is multiplied by a constant 𝑎, according to the chain rule, the derivative of the function will be 𝑎 times itself. 2𝑥

For example, consider the function 𝑒 . Its 𝑥

derivative would be 2𝑒 , meaning that the velocity vector will be twice the position vector.

Deriving eix = cosx+isinx What if the exponent 𝑥 in the 𝑥

exponential function 𝑒 is multiplied by the Now, consider the unit circle in the complex plane. Since the vertical axis represents the imaginary part of the complex number in the complex coordinate, 𝑠𝑖𝑛(θ) the vertical coordinate of the unit circle will represent the imaginary part. Similarly, the horizontal axis represents the real part of the complex number and thus 𝑐𝑜𝑠(θ) represents the coordinates for the real part. Thus, the unit circle in a complex plane can be represented as below.

𝑖𝑥

imaginary unit 𝑖? The derivative of 𝑒 would 𝑥

be 𝑖𝑒 . Previously, we found that multiplying a complex number by 𝑖 can be visualized by a 90° counter-clockwise rotation of the point along the origin in the complex plane. Therefore, this means that the velocity vector is perpendicular to the position vector, with the same magnitude, for all values 𝑥. Again taking a physics approach, this can be seen as a circular motion, as the velocity of an object in a circular motion is always perpendicular to its radius. Therefore, 𝑖𝑥

𝑒 can be represented as below.

Similarly, if we view the 𝑥 - imaginary axis, it will draw a 𝑠𝑖𝑛𝑥 graph, as the imaginary part of this equation is 𝑠𝑖𝑛𝑥. In other words, 𝑠𝑖𝑛x describes the relationship between 𝑥 and the imaginary part of the equation.

𝑖𝑥

Thus, we can see that 𝑒 produces the same form in the complex plane as 𝑐𝑜𝑠𝑥 + 𝑖𝑠𝑖𝑛𝑥. Therefore, the formula 𝑖𝑥

𝑒 = 𝑐𝑜𝑠𝑥 + 𝑖𝑠𝑖𝑛𝑥 can be derived geometrically.

5 Three-Dimensional Representation of Euler’s Formula

To satisfy all these conditions, the 𝑖𝑥

three-dimensional shape of 𝑒 must be a helical form. When viewed from the front, this is a circle. When viewed from the side it shows 𝑐𝑜𝑠𝑥, and when viewed from the top, it shows 𝑠𝑖𝑛𝑥.

𝑖𝑥

Now that we know 𝑒 = 𝑐𝑜𝑠𝑥 𝑖𝑥

+ 𝑖𝑠𝑖𝑛𝑥, we can find the graph 𝑒 in the three-dimensional plane of 𝑥 - real - imaginary axis. In this equation, when we view the plane in 2d view that shows the imaginary and real parts, it will draw a circle of radius 1, as explained previously.

Application of eix Due to its helical form that resembles 𝑖𝑥

However, if we view the 𝑥 - real axis, it will draw a 𝑐𝑜𝑠𝑥 graph, as the real part of this equation is 𝑐𝑜𝑠𝑥. In other words, 𝑐𝑜𝑠𝑥 describes the relationship between 𝑥 and the real part of the equation.

a helix and spiral, 𝑒 is applied in many fields of science for modeling. For example, the electric and magnetic fields of circularly polarized electromagnetic waves follow a helical path as they propagate. The mathematical description of these waves involves complex exponentials, 𝑖𝑥

much like 𝑒 , especially in describing the phase relationships between different components of the field. 𝑖𝑥

Another example of which 𝑒 is used is the mathematical modeling of the structure of DNA. DNA has a double helix shape, and the two helices of the sugar-phosphate backbone have a phase difference of and twist around each other. The structure of DNA

𝑖𝑥

can thus be described with two helices, 𝑒 and 𝑒

𝑖(𝑥−𝜋)

as shown below.

Conclusion

Overall, we were able to derive Euler’s formula, through our understanding of the unit circle, the complex plane, and the exponential 𝑖𝑥

function 𝑒 . Due to its particular shape that it produces in a three-dimensional plane, the formula is not a mere mathematical abstraction, but a key to describing and understanding the world around us.

The Application of Physics and Calculus in Rollercoasters

By Haein Kang 36994@jisedu.or.id

Applied Mathematics Field of Application: Physics

Introduction

Rollercoasters. Who doesn’t like rollercoasters? The thrills, the loops, the fear all jumbled into one. But did you know that because roller coasters do not have engines, mathematical concepts are crucial to keep the roller coaster going (and you alive)? Hence, let’s look into this deeper. Rollers coasters aren’t just made by drawing random curves and making the slope and steep as possible to get people as scared as possible. While designers do first start by sketching a roller coaster showing the placement of hills, twists and turn, they have to utilize math through technical drawings such as in computer aided design programs such as CAD. Math is needed to calculate the exact slopes of roller coaster hills, which will help us to accurately determine the speeds that will be generated at various points along the track. Because roller coasters aren’t run by engines, there has to be precise calculations on the needed speed and height to keep the roller coaster to keep moving. Hence, the speed that is needed is dependent on the size of the loops such as looking at the height.

Application of physics

The roller coaster usually starts with a long and slow lift up a steep hill. While it is used to build up fear and tension for the people, it also serves as the coasters engine. While it be difficult for you to feel, there is actually a conversion between the types of

energies during the process from kinetic energy to potential energy. Think of kinetic energy as energy of an object (in this case the rollercoaster) due to its motion, and think of potential energy as the energy that is stored in an object. In this case, the potential energy is gravitational potential energy, where the total potential energy is given by U=mgh (m represents the mass of the 2

train, g is a constant of 9.8 𝑚/𝑠 known as the acceleration due to gravity and h is the vertical height from the ground to the top of where the rollercoaster is located). Gravitational potential energy, U, can be thought of as the energy an object possesses due to its position above earth. To think of it simply, if you were standing on type of the eiffel tower, you would have a greater gravitational potential energy compared to if you were standing on the ground. The equation for kinetic energy is 2

given by K= 0. 5𝑚𝑣 , where m is the mass of the train and v is its velocity, which is the rate at which an object changes its position. In short, the greater the magnitude of the velocity, the faster the roller coaster is traveling. As the roller coaster climbs up, and up, and up, the kinetic energy of the roller coaster converts more to gravitational potential energy. That is why you will often feel the rollercoaster slowing down, and most likely stop for a short while at the pinnacle of the roller coaster.

stop for a short while at the pinnacle of the roller coaster.

50𝑚𝑔 = 2 𝑚𝑣 𝑣 =

50𝑚𝑔×2 2 𝑣 = 100𝑔 𝑚

𝑣 = 10 𝑔 𝑚/𝑠 Hence, if a roller coaster climbs to an extremely high hill, the greater the gravtiaitional potential energy that the roller coaster will have. However, what this indicates is that the kinetic energy would also have a large magnitude as the gravitational potential energy will be converted to kinetic energy. As shown by the equation of kinetic energy, if the magnitude of the kinetic energy is greater, the velocity will also be greater, hence the rollercoaster will travel faster. Therefore, math is always involved through the efficient use of these equations to take all of these factors into account. To get the calculations, we follow the law of conservation of energy which states that the total amount of energy must always be the same. 1

𝐾 + 𝑈 = 2 𝑚𝑣 + 𝑚𝑔ℎ

A sample question:

Suppose the first hill is 50 meters above ground, how fast will the train be going at the bottom?

Note that at the bottom, when the potential energy has been completely converted into kinetic energy, the roller coaster will reach a high speed as that is when the potential energy is the lowest. Hence, because energy is conversed, the rest will be kinetic energy. However, note that there are limitations to solely relying on these equation as some energy is lost to frictional forces by the track, so when potenntial energy is converted to kinetic energy you get less than you started with. In case you are wondering what friction is, think of friction as the force that resists motion. For instance, the reason that you are able to slide in an ice skating rink with little to no resistance is because there is little friction. That is why each hill on the track tends to be lower than the previous one or the train could roll back and get stuck if the kinetic energy is not enough to be converted into gravtiational potential energy.

At the top of the hill, where rollercoaster is assumed to be at a temporary stop, total energy is: 2 2 1 1 𝑚𝑣 + 𝑚𝑔ℎ = 2 𝑚(0) + 𝑚𝑔50 = 50𝑚𝑔 2

. At the bottom of the hill, the height will be 0, indicating that there is no gravitational potential energy. 2 2 2 1 1 1 𝑚𝑣 + 𝑚𝑔ℎ = 2 𝑚𝑣 + 𝑚𝑔(0) = 2 𝑚𝑣 2

. Because the total energy is conserved, we know that total energy at the top of the hill = total energy at the bottom of the hill.

Loops

The loop must be built with extreme precision. For instance, a loop that is too circular will require extremely high speeds that could result in a g-force that is too high for people to comfortably withstand. While changes in g-force is what makes rollercoasters fun, a high g-force can cause blood to rush down from the brain toward the feet, which can cause loss of vision or passing out.

normal. At the bottom, the roller coaster seats push up harder, increasing the g-force above 1 and making you feel heavier.

Application of Calculus

Calculus can be used to create and analyze curves, loops, and twists along the roller coaster track. It can also help with slope calculations and find the maximum and minimum points along the track. Maximum height: It is found when the slope goes from positive to negative or when its derivative function crosses from positive to negative. A perfect roller coaster loop is called a clothoid loop because it is shaped like a teardrop, which means that the radius changes and is shorter at the upper part of the loop than it is across the center. What this means is that the roller coaster can get through the loop at lower entry speeds. In a clothoid, the radius is the widest at the bottom of the rollercoaster. Remember, at the bottom the riders are traveling the fastest (highest kinetic energy and lowest gravitational potential energy combo), which helps to reduce the force when the passengers are moving the fastest.

G-whiz

Accelerations are given in g-forces, which are measured relative to gravitational free fall. As the train crests over a hill, it accelerates away from your body, reducing the push of the seat beneath you. The resulting g-force is less than 1g, so you feel lighter than

Speed: Find the derivative of distance with respect to time, as the derivative of displacement with respect to time is velocity. 𝑑 (𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒) = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑑𝑡

Magnitude of acceleration: Find the derivative of speed with respect to time as the derivative of velocity with respect to time is acceleration. 𝑑 (𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦) = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑡

Example question: If the velocity of the roller coaster is given by 2

𝑣(𝑡) = 0. 9𝑡 − 10𝑡 + 21, what is the acceleration at 2.8 seconds? 2

𝑣'(𝑡) = 𝑎(𝑡) = 1. 8𝑡 − 10 2

𝑎(2. 8) = 1. 8(2. 8) − 10 =− 4. 9𝑚/𝑠

The Golden Ratio in Architecture

By Doyeon Kim 70710@jisedu.or.id Applied Mathematics Field of Application: Architecture

Introduction

Throughout the history, the golden ratio has largely been used in diverse fields. Specifically, this unique mathematical proportion appears in nature, art, and architecture, offering a sense of balance. The golden ratio has been applied in architecture to create stabilized designs, from ancient structures like the Parthenon to modern buildings.

ratio of ‘a’ to ‘b’ is the same as the ratio of the whole length ‘a+b’ to ‘a’. It can expressed as

But why is this specific ratio frequently used? This article will explore the mathematical foundations behind the golden ratio and how the golden ratio is applied to architecture.

As we set that a/b is the same as ϕ, we can know b/a is 1/ϕ. Therefore we can change the equation as ..

𝑎 𝑏

𝑎 +𝑏 𝑎

= ϕ

And the (a+b) /a can be modified to the below equation.

𝑎 𝑎

𝑏

+ 𝑎 = ϕ

What is the golden ratio?

The Golden ratio, denoted by the Greek Letter ϕ (phi), is a constant equal to 1.618. It is derived by dividing a line into two parts, the ratio of which is the same as the ratio of the whole line to the larger part of the line.

For example, a line divides into two parts, ‘a’ and ‘b’ where ‘a’ is longer than ‘b’. As the

1+ ϕ = ϕ 2

ϕ + 1 =ϕ

Moreover, the golden ratio can be derived by the Fibonacci sequence. Fibonacci sequence is a sequence where each term is the sum of the two preceding terms. When we have two successive sequences, the ratio between two terms has a similar value to the golden ratio. The Fibonacci sequence is 0,1,1,2,3,5,8,13,21.. .

Applications in Architecture

The golden ratio is frequently seen in architectural structures. In this part, we will look at the example of using the golden ratio. The Great Pyramid of Giza: The golden ratio can be observed in the design of the Great Pyramid of Giza, where the dimensions of the pyramids is close to the golden ratio. This is a planet figure of the Great Pyramid of Giza.

By rearranging the terms, we can get the below equation. 2

ϕ − ϕ− 1= 0 Using a quadratic formula, we finally find the value ϕ approximately equal to 1.618. Therefore, the Fibonacci sequence is related to the golden ratio. The equation has two roots, as its degree is 2. However, the value of ϕ must be greater than 0; the answer should be one.

= ϕ=

1± 5 2

= 1. 618

1± 1+4 2

And this is the triangle when the folded planet figure is viewed from the front. To demonstrate the golden ratio, we can divide the slant height of the pyramid, 187m by half of the base length, 115m). This yields a value of 1.618, which is a close value to the golden ratio.

Moreover, we can also check the ratio between the sum of the slant height and half base length and the bigger value, slant height. The calculation 187+115 115

Also, by using the Fibonacci sequence, the golden rectangle and spiral can be expressed in one diagram, which is shown below.

results in 1.618, which is the golden

ratio. Therefore, it shows that the Great Pyramid was built with the golden ratio. Although it is debated whether the ancient people intentionally designed the pyramid using the golden ratio or not, the geometric relationship has intrigued architects, mathematicians, and others. The pyramid’s design suggests we have a better understanding of ratio and it is recorded as a fascinating architecture. The Parthenon: Additionally, the golden ratio is applied to the design of the Parthenon, which is the proportion of the width and height.

If we apply the golden ratio rectangle and spiral into the Parthenon, it shows a perfectly matching shape. It proves that the architects in the past planned a blueprint of the Parthenon based on the golden ratio to achieve visual balance. This is the front view of the Parthenon. The red line is the dimension of the base of the Parthenon, which is 30.9066 meters. And the width of the cellar is 19.2 meters. Below, by calculating the ratio of the base to the width, it shows that the ratio is similar to the golden ratio, 1.618. 𝑇ℎ𝑒 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑇ℎ𝑒 𝑤𝑖𝑑𝑡ℎ 𝑜𝑓 𝑐𝑒𝑙𝑙𝑎𝑟

30.9066 19.2

= 1. 617

Explaining the Curves of Baseball: The Magnus Effect By Jinwoo Yang 69490@jisedu.or.id Applied Mathematics Field of Application: Sports, Aerodynamics

Introduction

Ball sports are one of the most popular entertainment among people. Players throw, kick, or hit an object. Sometimes we can observe that those objects curve, drop, or wiggle as they fly mid-air, arousing a spectacular movement. Soccer players perform no-spin kicks, baseball players throw sliders, and tennis players topspin serves to vary the travel path of the ball to trick the opponents. How is this possible? Those players are not all mighty gods or they do not have the green screen to edit the movements.

The Magnus Effect

How they perform this interesting ability is all due to the Magnus Effect.

The Magnus effect is a phenomenon in fluid dynamics that occurs when a spinning object moves through a fluid, such as air. This effect is named after Heinrich Gustav Magnus, a German physicist who first discovered it in the 1850s. The Magnus effect is commonly observed in sports like soccer, baseball, and tennis, where players intentionally spin the ball to alter its trajectory. The principle behind the Magnus effect is based on the difference in pressure on opposite sides of a spinning object. When a ball spins, it drags some of the air around it due to friction. If a ball spins counter-clockwise, for instance, the air on the left side of the ball moves in the same direction as the incoming air, while the air on the right side moves in the opposite direction. This difference in relative motion causes the air to flow faster on one side and slower on the other. According to Bernoulli's principle, faster-moving air results in lower pressure. Therefore, the side of the ball where the air moves faster will have a lower pressure compared to the opposite side. This pressure difference generates a force perpendicular to the direction of motion, known as the Magnus force, which causes the ball to curve. In simpler terms, the Magnus effect is an effect that explains how an object is

affected by the different air pressure, caused by its spin.

helps athletes improve their game by predicting and utilizing the impact of spin on the ball's path.

Further Study of the Magnus Effect

A study called the “Aerodynamics of Sports Balls” conducted by John D Barrow examined how varying spin rates and velocities influenced the trajectory of spherical objects in a controlled wind tunnel. The researchers placed balls with different textures and weights on rotating mounts, adjusting spin rates from low to high. High-speed cameras recorded the paths as each ball moved through the wind tunnel, allowing researchers to analyze the curvature of each trajectory. The results confirmed that higher spin rates increased the Magnus force, causing a more pronounced curve, especially at medium to high velocities. The study also noted that smoother balls exhibited a greater Magnus effect at similar spin rates than textured ones, suggesting that surface friction can moderate the effect. Additionally, it was observed that heavier balls required a higher spin rate to achieve the same degree of curvature as lighter ones. These findings support the role of the Magnus effect in sports, showing how spin and ball design can impact accuracy and control. Applications of the findings extend to sports equipment design, particularly in soccer, baseball, and golf, where controlling spin is essential for precision. The study underscores the importance of spin physics in engineering, aerodynamics, and sports science. In sports, players use this effect to control the ball’s movement, making it change direction unexpectedly. For example, in baseball, a pitcher can apply spin to a pitch by creating a spin on the baseball using the seams. By doing so, they can create a movement of a ball to go down or move sideways. Furthermore, sometimes the pitchers use the Magnus effect's other side of view of not adding spin to an object. By doing so, the baseball will be affected by the current of air, causing the ball to wiggle. Understanding the Magnus effect

Conclusion

Overall, the Magnus effect is a crucial concept that illustrates how spin influences the behavior of objects moving through a fluid.

A Bayesian Approach to Survey Data: Class and Food Preferences Analysis By Justin Wu (50134@jisedu.or.id) A. Introduction Bayesian probability, named after Thomas Bayes (1701-1761), is a method of mathematics which allows us to update our beliefs about likelihood of events based on new evidence. It is commonly used in statistics and machine learning for reasoning about uncertainties and predictions. In this paper, I apply Bayesian probability to analyze the results of a survey with two questions. The first question asks participants to select their favorite core class (Math, English, Science, or Social Studies), and the second question asks participants to choose their favorite food (Pizza, Burger, or Chicken). Using Bayes’ Theorem, we calculate the joint probability and reverse conditional probabilities, predicting how likely students are to choose certain foods prefer based on their class preferences and how class preferences vary among students with different food choices. While in this case study it is used to analyze the relationship between class and food preferences, Bayesian probability is also widely used in medical diagnosis where doctors issue diagnoses based on the probability of a specific diagnosis given the appearance of specific symptoms or test outcomes; in machine learning, Bayesian probability is used to calculate classification problems such as spam detection, weather forecasting, and financial forcasting. B. Survey Data and Analytical Procedures Survey results from 69 students regarding their favorite class (Math, English, Science and Social Studies) and favorite food (Pizza, Burger and Chicken) were used for demonstrating the Bayesian probability case study: 1. Class preferences (prior probabilities): o

P(Math) = 0.159

P(English) = 0.261

P(Science) = 0.333

P(Social Studies) = 0.246

2. Food preferences (conditional probabilities) for each class are provided: o

For Math enthusiasts: §

P(Pizza∣Math) = 0.545

P(Burger∣Math) = 0.364

P(Chicken∣Math) = 0.091

For English enthusiasts: §

P(Pizza∣English) = 0.833

P(Burger∣English) = 0.056

P(Chicken∣English) = 0.111

For Science enthusiasts:

P(Pizza∣Science) = 0.522

P(Burger∣Science) = 0.304

P(Chicken∣Science) = 0.174

For Social Studies enthusiasts: §

P(Pizza∣Social Studies) = 0.471

P(Burger∣Social Studies) = 0.412

P(Chicken∣Social Studies) = 0.118

3. Joint probabilities are calculated by multiplying the prior probability of each class by the corresponding conditional probability for the food preference. For Math enthusiasts: o

P(Math and Pizza) = 0.159*0.545 = 0.087

P(Math and Burger) = 0.159*0.364 = 0.058

P(Math and Chicken) = 0.159*0.091 = 0.014

For English enthusiasts: o

P(English and Pizza) = 0.261*0.833 = 0.217

P(English and Burger) = 0.261*0.056 = 0.014

P(English and Chicken) = 0.261*0.111 = 0.029

For Science enthusiasts: o

P(Science and Pizza) = 0.333*0.522 = 0.174

P(Science and Burger) = 0.333*0.304 = 0.101

P(Science and Chicken) = 0.333*0.174 = 0.058

For Social Studies enthusiasts: o

P(Social Studies and Pizza) = 0.246*0.471 = 0.116

P(Social Studies and Burger) = 0.246*0.412 = 0.101

P(Social Studies and Chicken) = 0.246*0.118 = 0.029

4. Posterior Reverse probabilities are calculated to determine favorite class given favorite food. Figure 1 below tabulates the marginal probability, conditional probability, joint probability and reverse conditional probability of the data discussed in this article. Figure 2 is a conditional eikosogram for the 4 classes/events..

Figure 1. Bayesian probability data Prior Model

Math

English

Science

Social Studies

Marginal probability

0.159

0.261

0.333

0.246

0.545 0.364 0.091 1.000

0.833 0.056 0.111 1.000

0.522 0.304 0.174 1.000

0.471 0.412 0.118 1.000

Total

0.174 0.101 0.058

0.116 0.101 0.029

Total 0.594 0.275 0.130

Posterior Model (Reverse Conditional Probability) P(Class|Pizza) Pizza 0.146 0.366 0.293 P(Class|Burger) Burger 0.211 0.053 0.368 P(Class|Chicken) Chicken 0.111 0.222 0.444

0.195 0.368 0.222

Total 1.000 1.000 1.000

Likelihood Model Conditional probability P(Pizza|Class) Pizza P(Burger|Class) Burger P(Vhivkrn|Class) Chicken

Joint Probability P(Pizza∩Class) Pizza P(Burger∩Class) Burger P(Chicken∩Class) Chicken

0.087 0.058 0.014

0.217 0.014 0.029

Figure 2. Conditional eikosogram for 4 events (Math, Engl, Science and SS)

C. Bayes' Theorem Bayes' Theorem is a fundamental concept in probability and statistics, expressed as: "(% ∣ $) ⋅ "($) "($ ∣ %) = "(%) Where: •

P(A) is the prior probability, the initial belief of event A.

•

P(B∣A) is the likelihood probability, the probability of event B given the occurrence of event A.

•

P(B) is the marginal likelihood, the total probability of observing the probability of B, regardless of A.

•

P(A∣B) is the posterior probability, the updated belief of event A after considering the event B.

Using Bayes’ Theorem, we can calculate the reverse conditional probabilities, such as determining the class preference given a food preference using the survey data. D. Reverse Conditional Probability: Favorite Class Given Favorite Food We use Bayes' Theorem to calculate the reverse conditional probabilities, which allow us to determine how likely a respondent is to prefer a certain class given their food preference. For example, given that someone prefers Pizza, what is the probability that their favorite class is Math, English, Science, or Social Studies? •

Marginal probabilities for food preferences: o

P(Pizza) = 0.087 + 0.217 + 0.174 + 0.116 = 0.594

P(Burger) = 0.058 + 0.014 + 0.101 + 0.101 = 0.275

P(Chicken) = 0.014 + 0.029 + 0.058 + 0.029 = 0.130

Calculating Posterior Probabilities for Class Preferences Using Bayes' Theorem, we calculate the Posterior reverse conditional probabilities: •

For Pizza lovers: P(Math∣Pizza) =

!(!#$$% | (%)*) !((%)*)

P(English∣Pizza) = P(Science∣Pizza) =

!(!#$$%)

,../.∗,.1.2 ,..2/

!(!#$$% | 3456#7*) !(3456#7*) !(!#$$%) !(!#$$% | <=#>4=>) !(<=#>4=>)

P(Social Studies∣Pizza) =

!(!#$$%)

≈ 0.146

,.899∗,.:;1

,..2/ ,..::∗,.999 ,..2/

≈ 0.366 ≈ 0.293

!(!#$$% | <?=#%6<)@A#>7) !(<?=#%6<)@A#>7) !(!#$$%)

,./B∗,.:/; ,..2/

≈ 0.195

•

For Burger lovers: P(Math∣Burger) =

!(C@D5>D | (%)*) !((%)*) !(C@D5>D)

P(English∣Burger) = P(Science∣Burger) =

,.9;/∗,.1.2 ,.:B.

!(C@D5>D | 3456#7*) !(3456#7*) !(C@D5>D) !(C@D5>D | <=#>4=>) !(<=#>4=>) !(C@D5>D)

P(SocialStudies∣Burger)=

•

= =

≈ 0.211

,.,.;∗,.:;1 ,.:B. ,.9,/∗,.999 ,.:B.

≈ 0.053 ≈ 0.368

!(C@D5>D | <?=#%6<)@A#>7) !(<?=#%6<)@A#>7) !(C@D5>D)

,./1:∗,.:/; ,.:B.

≈0.368

For Chicken lovers: P(Math∣Chicken) =

!(E*#=F>4 | (%)*) !((%)*)

P(English∣Chicken) = P(Science∣Chicken) =

!(E*#=F>4)

,.,21∗,.1.2 ,.19,

!(E*#=F>4 | 3456#7*) !(3456#7*) !(E*#=F>4) !(E*#=F>4 | <=#>4=>) !(<=#>4=>)

P(Social Studies∣Chicken)=

!(E*#=F>4)

= =

≈ 0.111

,.111∗,.:;1 ,.19, ,.1B/∗,.999 ,.19,

≈ 0.222 ≈ 0.444

!(E*#=F>4 | <?=#%6<)@A#>7) !(<?=#%6<)@A#>7) !(E*#=F>4)

,.118∗,.:/; ,.19,

≈ 0.222

E. Applications of Bayesian Probability in Machine Learning Bayesian probability plays a crucial role in many machine learning algorithms, particularly in classification problems. One common applications include: 1. Spam Detection: In email filtering, Bayesian classifiers can predict whether an email is spam or legitimate based on the presence of certain keywords or features. 2. Medical Diagnosis: Bayesian methods are used to update the likelihood of a patient having a certain condition based on symptoms and test results. 3. Weather Forecasting: Bayesian model uses current weather information to predict future weather conditions. 4. Financeial/Marketing decision: In e-commerce, Bayesian methods can be used to recommend products based on a user's past behavior and preferences. F. Conclusion By applying Bayesian analysis to survey data, we were able to determine the conditional (likelihood of food preferences given class preferences) and reverse conditional probabilities (the likelihood of class preference given food preferences). This case study illustrates the practical application of Bayesian reasoning which allows us to make predictions and better understand how student’s choices in one area (like favorite food) might influence their preferences in another area (like favorite class). Bayesian analysis is also a powerful tool in machine learning,

medical diagnostics, and other decision-making processes that rely on handling uncertainties and updating beliefs with new information. G. References Wu, J. A, (2022). AP Statistics Survey Project, Food and Class Choices. Youtube. https://www.youtube.com/watch?v=2VxaA2-wfSM&t=4s