The Magic of Math — Everyday Wonders Hidden in Numbers
Unlock the Power of Math Without the Confusing Jargon
Mathematics is often thought of as a rigid, predictable science filled with rules and formulas. But beneath the surface lies a world of wonder, full of mind-bending concepts that defy intuition and reveal hidden patterns in nature, art, and even human behavior.
In this article, we’ll explore some of the most fascinating mathematical concepts, breaking them down in simple terms so anyone can appreciate their power and elegance.
Chaos Theory and the Butterfly Effect
Chaos Theory is a branch of mathematics that studies systems that appear to be disordered or unpredictable but are actually governed by underlying patterns and rules. These systems are called chaotic systems, and they are extremely sensitive to initial conditions, meaning tiny changes at the start can result in wildly different outcomes over time. Some examples include:
Small changes in the atmosphere, like a shift in temperature or wind, can lead to completely different weather patterns in the future. This is why weather forecasts become less accurate the further out you go.
In biology, small variations in the birth or death rate can cause huge swings in animal populations over time.
Small investor decisions can ripple through the market and lead to unexpected highs or crashes.
The Butterfly effect is a metaphor used to explain the concept of sensitivity to initial conditions. It suggests that something as small as the flap of a butterfly’s wings in Brazil could eventually set off a chain of events that causes a tornado in Texas. The idea is not that a butterfly literally causes a tornado, but that tiny, seemingly insignificant actions can have far-reaching and unpredictable consequences.
The term was coined by meteorologist Edward Lorenz in the 1960s. While working on weather prediction models, Lorenz discovered that even very small changes in his starting data (like rounding off a number from 0.506127 to 0.5061) could lead to drastically different weather outcomes. This realization led him to conclude that long-term weather prediction is inherently chaotic.
The hallmark of chaos is that even a tiny difference at the start of a process can make a big difference in the final outcome. This is often illustrated using the Butterfly effect.
To understand how chaos works, let’s break down the essential elements:
Imagine throwing a ball into a complex system of hills and valleys. If you throw it with just a tiny difference in speed or angle, it could end up in a completely different valley. The rules of physics are still being followed, but the outcome is unpredictable because of the sensitivity to your initial throw. This means that it’s deterministic but unpredictable.
A high birth rate one year can lead to overpopulation, causing resources to become scarce the next year, which in turn causes a drop in population, followed by a recovery as resources become more abundant again. The system keeps adjusting based on its own output. This means that there is a feedback loop in chaotic systems.
One of the simplest examples of a chaotic system is the double pendulum. A double pendulum consists of one pendulum hanging from the end of another. When you set it in motion, the movement of the two pendulums quickly becomes unpredictable, even though both are following the laws of physics.
If you start the double pendulum in slightly different positions, you’ll see dramatically different results. This is chaos in action — tiny changes at the beginning create huge differences in how the system behaves over time.
The Butterfly effect shows that small actions can have large, unpredictable impacts. It’s a powerful metaphor for how our world works, where countless small influences shape the course of events in unexpected ways.
The economy and stock markets are also chaotic systems. Small changes, like a rumor about a company’s earnings or a shift in government policy, can cause major ripples in financial markets.
One way to visualize chaos is through fractals, which are complex patterns that look the same at every scale. Fractals are often found in nature, like the branching of trees, coastlines, or even clouds. These patterns can help us understand how chaotic systems evolve, even if we can’t predict their exact outcomes. Even though chaotic systems are unpredictable in the long run, we can still make short-term predictions. For example, while we can’t predict the exact weather months in advance, we can often give a fairly accurate forecast for the next few days.
The Monty Hall Problem
The Monty Hall problem is one of the most famous and counterintuitive puzzles in probability theory. It’s named after Monty Hall, the host of the 1960s-70s game show Let’s Make a Deal.
At first glance, the problem seems simple, but its solution is surprising and has baffled many, even professional mathematicians. Let’s walk through the problem step-by-step, and I’ll explain why the solution works.
Imagine you’re on a game show. The host (Monty Hall) presents you with three doors:
Behind one door is a car (the prize you want).
Behind the other two doors are goats (which you don’t want).
You don’t know what’s behind any of the doors, but Monty does. Here’s how the game proceeds:
You choose a door — let’s call it Door 1.
(At this point, you have a 1 in 3 chance of picking the car.)Monty, who knows what’s behind each door, opens one of the other two doors, revealing a goat. For example, let’s say he opens Door 3.
Monty then gives you a choice:
Stick with your original door (Door 1)
Or switch to the remaining unopened door (Door 2).
Should you stick with your original choice, or switch to the other door to maximize your chances of winning the car?
Most people’s first instinct is to think, “There are two doors left, so the car must be behind one of them. Since I don’t know which, there’s a 50/50 chance it’s behind either door. It doesn’t matter if I switch or not.”
This seems reasonable, but it’s wrong. The odds are not 50/50. In fact, switching doors dramatically increases your chances of winning. Let’s break it down.
Here’s how the probabilities work out:
When you first pick a door, you have a 1/3 chance of picking the car.
This means that you also have a 2/3 chance of picking a goat.
Now, let’s think about what happens after Monty reveals one of the goats.
If you originally picked the car (a 1/3 chance), Monty reveals a goat behind one of the other doors, and switching would make you lose (because the car is already behind your chosen door).
But if you originally picked a goat (a 2/3 chance), Monty is forced to reveal the other goat, and switching would make you win (because the car must be behind the remaining door).
Therefore, switching gives you a 2/3 chance of winning the car, while sticking with your original choice only gives you a 1/3 chance.
The key to understanding this is realizing that Monty’s action — revealing a goat — doesn’t give you new chances, but it filters out one bad option for you. Let’s look at the two possibilities:
If you picked the car (1/3 chance): Monty will reveal a goat, and switching will cause you to lose.
If you picked a goat (2/3 chance): Monty will reveal the other goat, and switching will make you win.
Prime Numbers and Cryptography
A prime number is a whole number greater than 1 that cannot be divided evenly by any other numbers except 1 and itself. Some examples of prime numbers are 2, 3, 5, 7, 11, and 13.
Prime numbers are the building blocks of all whole numbers. Any whole number can be broken down into a product of primes. For example:
15 = 3 × 5
20 = 2 × 2 × 5
This process of breaking down a number into primes is called prime factorization. It’s kind of like how molecules can be broken into atoms, and primes are the atoms of numbers.
In the world of cryptography (the practice of secure communication), prime numbers are crucial because of one of their properties: it’s easy to multiply two large primes together, but very difficult to break the result back into those primes.
Let’s use an example:
If you multiply two small prime numbers, say 17 × 19 = 323, that’s easy to do.
But, if I just gave you 323 and asked you to figure out which primes I multiplied to get that number, it would take a bit of work, especially as the numbers get much larger.
This “one-way” difficulty is the foundation of modern cryptography. Encryption algorithms take advantage of this idea to secure data. One of the most widely used cryptography methods is called RSA encryption. Here’s a simplified explanation of how it uses prime numbers to encrypt information:
First, two large prime numbers are chosen. These primes are typically hundreds of digits long. Let’s call them P and Q. These two numbers are multiplied together to produce a large number N, which is called the modulus. N = P × Q. It becomes becomes part of both the public and private keys used for encryption and decryption.
A number called the public key exponent is chosen, and it’s made public along with N. The private key exponent, which is related to the prime numbers P and Q, is kept secret. This private key is what can decrypt the message.
Let’s say you want to send a message, like “HELLO,” securely. The message is converted into numbers (each letter becomes a number). Then, using the public key (which includes N and the public key exponent), the message is transformed into an encrypted form that looks like a jumbled, incomprehensible string of numbers.
The only way to decrypt this jumbled string back into the original message is to use the private key, which relies on knowing the original primes P and Q. Without the private key, even if someone intercepts the encrypted message, they can’t easily figure out how to decrypt it because they’d need to factor the large number N back into P and Q— which, for very large numbers, is practically impossible with current technology.
Prime numbers and RSA encryption are behind many of the security protocols we use every day:
When you enter your credit card details on a website, RSA encryption ensures that your information is secure as it travels over the internet.
Many messaging services use some form of encryption (like RSA) to make sure that only you and the person you’re communicating with can read your messages.
RSA is also used to authenticate digital documents and verify identities online, ensuring that messages or transactions haven’t been tampered with.
The Number 9
Multiplying any number by 9 and then adding the digits of the result will always sum to 9. For example, 9 x 5 = 45 (4 + 5 = 9), 9 x 3 = 27 (2 + 7 = 9).
In geometry we tend to find it hidden in a many places, such as:
The circle. It has 360 degrees (3 + 6 + 0 = 9)
The circle cut in half. Each half is 180 degrees (1 + 8 + 0 = 9)
The circle cut in quarters. Each quarter is 90 degrees (9 + 0 = 9)
The circle cut in 8 pieces. Each part is 45 degrees (4 + 5 + 0 = 9)
The circle cut in 16 pieces. Each part is 22.5 degrees (2 + 2 + 5 = 9)
The circle cut in 32 pieces. Each part is 11.25 degrees (1 + 1 + 2 +5 = 9)
A regular polygon inside a circle. Each angle is 60 x 3 (180 = 1 + 8 = 9)
A square. Each angle is 90 x 4 (360 = 3 + 6 + 0 = 9)
The following figures and their angles.
From left to right: Pentagon, Octagon, Decagon.
Pentagon = 108 = 1 + 0 + 8 = 9 // 72 = 7 + 2 = 9
Octagon =135 = 1 + 3 + 5 = 9 // 45 = 4 + 5 = 9
Decagon = 144 = 1 + 4 + 4 = 9 // 36 = 3 + 6 = 9
Also, If we add the digits that come before the number 9 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36). Then we’ll have as usual 3 + 6 = 9.
Infinity and the Concept of Larger Infinities
Aleph Null is a beautiful concept. It is the smallest infinite number. I know what you’re thinking, infinity should be just one concept, not many infinite numbers. After all, if there is an infinity larger than the other infinity, the first one is definitely not an infinity.
Let’s suppose that we have a basic idea on what infinity is (discussed below, 12 in the list). Aleph null is how many natural numbers there are (0, 1, 2, 3, etc.). This concept or number is huge in size and well, infinite.
What if we count all of the natural numbers two or three times? After finishing the first set, we’ll have numbers that extend beyond the natural numbers in order. So, we will need the order of numbers, otherwise known as ordinality. The next number after Aleph Null is omega (ω), then comes ω + 1.
These two last numbers are not cardinal numbers but ordinal, i.e. they represent their position compared to a horizontal axis. The below graph is a simplistic representation. Each set can represent the set of natural numbers in existence, and each set has the cardinality ℵ0. Adding one after the first set does not change the cardinality (You can just change the order and you’ll still be left with Aleph Null cardinality).
It helps to think of them as ordinals (order). Hence, the first ordinal transfinite number after a set is the one we’ve discussed above “ω”
A matchstick representation. Source of the image: Wikipedia
You do not have omega apples, but you can be finishing omega in the race (If you’re really bad)
Interestingly, ω + 1 isn’t necessarily bigger than ω, it just comes after it. This is all a bit too much to take in, so puttings things in perspective should help. Here’s what we should know:
Infinity and Aleph Null are two different things. The former is simply an extreme limit idea lying on the number axis while the latter is simply the size of the set (cardinality).
Cardinality is the size of a set and cardinal numbers represent quantity (1, 2, 459, 1002, etc.) while Ordinality is the order of a set and ordinal numbers represent order (1st, 2nd, 66th, etc.).
Just as there are infinite cardinals, there are infinite ordinals as well and the first infinite (uncountable) ordinal number is the one we have discussed above, omega ω.
Following this logic, Aleph one is the cardinality of omega ω.
Aleph Null is just the first of a huge set of other “Alephs”. Vsauce has made an amazing video discussing this concept, which I heavily recommend.