There has never been a math book this interesting.
Let's start with the simplest mathematical grammar.
A bold story that transcends axioms and probability into higher dimensions.

Reading complex numbers, calculations, and formulas tangled with symbols at a glance is no longer math! The young math storyteller behind "The Cheeky Math Book" has weaved mathematics, often perceived as difficult and tedious, into the world's most entertaining story.
Examining probability through the pigeonhole principle, understanding fixed points with the powder in a mug, wondering whether a straw has 1, 2, or 0 holes, and the fact that 0.9999... and 1 are the same number all become very interesting topics when approached through the story of 'Dimension'.
It stimulates your brain with stories you've never seen before and helps you realize your hidden math skills.

This book explains issues such as "the difference between convex and concave," "why higher dimensions are difficult to understand," and "when calculus is necessary" with clear explanations and easy-to-understand illustrations.
It will open the door to mathematics with its minimal theorem and rigorous standards, and completely change the boring world of mathematics that you knew.

What is mathematics? We all studied it in school, but ironically, very few people truly understand what it is.
Too many people have the misconception that mathematics is just about calculating numbers.
This misconception is evident in the media.
Most math geniuses in movies and dramas are portrayed as human computers who can quickly perform complex calculations.
They calculate the mass of the basketball and the acceleration due to gravity in their heads right before throwing it, and they succeed in making a perfect three-point shot.
But the idea that mathematicians are good at calculations is a huge mistake, no different from the idea that pianists are good at building pianos.
In fact, pure mathematics is one of the fields in the natural sciences that requires the least amount of calculation.
This misconception is particularly unfortunate because it leads people to avoid mathematics, thinking of it as a subject filled with tedious calculations and difficult numbers.
Mathematics is absolutely not this kind of subject.
That's why there's no complicated numerical calculations in this book.
--- pp.14~15

All languages ​​are made up of symbols and grammar.
English uses the Latin alphabet, Korean uses Hangul, and Chinese uses Hanja.
By arranging these symbols according to the grammar of each language, a sentence is completed.
Likewise, mathematics is made up of several symbols and grammar.
To be precise, mathematics is a language consisting of only six symbols, twelve rules of inference, and a properly defined set of axioms.
The only thing that distinguishes mathematics from other languages ​​is that it is not a language for communicating with each other in everyday life, but a language for describing logical reasoning.
And for logical reasoning to be possible, we must be able to determine with certainty whether every sentence is true or false.
Without even an inch of ambiguity.
--- pp.24~25

There was a question that once heated up the Internet.
The question was, 'Is there one or two holes in a straw?'
Isn't it common on the Internet to get excited about such meaningless topics?
First, those who claim that there are two holes in the straw say that there is one hole for the drink to go in and one hole for the drink to come out, so there are two holes in total.
Some argue that a straw has only one hole because a straw is just one long hole.
Moreover, there is also a claim that the number of holes is 0.
They say that straws have no holes because they are made by rolling up a rectangle and not by poking holes in the walls with an object like an awl.
Surprisingly, all three claims have some truth to them.

The reason there is controversy over the number of holes in a straw is because everyone has a slightly different definition of a hole.
So what is the correct definition of a hole? From a linguistic perspective, this question is meaningless.
The definition of vocabulary varies slightly from person to person, so it is difficult to say which is better.
All of them are correct.
But the mathematical perspective is different.
Mathematics, which likes to clearly define all its terms, has a strict definition of hole, and only one is accepted as correct.
But before we look at how we define a hole in mathematics, let's try a logical approach.
--- pp.42~43

So what does the universe look like? One way to calculate the curvature of the universe, as discussed earlier, is to draw a large triangle in space and find the sum of its three interior angles.
But since we can't actually draw a large triangle in space, physicists have been analyzing the early appearance of the universe from the cosmic microwave background radiation, and from that, they have been able to derive the spatial relationships of points in the universe.
This data allowed us to find the sum of the three angles of a triangle that would answer the question, "What if you actually drew a large triangle in space?"
As a result, the universe is said to be flat within an error of only ±0.4 percent.
A margin of error of ±0.4 percent could make it flatter than your desk.
It's almost perfectly flat.
The fact that the universe is flat may be a bit disappointing for those who were expecting a universe with a curious structure.
But the fact that the universe is flat is even more shocking.
As mentioned earlier, the structure of the universe is determined by the total amount of matter and energy it contains, and for the universe to be flat, these factors must align perfectly.
Physicists are puzzled by how the universe can have such a perfectly Euclidean structure, despite such a low probability.
It's a chilling thought that perhaps some transcendent being created the universe so perfectly.
--- p.84

The minimum number of directions required to express the position of an object is called the dimension of that space.
On a plane, any position can be expressed in two directions (horizontal/vertical).
If you express it as '+2m horizontally, -1m vertically', you will be able to express all locations on the plane.
So the plane is two-dimensional.
Meanwhile, a solid requires three directions (width/length/height).
Therefore, the solid is three-dimensional.
What do 1D and 0D look like? When we say that an object can only move in one direction, we mean that it moves only in a straight line.
That is, the first dimension is a straight line.
Meanwhile, zero dimensions mean that objects cannot move in any direction.
The 0th dimension is a point, meaning that the object is fixed in one place.
--- pp.108~109

From the perspective of three-dimensional creatures like us, we see the treasure (purple square) along with the alarm device (blue square) as shown in the picture above.
However, for a 2D creature, the treasure will be completely hidden by the alarm device and will not be visible.
If you look at this structure from their perspective, all you will see is the blue line that marks the structure's boundary.
You can't help but be unaware that there is a treasure inside.
However, we can look down on objects in a three-dimensional direction (height) that does not exist in two dimensions, so we can check the outside (frame) and inside (treasure inside) of the alarm device at the same time.
Likewise, while we, as three-dimensional creatures, can only see the outside of the alarm device, a four-dimensional creature looking at the next cube would be able to see both the inside and the outside of the cube at once, in a way that we cannot easily imagine.
Furthermore, to a four-dimensional being, our faces and the organs inside our bodies will be visible all at once, and we will be able to see who lives in which house, and even the structure of the Earth at a glance.
A space where you can take out items trapped inside a box, and even a space where you can see both the inside and outside at the same time.
The fourth dimension exudes a strange sense of mystery and stimulates our imagination of the vastness of space beyond our perception.
If you add a new direction to the fourth dimension, you get the fifth dimension, and you can think of it this way up to the sixth and seventh dimensions, but in this book, we will focus on the fourth dimension.
(It's fun enough in 4 dimensions!)
--- pp.113~114

So far, we have focused on algebraic objects, such as the set of natural numbers or the set of integers.
The set of integers is twice as large as the set of natural numbers, but the bases are the same.
The same logic applies to geometric objects.
For example, two spheres have twice as many points as one sphere, but both two spheres and one sphere have points of base ?1.
Then, wouldn't it be possible to cut a single sphere into several pieces and then reassemble them into two? Just as the Hilbert Hotel, which already has all its rooms occupied, offers an infinite number of additional rooms.
Stefan Banach and Alfred Tarski, who pondered this problem, showed that it was indeed possible.
Banach and Tarski's conclusion was so counterintuitive that it was called a "paradox" even though it was a correct theorem.
--- p.174

Have you ever imagined as a child that if you dug deep into the ground, you'd eventually emerge on the other side of the Earth? Later, as you studied Earth science, you'd learn that such a tunnel was impossible, but it's still a fascinating fantasy.
One of the famous problems in physics is the problem of calculating how long it would take for a package to reach the other side of the Earth if it were dropped into a tunnel that passed through the center of the Earth.
Surprisingly, it only takes 42 minutes, regardless of the weight of the package.
If we can build such a tunnel with incredible technological prowess later on, it will be a revolutionary quick service.
Unfortunately, even if the technology to build such a tunnel were developed, it is unlikely that a trans-Earth quick service would be implemented in Seoul.
Because the opposite side of Seoul is the sea.
Argentina and Uruguay are not far away, but they are unfortunately out of sync.
The only place that has the potential to become Korea's first direct quick service location is Jeju Island.
Jeju Island is located in a very advantageous location because the other side is the border between Brazil and Uruguay.
It is said that when a tunnel is dug through the Earth, the two points at each end of the tunnel are in a relationship of antipodes.

--- pp.229~230

Timo, who loves books, got a job as a librarian.
One day, the library purchased 1,000 new books.
Timo was excited about the prospect of having access to more books, but when the 1,000 volumes actually arrived, he realized the problem.
Now Timo has to sort 1,000 books by library code number.
What algorithm would be the fastest way to sort books?
The first algorithm that came to Timo's mind was:
Compare the code numbers of the first and second books in the stack.
Of these, the book with the lower number is left as is, and the book with the higher number is compared with the third book.
Similarly, we leave the lower-numbered books alone and compare the higher-numbered books with the fourth book.
If you continue like this, the book with the highest number will be moved to the very end.
Repeating this process one more time will move the second-largest book to the second-to-last position, and repeating this process a thousand times will sort all the books.
The sorting algorithm that does this is called bubble sort.
--- pp.276~277

Dimen, who had gone on a trip for the first time in a long time, felt tired while driving and decided to drink some coffee.
But I looked around and couldn't find any cafes.
As I wandered around, I suddenly saw a group of cafes that I hadn't seen before.
Starbucks, Coffee Bean, Tous Les Jours, Ediya… all kinds of cafes are gathered together.
Although it is fortunate that they found a cafe, Dimen's mind becomes dissatisfied.
If cafes are evenly distributed throughout the neighborhood, consumers will be able to find them more easily, and businesses will be able to avoid competition.
Cafes aren't the only industry that likes to be crowded together.
Whether it's restaurants, hospitals, real estate, or hotels, businesses prefer to be clustered in one place rather than spread out evenly.
Why is that?
To answer this question, let's imagine a hypothetical town.
In this village, eight consumers live evenly spaced along a straight road.
If Dimen wanted to sell bungeoppang (fish-shaped bread) in this town, where would be a good location? Obviously, he should choose a central location, closest to the eight customers.
Dimen takes his place in the middle and starts selling bungeoppang diligently.
This is an oligopoly where only one company sells the product to everyone.
--- pp.298~299

It is often said that integration is the inverse operation of differentiation.
Just as division is the inverse operation of multiplication, and subtraction is the inverse operation of addition.
It is not a wrong statement, but it is not a desirable explanation when first explaining integration.
Because the definition of integration itself has nothing to do with differentiation.
The definition of division is the inverse operation of multiplication, and the definition of subtraction is the inverse operation of addition.
This is the very definition of division and subtraction.
However, integration is a concept originally conceived in a very different field from differentiation.
But then I realized that differentiation and integration were inverse operations.
The fact that integration is the inverse operation of differentiation is not a definition of integration, but a theorem that has been proven through mathematical proof.
Integration is a concept designed to find the area and volume of a shape.
We can easily find the area of ​​shapes drawn with straight lines, such as triangles or squares.
Even if the number of sides in a shape increases, you can find the total area by dividing it into several triangles and then adding the areas of each triangle.
--- p.344

Edward Norton Lorenz was a mathematician and meteorologist active in the mid-20th century.
Lorenz's interest was in using data to predict the weather.
One day in 1961, Lorenz was running a computer weather simulation using 12 variables, including temperature and humidity.
After obtaining the results, he ran the simulation again with the same initial values ​​(perhaps to check that there were no errors in the simulation results).
However, unexpectedly, the results of the second simulation were very different from the first simulation.
Even though both simulations started with identical weather conditions, after some time the first simulation outputted a sunny day, while the second outputted a cloudy day.
At first I thought it was a computer malfunction.
But no matter how much I looked, the computer was fine.
Only later did Lorenz realize why these results had occurred.
Lorenz used the report from the first simulation to set the initial values ​​for the second simulation.
However, the internal computer calculations of the simulation program consider up to 6 decimal places, but when outputting, only up to 3 decimal places are output.
The value that Lorenz set as the initial value for the second simulation was 0.506, which was calculated as 0.506127 in the first simulation.
The difference between the two initial values ​​was only 1/4000, but this error grew into a very large difference over time.
Lorenz named this phenomenon, where very small errors develop into large differences, chaos.
--- pp.361~362

Free will is a wonderful optical illusion.
Because so many external factors influence human consciousness (sensory information including sight and smell, the resulting electrical activity of neurons and chemical reactions of hormones in the body, the base pairs written in DNA and the resulting genetic traits, etc.), we simply mistakenly believe that our own decisions come from our own free will.
Even if you choose what to have for dinner tonight by flipping a coin to escape the shackles of determinism, the very idea that you will choose what to have for dinner by flipping a coin is already a decision.
Essentially, we are just protein balls rolling around in a pinball machine called the universe.
On the one hand, this conclusion seems to express a sense of helplessness about life.
So many people try to ignore this fact and live their lives.
But if we pause for a moment and slowly reflect on this fact, we can gain a new perspective on life.
I believe that these values, just as much as traditional values, make life even more beautiful.

--- p.377