The flow and core of middle and high school mathematics in one book!
A math guide that makes the world more interesting just by knowing it.

As data science and AI advance, the number of people wanting to learn mathematics is increasing, but it's practically impossible to study the vast scope of mathematics from scratch.
This book provides a brief overview of important mathematical terms and includes various illustrations to aid understanding, allowing those who are curious to learn in a short amount of time.
You don't have to read from the first page. You can open it and read from the part that interests you, like a dictionary.
Let's understand and apply fundamental mathematical concepts in practice through this book!

It is known that the name perfect number was given by Pythagoras, who said that 'the origin of all things is number'.
The perfect numbers discovered in ancient Greece are 6, 28, 496, and 8128.
All of the perfect numbers given as examples so far are even numbers.
It is not yet known whether there are odd perfect numbers.
It is also not clear whether the number of perfect numbers is infinite or finite.
Although the definition of a perfect number is simple, there are still many unresolved problems related to perfect numbers.
--- 「Chapter 1 '05.
From "Perfect Number"

After middle school math, the need to find specific numbers decreases because letters are used more frequently, but it is very important to calculate the final equation to find specific numbers.
That is why we rationalize the denominator to find a specific number.
Conversely, if you don't need to know the specific number, you don't need to rationalize the denominator.
When teaching students, they sometimes ask, “Why don’t you rationalize the denominator this time?” Remember that when solving problems that don’t require identifying specific numbers, you don’t need to rationalize the denominator.
--- 「Chapter 2 '02.
From "The Rationalization of the Denominator"

Something that is accepted as true regardless of the reason is called an axiom.
It can be understood that a theory is accepted as true even without the assumptions that form the theory's premise or without proof.
For example, 'No matter how large the natural number n is, there exists a natural number next to it, n+1', is also an axiom.
--- 「Chapter 3 '01.
From "Definition, Theorem, Formula, Proposition"

The meaning of the formula for the roots of a quadratic equation is that it guarantees that 'a quadratic equation must have a solution and that its value can be specifically found', including imaginary numbers that will be discussed later.
In general, in mathematics, you cannot tell whether a problem is solvable or not.
However, since there is a formula for the roots of a quadratic equation, we can see that it can definitely be solved.
When you study mathematics in college, you are introduced to the concepts of 'existence' and 'uniqueness'.
In other words, what is important is whether the solution exists (existence) and whether there is only one solution obtained regardless of the method used (uniqueness).
The ‘root formula of a quadratic equation’ contains both the important concepts of ‘existence’ and ‘uniqueness’.
--- 「Chapter 4 '06.
From "Root formula, discriminant, and conjugate"

It was thanks to a 'bug' that Descartes was able to come up with the concept of a coordinate plane.
One day, Descartes woke up and saw a bug on the ceiling. He wondered how he could properly convey the bug's location to his friend.
And I thought the other person would understand if I said that there was a bug at the 4th position to the right of the corner and 3rd position above.
--- 「Chapter 5 '01.
From "Coordinate Plane (Cartesian Plane)"

Deductive reasoning is a method of deriving a conclusion by using premises or general laws and specific facts.
A representative example is the syllogism.
For example, let's say the premise or general law is 'Humans (B) will die someday (C).'
If the concrete fact is 'Socrates (A) is human (B),' then the conclusion is 'Socrates (A) will die someday (C).'
--- 「Chapter 7 '07.
From "Deductive and Inductive Methods"

When you make a bundle by 'picking out' several objects without considering the order, each of them is called a combination.
Because the order is not taken into account, for example, the cases of picking 'A and B' and picking 'B and A' are considered the same.
In other words, 'People who like sushi are A and B' and 'People who like sushi are B and A' have the same meaning.
--- 「Chapter 8 '04.
Among the permutations and combinations (C) that include the same thing,

The importance of data is increasing day by day these days, and statistics is what helps us use data appropriately.
For example, it is difficult to understand the characteristics of numerical data, such as test scores, simply by looking at them.
So, with the given data, we can get the highest score, average score, and deviation value as needed, and make meaningful numbers to determine whether the test is passed or failed.
In other words, statistics can be seen as the process of turning simple data into valuable and meaningful information.
--- 「Chapter 9 '01.
From "Descriptive Statistics and Inferential Statistics"

The basis of differentiation is division, and the basis of division is subtraction.
That is why differentiation is useful when finding the ‘difference’.
In high school mathematics, differentiation is used to find increases and decreases and to draw graphs. This is possible because differentiation allows us to find differences.
--- 「Chapter 10 '02.
Among the average rate of change, instantaneous rate of change, differential coefficient, and derivative

In the past, trigonometric problems using inscribed quadrilaterals were frequently asked in Japanese university entrance exams.
Inscribed quadrilaterals can be used as a basic problem as well as an applied problem using auxiliary lines, so they would be an appropriate material for a test that measures a wide range of learning abilities.
However, learning the technique of utilizing auxiliary lines is not easy.
Moreover, it will feel even more difficult in situations where it is difficult to demonstrate one's usual abilities due to the tension, such as during an exam.
In such cases, we tend to resort to special tricks or techniques, and there is one theorem that is useful for applying to problems involving inscribed quadrilaterals.
This is Ptolemy's theorem.

--- 「Chapter 12 '06.
From "Ptolemy's Theorem"