Lesson 4 Real Numbers and Mappings - Kuina-chan Mathematics

March 20, 2026

Kuina-chan

Kuina-chan Math Lesson 4 explains functions and mappings that connect numbers! It is assumed you have read Lesson 1.

Lesson 3 defined integers “ $\mathbb{Z}$ ”. This time, we will explain rational numbers “ $\mathbb{Q}$ ” and real numbers “ $\mathbb{R}$ ”, which are so-called decimals, as well as functions and mappings.

1.Rational Numbers and Real Numbers

So far, we have dealt with integers, but from now on we will deal with more detailed “rational numbers” and “real numbers”. These are what we call “decimals”.

1.1Rational Numbers

A number that can be expressed as a fraction of “integerinteger” where the denominator is other than

is called a “rational number”. For example, “

”, “

”, and “

” are rational numbers. The decimal “

$\dots$ ” is also a rational number because it can be expressed as the fraction “

”.

At this time, we denote the set of all rational numbers as “ $\mathbb{Q}$ ”. That is, “ $\mathbb{Q}$

$\{$

$\dots$

$\dots$ $\}$ ”.

1.2Conversion from Decimal to Fraction

By the way, decimals where digits repeat like “

$\dots$ ” can always be converted into “integer

integer” fractions, so they are rational numbers. “

” is also a rational number because it repeats as “

$\dots$ ”.

The method for converting a repeating decimal like “

$\dots$ ” into a fraction is as follows.

Any repeating decimal can be converted to a fraction using this method.

1.3Irrational Numbers

On the other hand, non-repeating decimals are called “irrational numbers”. Irrational numbers cannot be expressed as “integer

integer” fractions. Examples of irrational numbers include Pi “

$\dots$ ” and the number that becomes

when squared, “

$\dots$ ”.

Rational numbers and irrational numbers together are called “real numbers”. We denote the set of all real numbers as “ $\mathbb{R}$ ”.

Supplement

In this definition of “real numbers”, we used the vague term “decimal”, but it can be defined more rigorously. There are several ways to define it, but simply put, if you line up infinitely many rational numbers, they may approach something arbitrarily close. That number may not be a rational number, and we define that as an irrational number. Rational numbers and irrational numbers together are real numbers.

Now, all natural numbers are included in integers. Also, all integers are included in rational numbers. Therefore, the inclusion relationship of the numbers introduced so far is as follows.

\mathbb{N} — Inclusion Relationship of Main Numbers

\subset — Inclusion Relationship of Main Numbers

1.4Main Operations

For rational numbers and real numbers, just like integers, addition “

”, subtraction “

”, multiplication “

$\cdot$

”, exponentiation “

$^{b}$ ”, and absolute value “

” are defined for two numbers

. Also, for

not equal to

, division “

” is also defined. However, if

, for example “

”, it is undefined.

Furthermore, “square root” is defined for real numbers. The “square root of ” is that satisfies “ $^{2}$ ”. For example, the “square roots of

” are the two numbers “

” because “

$^{2}$ ” and “

$^{2}$ ” hold true. Similarly, the “square roots of

” are “

”.

Of the square roots, the one that is

or greater is called the “positive square root” and is represented by the symbol “ $\sqrt{x}$ ”. That is, “ $\sqrt{9}$

” and “ $\sqrt{4}$

”.

Extending this, the value of satisfying “ $^{n}$ ” is called the “-th root of ”. And the

-th root of

(where

or greater) is expressed as “ $\sqrt[n]{x}$ ”. For example, since “

$^{4}$ ”, “ $\sqrt[4]{16}$

”.

Here are some values of positive square roots.

\sqrt{0} — Values of Positive Square Roots

Positive Square Root
$\sqrt{0}$
$\sqrt{1}$
$\sqrt{2}$ $\dots$
$\sqrt{3}$ $\dots$
$\sqrt{4}$
$\sqrt{5}$ $\dots$
$\sqrt{6}$ $\dots$
$\sqrt{7}$ $\dots$
$\sqrt{8}$ $\dots$
$\sqrt{9}$

If we graph the positive square root

$\sqrt{x}$ , it looks like the figure below. If

is less than

, there is no real number that becomes

when squared, so $\sqrt{x}$ is undefined.

By the way, $\sqrt{2}$ is an irrational number. It is simple, so let’s prove it.

In this way, the proof method of “assuming $\neg$

holds to intentionally derive a contradiction, and proving

by elimination” is called “proof by contradiction”.

2.High-Degree Equations

2.1Linear Equations

Let’s try solving real number equations. First, here is a simple problem below.

An equation of the form “ (where

$\neq$

)” is called a “linear equation”. A linear equation can be easily solved simply by adding or multiplying the same number to both sides.

2.2Quadratic Equations

Next is a slightly more complex problem below.

An equation of the form “ $^{2}$ (where

$\neq$

)” is called a “quadratic equation”. A quadratic equation can be easily solved if it can be transformed into the form “”, so we aim for this form.

First, expand the left side of the equation “

”. There is a rule “”, so by repeatedly applying this to remove the parentheses, we can transform it to “

$^{2}$

”. We got closer to the equation in the problem.

Comparing this “ $^{2}$ ” with the problem equation “ $^{2}$ ”, we can see that if we fit

into “

” and

into “

”, we can make the same equation. If we think of appropriate

such that “

” and “

”, we find that “

”. Fitting them in, we get “

$^{2}$

$\cdot$

”.

So, from the results so far, we found that the problem equation “

$^{2}$

” can be transformed into “

$^{2}$

$\cdot$

”, and this can be transformed into “

”. In other words, instead of the problem equation, let’s find

that satisfies “

”. This means that multiplying “

” and “

” gives

, so at least one of them must be .

Considering the case where “

” is

, we find

. Considering the case where “

” is

, we find

. Both do not become

simultaneously. Therefore, these are all the solutions. That is,

satisfying

$^{2}$

is .

2.3Quadratic Formula

By the way, the solution to a quadratic equation can also be solved by the following formula called the “quadratic formula”.

Indeed, we obtained the same solutions as before.

3.Mappings

Finally, we will solve functions and mappings.

A “mapping” is something that corresponds every element of a certain set to an element of a certain set, and is sometimes called a “function”. In the figure below, what corresponds to the collection of “arrows” connecting elements is the mapping.

A mapping

associating an element of set $\bm{A}$ with an element of set $\bm{B}$ is expressed as “ $\bm{A}$ $\rightarrow$ $\bm{B}$ ”. Also at this time, the element of set $\bm{B}$ corresponding to the element

of set $\bm{A}$ is expressed as “”. For example, in this figure, since element

$_{2}$ is associated with element

$_{1}$ , it becomes

$_{1}$

$_{2}$ .

When “

$\bm{A}$ $\rightarrow$ $\bm{B}$ ”, for any element

of set $\bm{A}$ , there exists exactly one corresponding element

in set $\bm{B}$ . It is never the case that there is no destination or multiple destinations.

Also, a mapping can associate between the same set. In other words, it can be “

$\bm{A}$ $\rightarrow$ $\bm{A}$ ”.

For example, for the set of all natural numbers $\mathbb{N}$ , “

” which doubles the element

of $\mathbb{N}$ becomes the mapping “

$\mathbb{N}$ $\rightarrow$ $\mathbb{N}$ ”.

3.1Surjection, Injection, Bijection

In “

$\bm{A}$ $\rightarrow$ $\bm{B}$ ”, if the elements of $\bm{A}$ correspond to all elements of $\bm{B}$ without leaving any out,

is called a “surjection”. Strictly speaking, when the collection of

for all elements

of the set $\bm{A}$ matches the set $\bm{B}$ ,

is a surjection.

Also, when the elements of $\bm{B}$ corresponding to each element of $\bm{A}$ have no duplicates,

is called an “injection”. Strictly speaking, for any two different elements

of $\bm{A}$ , if

and

are always different,

is an injection.

When the mapping

is both surjective and injective,

is called a “bijection”. At this time, the elements of $\bm{A}$ and the elements of $\bm{B}$ correspond exactly one-to-one.

The images of surjection, injection, and bijection are summarized in the figure below.

3.2Inverse Mapping

The mapping with the direction of correspondence reversed of mapping

is called the “inverse mapping” of

and is expressed as “ $^{-}$ $^{1}$ ”. Strictly speaking, although it is complicated to explain, when two mappings “

$\bm{A}$ $\rightarrow$ $\bm{B}$ ” and “

$\bm{B}$ $\rightarrow$ $\bm{A}$ ” always satisfy “

” for any element

of $\bm{A}$ and always satisfy “

” for any element

of $\bm{B}$ ,

is the inverse mapping of

“

$^{-}$ $^{1}$ ”.

For example, if we consider the mapping “

$\mathbb{Z}$ $\rightarrow$ $\mathbb{Z}$ ” defined by “

” and the mapping “

$\mathbb{Z}$ $\rightarrow$ $\mathbb{Z}$ ” defined by “

”, since the direction of correspondence is reversed, we can say that

is the inverse mapping of

“

$^{-}$ $^{1}$ ”.

By the way, if the mapping

is not a bijection, the inverse mapping does not exist for

. Also, if

is a bijection, the inverse mapping

$^{-}$ $^{1}$ of

always exists, and there is only one kind.

We explained real numbers and mappings this time. Next time, we will explain various figures such as triangles and circles!

Lesson 4Real Numbers and Mappings

1.Rational Numbers and Real Numbers

1.1Rational Numbers

1.2Conversion from Decimal to Fraction

1.3Irrational Numbers

1.4Main Operations

2.High-Degree Equations

2.1Linear Equations

2.2Quadratic Equations

2.3Quadratic Formula

3.Mappings

3.1Surjection, Injection, Bijection

3.2Inverse Mapping

Lesson 4
Real Numbers and Mappings