Episode 4 Real Numbers and Mappings - Kuina-chan Mathematics

September 16, 2025

Kuina-chan

In Episode 4 of Kuina-chan Mathematics, I explain functions and mappings that connect numbers to other numbers! This episode assumes you’ve read the series in order, beginning with Episode 1.

In Episode 3, I defined the set of integers $\mathbb{Z}$ . This time, I explain rational numbers $\mathbb{Q}$ , real numbers $\mathbb{R}$ , and the concepts of functions and mappings.

1.Rational and Real Numbers

So far, we’ve dealt with integers, but now let’s explore more detailed numbers: rational numbers and real numbers. These are what we usually call “decimals.”

1.1Rational Numbers

A rational number is a number that can be expressed as a fraction of integer over integer, where the denominator is not zero. For example,

, and

are rational numbers. Even a decimal like

$\dots$ is rational because it can be written as the fraction

We’ll denote the set of all rational numbers as $\mathbb{Q}$ . That is, $\mathbb{Q}$

$\{$

$\dots$

$\dots$ $\}$ .

1.2Converting Decimals to Fractions

By the way, repeating decimals like

$\dots$ can always be converted into a fraction of integer over integer, making them rational numbers. Even

is a repeating decimal (

$\dots$ ), so it’s rational.

Here’s how to convert a repeating decimal like

$\dots$ into a fraction:

Any repeating decimal can be converted into 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 a fraction of integer over integer. Examples include the number

$\dots$ (pi) and the number whose square is 2,

$\dots$ .

Combining rational and irrational numbers gives us real numbers. We’ll denote the set of all real numbers as $\mathbb{R}$ .

Note

In this definition of real numbers, I used the vague idea of “decimals,” but there are more rigorous ways to define them. One such method is to consider that when you line up rational numbers densely, they can get arbitrarily close to some number. That number might not be rational, and we define such numbers as irrational. Rational and irrational numbers together form the real numbers.

All natural numbers are included in the integers. All integers are included in the rational numbers. Therefore, the inclusion relationships among the types of numbers we’ve introduced so far are as follows:

\mathbb{N} — Inclusion Relationships Among Major Number Sets

\subset — Inclusion Relationships Among Major Number Sets

1.4Basic Operations

Rational and real numbers support the same operations as integers. For any two numbers

and

, we can define addition

, subtraction

, multiplication

$\cdot$

, exponentiation

$^{b}$ , and absolute value

. Also, for

$\neq$

, division

is defined. However, if

, such as

, the expression is undefined.

Additionally, real numbers allow us to define square roots. The square root of is a number

such that $^{2}$ . For example, the square roots of

are

and

, since

$^{2}$ and

$^{2}$ . Similarly, the square roots of

are

and

The positive square root is called the principal square root and is denoted by $\sqrt{x}$ . So $\sqrt{9}$

and $\sqrt{4}$

We can extend this idea: a number

that satisfies $^{n}$ is called the th root of . The positive

th root of

is denoted by $\sqrt[n]{x}$ . For example, since

$^{4}$ , we have $\sqrt[4]{16}$

Here are some values of principal square roots:

Principal 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 principal square root

$\sqrt{x}$ , it looks like the image below. When

is less than 0, there is no real number whose square is

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

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

This method of assuming $\neg$

and deriving a contradiction to prove

is called proof by contradiction.

2.Polynomial Equations

2.1Linear Equations

Now let’s try solving equations with real numbers. Here’s a simple problem:

An equation of the form (where

$\neq$

) is called a linear equation. You can solve linear equations easily by adding or multiplying both sides by the same number.

2.2Quadratic Equations

Next, let’s try a slightly more complex problem:

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

$\neq$

) is called a quadratic equation. If you can transform it into the form , it’s easy to solve. So let’s aim for that form.

First, expand the left-hand side of

. Using the rule , we apply it repeatedly to remove the parentheses:

$^{2}$

Now compare this with the problem equation

$^{2}$

. We want

and

. Trying some values, we find

and

work. Substituting gives:

$^{2}$

So the original equation

$^{2}$

can be transformed into

. This means we need to find

such that

. That is, at least one of the factors must be zero.

, then

. If

, then

. Both can’t be zero at the same time. So these are all the solutions. Therefore, the solutions to

$^{2}$

are .

2.3Quadratic Formula

By the way, you can also solve quadratic equations using the quadratic formula shown below:

Indeed, this gives the same solutions as before.

3.Mappings

Finally, let me explain functions and mappings.

A mapping is a way to assign each element of one set to an element of another set. It’s also called a function. In the diagram below, the arrows connecting elements represent a mapping.

If a mapping

assigns elements of set $\bm{A}$ to elements of set $\bm{B}$ , we write it as $\bm{A}$ $\rightarrow$ $\bm{B}$ . The element of $\bm{B}$ that corresponds to an element

of $\bm{A}$ is written as . For example, in the diagram, element

$_{1}$ corresponds to

$_{2}$ , so

$_{1}$

$_{2}$ .

$\bm{A}$ $\rightarrow$ $\bm{B}$ , for every element

in $\bm{A}$ , there is exactly one corresponding element

in $\bm{B}$ . There are no missing or multiple correspondences.

Mappings can also be defined between the same set. That is,

$\bm{A}$ $\rightarrow$ $\bm{A}$ is allowed.

For example, consider the mapping

on the set of natural numbers $\mathbb{N}$ . This is a mapping

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

3.1Surjective, Injective, Bijective

$\bm{A}$ $\rightarrow$ $\bm{B}$ , if the elements of $\bm{A}$ cover all elements of $\bm{B}$ without omission, then

is called surjective. More precisely, if the set of all

for

in $\bm{A}$ equals $\bm{B}$ , then

is surjective.

If each element of $\bm{A}$ maps to a unique element in $\bm{B}$ without duplication, then

is called injective. More precisely, for any two distinct elements

and

in $\bm{A}$ , if

$\neq$

, then

is injective.

If a mapping

is both surjective and injective, then it’s called bijective. In this case, the elements of $\bm{A}$ and $\bm{B}$ correspond one-to-one.

Here’s a diagram showing the concepts of surjective, injective, and bijective mappings:

3.2Inverse Mapping

The inverse mapping of a mapping

is a mapping that reverses the direction of correspondence. It’s denoted as $^{-}$ $^{1}$ . To explain it precisely: if we have mappings

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

$\bm{B}$ $\rightarrow$ $\bm{A}$ , and for every

in $\bm{A}$ ,

, and for every

in $\bm{B}$ ,

, then

is the inverse mapping of

, written as

$^{-}$ $^{1}$ .

For example, consider the mapping

on integers $\mathbb{Z}$ , and the mapping

on $\mathbb{Z}$ . Since the direction of correspondence is reversed,

is the inverse mapping of

, i.e.,

$^{-}$ $^{1}$ .

By the way, if a mapping

is not bijective, then it has no inverse mapping. If

is bijective, then it always has a unique inverse mapping

$^{-}$ $^{1}$ .

In this article, I explained real numbers and mappings. Next time, I’ll explain various geometric shapes like triangles and circles!

Episode 4Real Numbers and Mappings

1.Rational and Real Numbers

1.1Rational Numbers

1.2Converting Decimals to Fractions

1.3Irrational Numbers

1.4Basic Operations

2.Polynomial Equations

2.1Linear Equations

2.2Quadratic Equations

2.3Quadratic Formula

3.Mappings

3.1Surjective, Injective, Bijective

3.2Inverse Mapping

Episode 4
Real Numbers and Mappings