Numerical Sets: Natural Numbers, Integers, Rational Numbers, Irrational Numbers, and Real Numbers

Natural numbers are whole and positive numbers.
For a certain number to be considered a "natural number" it must be whole and positive, therefore, what is questioned to find out is the following:
"Is the number whole and positive?"
If the answer is positive: it is a natural number
If the answer is negative: it is a number that is not natural

Important note about zero: Whether $0$ is considered a natural number depends on the mathematical context. In some definitions, natural numbers start with $1$ $(\mathbb{N} = {1, 2, 3, 4, ...})$, while others include $0$ $(\mathbb{N}_0 = {0, 1, 2, 3, 4, ...})$. For elementary mathematics, we typically start with $1$.

For example:
Is the number $65$ a natural number?
Yes, it is whole and also positive, that means it is natural.

Is $-7$ a natural number?
No, it is negative.

Is $3.5$ a natural number?
No, it is not whole (it has a decimal part).

Is $0$ a natural number?
Depends on the definition - check your textbook or teacher's preference.

You can read again about natural numbers by clicking here.

Integers are those numbers that are not fractions. They can be positive or negative.
For a certain number to be considered an "integer" it cannot be a fraction or decimal.
Therefore, what is questioned to find out is the following:
"Is the number a fraction or decimal?"
If the answer is positive: it is not an integer.
If the answer is negative: it is an integer.

It is useful to know that $0$ is considered an integer.

The set of integers is represented as: $\mathbb{Z} = {..., -3, -2, -1, 0, 1, 2, 3, ...}$

For example:
Is the number $3.2$ an integer?
No, to be an integer it cannot be decimal.

Is $7$ an integer?
Yes, it is a whole number.

Is $-15$ an integer?
Yes, it is a whole number (negative integers are still integers).

Is \(\frac{4}{2}\) an integer?
Yes, because $\frac{4}{2} = 2$, which is a whole number.

Is $\frac{7}{3}$ an integer?
No, because $\frac{7}{3} = 2.333...$, which is not whole.

You can learn more about integers by clicking here.

Rational numbers, positive or negative, are those that can be represented as a fraction - numerator and denominator $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. They can be positive, negative, or zero.
With this understanding, we know that:

• Every integer is rational since every integer can also be represented as a fraction, and this makes it a rational number (for example: $5 = \frac{5}{1}), (-3 = \frac{-3}{1}$).
• Some decimal numbers are rational - specifically those that either terminate (end) or repeat in a pattern:
  • Terminating decimals: $0.5 = \frac{1}{2}), (0.75 = \frac{3}{4}$
  • Repeating decimals: $0.333... = \frac{1}{3}), (0.666... = \frac{2}{3}$

Important: Not all decimals are rational! Non-repeating infinite decimals (like $\pi = 3.14159...$) are irrational.

Therefore, the question to determine if a number is rational or not, is the following:
"Can this number be expressed as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$?"

If yes: it is a rational number
If no: it is not a rational number

The set of rational numbers is denoted by $\mathbb{Q}$.

For example:
Is the number $4$ rational?
Yes. It can be represented as a fraction.

Is $\frac{-7}{3}$ rational?
Yes, it's already in the required form

Is $0.25$ rational?
Yes, it equals $\frac{1}{4}$

Is $\sqrt{2} = 1.41421...$ rational?
No, it's a non-repeating infinite decimal

You can read again about rational numbers by clicking here.

Irrational numbers are those that cannot be represented as a fraction of two integers - numerator and denominator $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$ .
How can we recognize an irrational number?
If we have a decimal number with infinite digits to the right of the decimal point and these do not appear repetitive in a certain order (periodic), the number is irrational.

It is useful to know that $\sqrt 2$ is irrational.

For example:

$6.52495......$ is irrational.

$\pi = 3.14159265...$ - pi - the ratio of circumference to diameter, is irrational.

$e = 2.71828182...$ - Euler's number, is irrational.

$\sqrt{3} = 1.73205080...$ - square root of 3, is irrational.

$\phi = 1.61803398...$ - golden ratio, is irrational.

You can learn more about irrational numbers by clicking here.

Real numbers include all rational and irrational numbers - essentially, any number that can be represented on the number line. They represent any measurable quantity, whether positive, negative, or zero.

• A real number can be positive, negative, or zero.
• A real number can be decimal (both terminating and non-terminating).
• A real number can be a fraction.
• Include all irrational numbers like $\pi$, $\sqrt{2}$, and $e$ .

In fact, real numbers make up the largest numerical set since they include all others: $[\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}]$

This means:

• All natural numbers are integers
• All integers are rational numbers
• All rational numbers are real numbers
• All irrational numbers are also real numbers
• It is worth knowing that $0$ is also a real number.

Real numbers are denoted by the symbol $\mathbb{R}$.

Examples of real numbers:

• $5$ (natural, integer, rational, real)
• $-3$ (integer, rational, real)
• $\frac{2}{7}$ (rational, real)
• $0.25$ (rational, real)
• $\pi = 3.14159...$ (irrational, real)
• $\sqrt{5} = 2.23606...$ (irrational, real)
• $0$ (integer, rational, real)

You can read again about real numbers by clicking here.

Analyze each number in the following group and define if it is natural, integer, rational, irrational, or real.
Keep in mind that numbers can belong to several categories.

Given numbers: .......2.867525895,12,−3, \frac{2}{3

Solution:

To avoid mistakes when determining, it is convenient that we briefly write down the requirements for each numerical set:
Natural number –> positive whole number
Integer number –> whole numbers
Rational number –> can be expressed as $\frac{a}{b}$ where $a, b$ are integers and $b \neq 0$
Irrational number – >cannot be expressed as a fraction of integers
Real number –> any number on the number line (includes all rational and irrational numbers)

Now we will review each number and see how many sets each one corresponds to:

\frac{2}{3
Rational and real.

• It's already expressed as a fraction of two integers
• Not natural (it's less than 1) or integer (it's not whole)

$-3$
Integer, rational, and real.

• It's a whole number (integer)
• Can be written as $\frac{-3}{1}$ (rational)
• Not natural (it's negative)

$12$
Natural, integer, rational, and real.

• It's a positive whole number (natural and integer)
• Can be written as $\frac{12}{1}$ (rational)
• Belongs to all categories except irrational

$2.867525895......$
Irrational and real.

• The decimal continues infinitely without repeating
• Cannot be expressed as a fraction of integers
• Not natural, integer, or rational

If you are interested in this article, you may also be interested in the following articles:

Angle Notation

Graphical Representation of a Function

Algebraic Representation of a Function

Domain of a Function

Indefinite Integral

Numerical Value Assignment in a Function

Variation of a Function

Increasing Function

Decreasing Function

Constant Function

Intervals of Increase and Decrease of a Function

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