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

It is common to distribute numbers into different categories:

Natural numbers - Whole and positive numbers. These are the ones we use daily to count or number elements, for example $2,10,17,100$, etc.

• Integers - Contain whole numbers (no fractions), both positive and negative, and zero.
• Rational numbers - Are numbers that can be represented as the quotient (result of division) of two integers.
• Irrational numbers - Are numbers that cannot be represented as the quotient (result of division) of two integers.
• Real numbers - Are numbers that represent a specific size, whether positive or negative.

## Number Sets: Natural Numbers, Integers, Rational Numbers, Irrational Numbers, and Real Numbers

### Natural 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

It is useful to know that $0$ is considered a natural number.

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

### Whole Numbers

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.

For example:
Is the number $3.2$ an integer?
No, to be an integer it cannot be decimal, it must be whole like, for example, the $7$.

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### Rational Numbers

Rational numbers, positive or negative, are those that can be represented as a fraction - numerator and denominator.
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.
• A decimal number is also rational
since every decimal number can also be represented as a fraction, and if it can be written as a fraction, it is rational.

Therefore, the question to determine if a number is rational or not, is the following:
"Can this number be represented as a fraction of two integers, numerator and denominator?"
If the answer is positive: it is a rational number.
If the answer is negative: it is not a rational number.
For example:
Is the number $4$ rational?
Yes. It can be represented as a fraction.

## Irrational Numbers

Irrational numbers are those that cannot be represented as a fraction of two integers - numerator and denominator.
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 $2$ is irrational.

For example:

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

### Real Numbers

A real number is any that represents a certain size and is found on the number line.

• A real number can be decimal
• A real number can be both positive and negative
• A real number can be a fraction
• Real numbers are denoted by the letter (R)

In fact, real numbers make up the largest numerical set since they include all others: natural, integers, rationals, and irrationals.
It is worth knowing that $0$ is also a real number.

Let's look at some examples:

• $-3$ is a real, rational, and integer number
• $½$ is a real and rational number
• $4+$ is a real, rational, integer, and natural number

### Complete Exercise for Advanced Level:

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.

$.......2.867525895,12,−3,23$

Solution:

To avoid mistakes when determining, it is convenient that we briefly write down the requirements for each numerical set:
Natural number –>positive and integer
Integer number –> only integer
Rational number –> can be represented as a fraction
Irrational number – >cannot be represented as a fraction
Real number –> any number on the number line

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

$23$
Rational and real.

$-3$
Integer, rational, and real.

$1212$
Natural, integer, rational, and real.

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