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

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.

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.

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 $7$.

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.
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.

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.
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.

You can learn more about irrational numbers by clicking here.

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.

You can read again about real numbers by clicking here.

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

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.

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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