## 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.

You can read again about natural numbers by clicking here.

### 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 $7$.

You can learn more about integers by clicking here.

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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.

You can read again about rational numbers by clicking here.

## 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.

You can learn more about irrational numbers by clicking here.

### 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.

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

### 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.

**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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