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**Rational Numbers|Definition & Meaning**

**Definition**

A number is **rational** if it is possible to represent it as a** fraction** (equivalently, a ratio or division) of **two integers**. For example, we can write 2 as 4/2 or 2/1, so 2 is a rational number. Any number that cannot be represented in this form is called an irrational number (e.g., pi and Euler’s number e).

Figure 1: Diagram showing rational numbers as a set of N, W and Z.

The set of **rational** numbers also contains all of the **integers**, which can each be expressed as a **ratio** of two numbers where the **numerator** can be an **integer **and the **denominator** can be equal to 1. **Rational** numbers can be **repeating** or **terminating** decimals in **decimal** form.

The word “**ratio**” is where the word “**rational**” was first derived from. **Rational** numbers are therefore closely tied to the idea of **fractions**, which stand for ratios.

In other terms, a **number** would be a **rational** number if it can be written as a **fraction** in which the **numerator** and **denominator** are both **integers**.

All **natural**, **whole**, and **integer **numbers are included in the set of **rational **numbers. The set of **rational **numbers is denoted by **Q**.

**Identification of Rational Numbers**

The following attributes make it simple to recognize rational numbers:

**Rational**numbers include all**integers**,**whole**numbers,**natural**numbers, and**fractions**containing integers.- We can tell that a
**number**is**rational**if its**decimal**form is**terminating**or**recurring**, as in the cases of 7.5 and 4.272727. **Irrational**numbers are those in which it appears that the**decimals**never finish or don’t repeat. Consider the number 5, which is**irrational**and**equal**to 2.2360679774997896409173.- Checking if a
**number**can be represented as the**ratio**of**p/q**, where**p**and**q**belong to the set of**integers**and q must not be equal to**0**, is another method for determining whether or not a**number**is**rational**.

**Terminating Decimal Numbers Are Rational**

A **rational** number is any **decimal** that **terminates** or concludes at some point.

Consider the **decimal** value 0.5 as an illustration. This is a **rational** number since it may be changed to 1/2.

It is completely possible to **transform** even lengthier **terminating** decimal **integers** into **fractions**.

For instance, 0.0001 is a **rational **number since it can be written as 1/10,000.

A **decimal **number is, therefore, **rational** if its final value is reached without **approximation **or **rounding**.

**Non-terminating Decimal Numbers With Indefinite Repetition Are Decimal**

**Rational **numbers are those **numbers **that never end and have **patterns**. Since the pattern must continue forever, this is a little problematic.

Take the number **0.33333**, for instance. Despite the fact that this is frequently abbreviated as 0.33, the sequence of 3s that follow the **decimal **point repeats **indefinitely**. This indicates that the number is **rational **and can be expressed as the **fraction **1/3.

Consider using a more challenging **number**, such as 0.142857142857. Again, the pattern of 142857 that follows the **decimal **point repeats **indefinitely**, and the **number **can be expressed as the **rational **fraction 1/7.

There are **decimal **numbers, though, that never end and don’t have **recurring **patterns. They are not **rational **numbers.

**Properties of Rational Numbers**

A **rational **number will adhere to all the characteristics of the **real number **system because it is a **subset **of the **real **number. The following are some crucial characteristics of **rational **numbers:

- If we
**multiply**,**add**, or**subtract**any two**rational**numbers, the outcome is always a**rational**number. - If we use the same factor to
**divide**or**multiply**the**numerator**and**denominator**, the answer is still a**rational**number. - A
**rational**number itself is what is obtained if we add**zero**to it. - The operations of
**addition**,**subtraction**, and**multiplication**on**rational**numbers are**closed**. **Commutative addition**and**multiplication**apply to**rational**numbers.

### Commutative Property

According to the **commutative **law of **addition:**

A + B = B + A

According to the **commutative **property of **multiplication:**

AB = BA

### Associative Property

The **associative **property is followed by **rational **numbers while **adding **and **multiplying**. Assuming X, Y, and Z are **rational, **for **addition**:

X + (Y + Z) = (X + Y) + Z

In the case of **multiplication**, we have:

X (YZ) = (XY) Z

### Distributive Property

If P, Q, and R are three **rational **integers, according to the **distributive **property, then:

P (Q + R) = (PQ) + (PR)

We can apply it in the same way for **multiplication over subtraction**. However, remember that **division is not distributive** over the set of rational numbers (consider the case where the denominator becomes zero).

**Irrational Numbers**

**Irrational **numbers are any **numbers **that do not fit the criteria of a **rational **number. A **decimal **can be used to represent an **irrational **number even though it cannot be expressed as a simple **fraction**. After the **decimal **point, there are an **infinite **number of **non-repetitive **digits.

Many **square **roots (√2 = 1.41421356237…) and **pi **(π = 3.1415926536…) have digits that extend **indefinitely **past the **decimal **point. However, as they lack **infinitely repeating patterns**, they are regarded as **irrational**.

A **separate **set, the set of **irrational **numbers, does not at all constitute any of the remaining **sets **of numbers. **Q’** represents the set of **irrational **numbers.

**Solved Examples Using the Definition of Rational Numbers**

**Example 1**

Assess whether 8/10, 70/1408, 25, and √7 are **rational **or **irrational**.

**Solution**

Since a **ratio **can be employed to express a **rational **number, which shows that it may be written as a **fraction **with a **whole number **as both the **denominator **and the **numerator**.

- Due to it being stated as a
**fraction**, 8/10 is a**rational**number. Since 8/10 = 0.8. - 70/1408 is a
**fraction**and hence a**rational**number. - As well as being
**rational**, 25 can also be expressed as 25/1. Once more, a**rational**number. - √7 has a value of 2.645751311….. Because it is
**non-terminating**, it cannot be expressed as a**fraction**. Therefore, the number is**irrational**.

**Example 2**

Find three **rational **numbers between 2/5 and 2/3.

**Solution**

Let’s **multiply **2/5 by 3/3 and 2/3 by 5/5. Hence, we get:

6/15 and 10/15

Since there is a **sequence**, the other three **numbers **are:

6/15, **7/15**, **8/15**, **9/15**, 10/15

**Example 3**

Prove the **associative **property under **multiplication **when:

A = 8

B = 3

C = 12

**Solution**

**Associative **property under **multiplication **is:

A (BC) = (AB) C

8 (3×12) = (8×3) x 12

8 (36) = (24) 12

288 = 288

Hence, proved.

*All images/mathematical drawings were created with GeoGebra.*