In this math chapter, we will try to answer the question if every rational number is an integer ?

To understand the topic you should have basic understanding of the concept of integer and rational numbers.

Let’s first revise both the concepts.

## What are integers ?

The **numbers which do not contain any fraction or decimal** are called **integers**.

Integers can be **positive or negative numbers**.

Numbers like -25, 6, 31, -102 & 356 are examples of integers.

Numbers like 23.5, 100.2 , \mathtt{\frac{2}{3} \ \&\ \sqrt{2}} are not part of the integers.

To learn about the **integers in detail**, click the red link.

## What are rational numbers ?

The **numbers that can be represented in the form of P / Q** are called **rational numbers.**

Where, P & Q are integer numbers except 0.

Numbers like 2.5, 13.76, \mathtt{\frac{16}{3} \ \&\ \frac{21}{4}} are all examples of rational numbers.

Note that the rational numbers can be integer or decimal numbers.

To learn about** rational numbers in detail**, click the red link.

## Is every rational number part of an Integer ?

The answer is NO !!

**Rational numbers can be both integers or decimals**, depending on the condition.

(i) If the rational number P / Q can be reduced to number with denominator 1, then it is a integer.

(ii) If the rational number P / Q is simplified to produce decimal number, then the rational number is a decimal.

Let us understand the above concept with example.**Example 01**

Check if rational number \mathtt{\frac{26}{1} \ } is an integer.

**Solution**

The number \mathtt{\frac{26}{1} \ } contains denominator 1, so it can be written as simply 26.

\mathtt{\frac{26}{1} \ \Longrightarrow \ 26}

Hence, the above rational number is an integer.

**Conclusion**

Any rational number with 1 as denominator** is an integer**.

**Example 02**

Check if rational number \mathtt{\frac{24}{4}} is an integer.

**Solution**

The number \mathtt{\frac{24}{4}} can be simplified further.

Divide both numerator and denominator by 4.

\mathtt{\Longrightarrow \frac{24}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{24\div 4}{4\div 4} \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{6}{1}}

After the division, we get the fraction with denominator 1. The fraction \mathtt{\frac{6}{1}} can be simply written as 6.

Hence, the above rational number is an integer.

**Conclusion**

If the rational number can be reduced to fraction with denominator 1 then** it is an integer**.

**Example 03**

Check if the rational number \mathtt{\frac{5}{2}} is an integer.

**Solution**

The number \mathtt{\frac{5}{2}} cannot be further simplified.

Since further simplification is not possible and the fraction contain denominator 2, the given rational number is not an integer.

**OR**

Simply find the value of fraction after division.

If the final value is decimal number, then the given rational number is not an integer.

In this case;

\mathtt{\frac{5}{2} \ \Longrightarrow \ 2.5}

After division of \mathtt{\frac{5}{2}} we get decimal number 2.5

**Hence, it is not an integer.**

I hope you understood the above three examples, let us solve some practice problems.

## Rational Number is an Integer or not ? Practice Problem

(i) \mathtt{\frac{10}{7}}

**Solution**

The fraction cannot be simplified further.

Since, the fraction contain 7 as denominator, the given number is not an integer.

**Alternate explanation**

Dividing the above fraction we get;

\mathtt{\frac{10}{7} \ \Longrightarrow \ 1.43}

Since the given rational number is divided to get fraction 1.43, the **given number is not an integer**.

(ii) \mathtt{\frac{104}{2}}

**Solution**

Dividing the numerator by denominator, we get;

\mathtt{\frac{104}{2} \ \Longrightarrow \ 52}

On division we get the integer 52.

Hence, the above **rational number is an integer**.

(iii) \mathtt{\frac{-\ 853}{1}}

**Solution**

Since the above rational number contain denominator 1, this can be written as;

\mathtt{\frac{-\ 853}{1} \ \Longrightarrow \ -853}

Hence, the above **rational number is an integer.****Note:**

Integer includes negative number.

(iv) \mathtt{\frac{55}{30} \ }

The rational number can be further simplified.

Dividing numerator and denominator by 5, we get;

\mathtt{\Longrightarrow \frac{55}{30} \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{55\div 5}{30\div 5} \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{11}{6}}

The fraction cannot be simplified further.

Hence the given **rational number is a fraction and not integer.**

(v) \mathtt{\frac{660}{10} \ }

The above rational number can be further simplified.

Divide numerator and denominator by 10.

\mathtt{\Longrightarrow \frac{660}{10} \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{660\div 10}{10\div 10} \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{66}{1}}\\\ \\ \mathtt{\Longrightarrow \ 66}

Hence, the above **rational number is an integer.**