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.