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Is 37.5 a rational or irrational number?

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  • Last Updated : 18 May, 2022

A number system in mathematics is a system used to represent numbers on different bases. In real life, we use the number system of base 10 and in this number system, there are different types of numbers or it can be said the numbers are categorized based on their different properties. Rational and irrational numbers are different numbers in the number system, let’s learn about these terms in detail,

Rational numbers

Rational numbers are of the type p/q, where p and q are integers and q ≠ 0. Most people have difficulty distinguishing between fractions and rational numbers due to the underlying structure of numbers, the p/q form. When a rational number is divided, the result is in decimal form, which can be either ended or repeated. 3, 4, 5, and so on are instances of rational numbers since they can be written in fraction form as 3/1, 4/1, and 5/1. Below are some examples,

  1. The number 5 is expressed as 5/1, where 5 and 1 are both integers.
  2. 0.6 can be written as 3/5, 6/10, or 60/100, as well as in all termination decimal forms.
  3. √49 is a rational number since it can be reduced to 7 and expressed as 7/1. 
  4. 0.66666 is a rational number with repeating decimals. 

Irrational Number 

Irrational numbers are real numbers that can’t be written as a fraction, p/q, where p and q are integers. The denominator q does not equal zero (q ≠  0). Furthermore, an irrational number’s decimal expansion is neither ending nor repeated. For example, 

  1. √2 and √3, etc. are irrational numbers.
  2. 4/0 is an irrational number having a zero denominator.
  3. π is an irrational number with the value 3.142…and it generates a never-ending and non-repeating number as a result.
  4. 0.3131548525… is a irrational number since it is not repeating or ending.

Is 37.5 a rational number?

Solution: 

Rational numbers are of the type p/q, where p and q are integers and q ≠ 0. Most people have difficulty distinguishing between fractions and rational numbers due to the underlying structure of numbers, p/q form. A rational number is a type of real number with the form p/q, where q≠0. When a rational number is divided, the outcome is a decimal number, which can be either ending or repeating.

Here, Given 37.5 can be expressed as rational number in form of p/q. Therefore, 37.5 can be written as 375/10 or its showing 37.5 is terminating after decimal hence it is a rational number  .

Sample Questions

Question 1: How is 2 .57 a rational number?

Answer:  

A rational number is a type of real number with the form p/q, where q≠0. When a rational number is divided, the outcome is a decimal number, which can be either ending or repeating.

Here, the given number, 2 .57 can be expressed  in the form of p/q, and we can write 2.57 as 257/100. Hence, 2.57 is a rational number.

Question 2:  Determine whether 5.22222… is a rational number.

Answer:  

A rational number is a type of real number with the form p/q, where q≠0. When a rational number is divided, the outcome is a decimal number, which can be either ending or repeating.

Here, the given number, 5.22222… has recurring digits. Hence, 5.22222… is a rational number.

Question 3: Is 6.24  a rational number or an irrational number?

Answer: 

A rational number is a type of real number with the form p/q, where q≠0. When a rational number is divided, the outcome is a decimal number, which can be either ending or repeating.

Here, the given number, 6.24 can be expressed in the form of p/q as 6.24 = 624/100 = 312/50 and has terminating digits. Hence, 6.24 is a rational number.

Question 4: Determine whether 8.25655848 is a rational number or an irrational number.

Answer: 

A rational number is a type of real number with the form p/q, where q≠0. When a rational number is divided, the outcome is a decimal number, which can be either ending or repeating.

Here, the given number 8.25655848 is an irrational number as it has non terminating and non recurring digits.

Question 5: Is 7.89 a rational number or an irrational number?

Answer: 

A rational number is a type of real number with the form p/q, where q≠0. When a rational number is divided, the outcome is a decimal number, which can be either ending or repeating. Here, the given number, 7.89 can be expressed in the form of p/q as,

7.89 = 789/100  

Hence, 7.89 is a rational number.

Question 6: Determine whether 9.196196… is a rational number.

Answer:

A rational number is a type of real number with the form p/q, where q≠0. When a rational number is divided, the outcome is a decimal number, which can be either ending or repeating.

Here, the given number 9.196196 is an rational number as it has recurring digits after decimal.

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