Is 5 pi a rational number?

In the Number system, Natural numbers are the numbers that start from 1 and go up to Infinity. For example, (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21… ∞) are Natural Numbers. Let’s take a look at what are rational numbers,

Rational Numbers  

A rational number is represented or expressed in a p/q form where q is not equal to 0. A set of rational numbers includes positive, negative numbers, and zero and it is denoted by Q. We can also write the rational numbers in fraction form. When a number is expressed in a p/q form or in fraction form where both the numerator and the denominator part are integers, then the number is known as a rational number. Examples: 3/2, -2/7, 1/10, -7/10, 12/99.

Zero “0” is also a rational number because it can be represented in many forms such as 0/1, 0/2, 0/3, etc. But, 1/0, 12/0, 30/0, etc. are not rational numbers because they give infinite values.

Note: You can express rational numbers in decimal form.  

Types of Rational Numbers   

1. Natural Numbers: All natural numbers are rational numbers because they can be written as in p/q form. For example – 2 can be expressed in 2/1 (p/q) form. Some examples are, 1, 2, 3, 4, 5 …. etc.
2. Terminating Decimals: Rational numbers can also be expressed in decimal form because decimal numbers can be represented in p/q form. For example, 1.1 can be written as 1.1 = 11/10. All decimals which have terminating decimals are also rational numbers. Some examples are, 0.26, 0.8130, 0.7598 etc. 
3. Non-Terminating Decimals: Non-terminating decimals are those which keep on continuing after the decimal point or continue endlessly such as 0.11111…, 0, 14141414… are also rational numbers. Since 0.33333… can be written as 1/3, therefore it is a rational number. Some examples are 0.88888…, 0.121212… etc. 
4. Fractions: When a number is expressed in a p/q form or in fraction form where both the numerator and the denominator part are integers, then it is a rational number. Some examples are 3/4, 2/7, 7/10, -7/10, 14/99 (all are in p/q form)  
5. Whole Numbers: All whole numbers are rational numbers because the whole numbers can also be expressed in p/q fraction form. Some examples are 0 is a rational number because it can be written as a fin 0/1, 0/-2,… etc.

Methods to identify rational numbers 

There are some conditions to check whether a number is a rational number or not. They are,

1. Always rational numbers are represented in the p/q form, where q≠0. For example, 1/4 , 2/7,  3/10, -7/10, 0/1 etc.
2. A rational number can be simplified further and represented in decimal form. For example, 0.7, -0.165, 3.75, -1.0 etc.

Is 5π a rational number?

Solution : 

No 5π is not a rational number. 5π  is an irrational number.

Explanation: A rational number is a number that is represented in fraction form i.e. p/q form. Repeating decimals are considered rational numbers because they can be written in the ratio form of two integers. Pi (π) is an irrational number that has a value of 22/7 or approximately equals 3.14159265359… in decimal. It is a non-terminating non-repeating decimal number. 

When any number is multiplied with pi it always gives non-terminating non-repeating decimal numbers. That’s why 5π is not a rational number. Hence 5π is an irrational number.

Sample Problems

Question 1: Check whether Mixed Fraction 1 5/4 is a Rational Number or Not?

Answer:

The Simplest Form of Mixed fraction 1 5/4 is 9/4. It is in p/q form. Thus, 9/4 is a Rational Number.

Question 2: How do you Identify a Rational Number?

Answer: 

When a number is expressed in the p/q form where p, q are integers and q is non-zero. Then it is a Rational Number.

Question 3: Is 8.3 is a rational number?

Answer:

Yes, 8.3 is a rational number.

Explanation: A rational number is any number that can be expressed as the quotient of two integers. In other words, a rational can be expressed in p/q form. If a decimal representation terminates or recurs, then it is also expressible in the form  p/q for some integers p and q.

Question 4: What do you mean by terminating decimal for example?

Answer:  

A decimal that can be expressed in a finite number of figures or those numbers which come to an end after a few repetitions after the decimal point are called terminating decimals.

Example : 0.2, 1.224, 123.456, etc.

Question 5: Define non – terminating decimal for example?

Answer:

Non-terminating decimals are those which keep on continuing after the decimal point or continue endlessly. They don’t come to end or if they do it is after a long interval. Then it is known as non – terminating decimals.

Example: 3.14159265358979323846…) is an example of a non-terminating decimal as it keeps on continuing after the decimal point.

Question 6: Explain why 11 is not an irrational number?

Answer:

11 is not an irrational number because an irrational number is a real number that cannot be expressed as a/b where a and b are integers. So 11 is not an irrational number. It is a rational number.