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Is pi a rational or irrational number?

  • Last Updated : 10 Dec, 2021

A number system or numeric system is a way of representing numbers in arithmetic or algebraic form in figures or words. These numbers present in the number system are divided into different types like real numbers, prime numbers, even numbers, odd numbers, rational numbers, irrational numbers, etc. These numbers are a significant part of mathematical operations for the conduction of calculations.

Numbers

Numbers are the arithmetic values used in different mathematical operations. They are generally used to label fixed quantities, make measurements, sales, trading, etc. Numbers have been a huge part of the economic and financial world. Integers, whole numbers, natural numbers, rational numbers, etc are some of the examples of sets of numbers.

Types of numbers

Apart from the four different types of numbers, those are, decimal numbers, hexadecimal numbers, binary numbers and octal numbers. Numbers are also classified into other types based on their characteristics. They are categorized into sets by the number system. The types are described below:

  • Natural numbers: Natural numbers are the set of numbers counting from 1 to infinity. The set of natural numbers is represented by ‘N’. It is the numbers we generally use for counting. The set of natural numbers by is given N = 1, 2, 3, 4, 5, 6, 7,…
  • Whole numbers: Whole numbers are the set of natural numbers including zero, which counts from 0 to infinity. Whole numbers do not include fractions or decimals. The set of whole numbers is represented by ‘W’. The set of whole numbers is given by W=0,1,2,3,4,5,…
  • Integers: Integers are the set of numbers including all the positive natural numbers, zero as well as all negative counting numbers which count from negative infinity to positive infinity. The set doesn’t include fractions and decimals. The set of integers is denoted by ‘Z. The set of integers is given by  Z=..,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …
  • Decimal numbers: Any numeral value that consists of a decimal point is a decimal number. It can be expressed as 2.5, 0.567, etc.
  • Rational numbers: Rational numbers are the numbers that can be expressed as the ratio of two integers. It includes all the integers and can be expressed in terms of fractions or decimals. It is denoted by ‘Q’.
  • Irrational numbers: Irrational numbers are numbers that cannot be expressed in fractions or ratios of integers. it can be written in decimals and have endless non-repeating digits after the decimal point. It is denoted by ‘P’.

Rational numbers

Rational numbers are the set of numbers that can be expressed as fractions of two integers and can be written as a positive number, negative number, prime, and even a zero.

Rational numbers are expressed in the form of p/q, where q ≠0.

Rational numbers can be expressed as fractions, decimals, and even zero. All the numbers with a non-zero denominator which can be written in p/q form are rational numbers. For example, 4/5 is a rational number that expresses that integer 4 is divided by integer 5.

A rational number is a fraction or a  ratio of two integers which can be written in the form of p/q where q is not equal to zero. Hence, any fraction with a non-zero denominator is a rational number. For example, 4/5 is a rational number where 4 is an integer being divided by a non-zero integer that is 5. A rational number can be also written in the decimal form if the decimal value is definite or has repeating digits after the decimal point. For example, 0.8 is a rational number. As the value 0.8 can be further expressed in the form of ratio or fraction as p/q

0.8 = 4/5

Which is a ratio of two definite integers 4 and 5.

Irrational numbers

Irrational numbers are the set of numbers that cannot be expressed in fractions or ratios of integers. it can be written in decimals and have endless non-repeating digits after the decimal point.

Irrational numbers cannot be  expressed in the form of p/q, where q ≠0.

For example 0.1211212111122… is an irrational number that is non-terminating.

Is π a rational or irrational number?

Answer:

π is a mathematical expression whose approximate value is 3.14159365… The given value of π is expressed in decimal which is non-terminating and non-repeating. As the value is non-terminating it shows the nature of irrational numbers. Hence, π is not a rational number. It’s an irrational value.

Sample Problems

Question 1: Is 22.7 a rational number?

Answer:

22.7 can be written infraction form as 227/10 which is in the form of p/q and q is not equal to zero. Hence, 22.7 is a rational number.

Question 2: 0 is a rational number, how?

Answer:

0 is also included in rational number as it has a non-zero denominator. If we express 0 in the form of p/q

0 = 0/1

Where 0 is an integer and divided by integer 1.

Question 3: Is 0.5 is a rational number?

Answer:

0.5 can be written infraction form as 1/2 which is in the form of p/q and q is not equal to zero. Hence, 0.5 is a rational number.

Question 4: Is 33.5 a rational number?

Answer:

33.5 can be written infraction form as 335/10 which is in the form of p/q and q is not equal to zero. Hence, 33.5 is a rational number.

Question 5: Mention the properties of a rational number.

Answer:

The general properties of rational numbers are,

  • Closure property
  • Commutative Property
  • Associative Property
  • Destructive property
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