Various types of numbers in mathematics have number systems. Some of them are natural numbers, whole numbers, real numbers, integers, rational numbers, irrational numbers, and so on. Here, we will learn the definition of irrational numbers along with their examples. Also, know the steps to identify an irrational number.
Irrational Numbers Definition
Irrational nhttps://ccssanswers.com/umbers are the numbers that cannot be expressed in a fractional form i.e in \(\frac { a }{ b } \) where p and q are integers and q is not equal to zero. Those irrational numbers neither terminate nor repeat. They are also known as non-terminating non-repeating numbers.
Example: All square roots of prime numbers are irrational numbers. A number √x where x is a positive integer and it is not a perfect square is an example of an irrational number. Some of the famous examples are Pi, Euler’s Number e, golden ratio φ, √5.
Irrational Number Examples
Provided below are some specific irrational numbers that are commonly used.
- π is an irrational number. π = 3.14159265.. The decimal value never stops. π is closer to the fraction \(\frac { 22 }{ 7 } \).
- Euler’s number e is an irrational number. The value of e = 2.718281…
- Golden ratio, φ = 1.61803398874..
- √3 is an irrational number. √3 = 1.73205080757
Applications of Irrational Numbers
Irrational numbers have a lot of practical applications in our daily life. In a few cases, irrational numbers are not directly used, but their components are used in other concepts. Some of the applications are as follows:
- Construction: In construction, where there is a need to build structures that are cylindrical in shape, we use the irrational number pi.
- Money: Irrational numbers are used for calculating the compound interest on loans.
- Design and Engineering: The concept of Euler’s number is used in the engineering and design fields indirectly but not directly.
- Irrational numbers are used to find the gravity equations.
Question’s on Irrational Numbers
Question 1:
An amount of $5,000 is given to Amesh by his friend for a tenure of 3 years at an interest of 2% per annum compounded annually. Calculate the amount of Amesh needs to return his friend after 3 years.
Solution:
Given that,
Principal = $5000
Time = 3 years
Interest rate r = 2% per annum
Amount = p(1 + \(\frac { r }{ 100 } \))t
= 5000(1 + \(\frac { 2 }{ 100 } \))³
= 5000(1.061208)
= 5306.04
Hence the amount that Amesh needs to return to his friend is $5306.4.
Question 2:
Which of the following are rational numbers or irrational numbers?
2, -.45678…, 6.5, √3, √2
Solution:
Rational numbers have terminating decimals. So, rational numbers are -2, 6.5
Irrational numbers have non-terminating decimals. So, the numbers are -.45678.., √3, √2
FAQ’s on Definition of Irrational Numbers
1. Which is an irrational number?
Irrational numbers are numbers that cannot be represented as a ratio of who integers. This is the opposite of rational numbers. The examples are pi, √9, Euler’s number e and so on.
2. What is the definition of irrational numbers and rational numbers?
Rational numbers are numbers that can be expressed as fraction. Irrational numbers are numbers that cannot be represented as a fraction or ratio of two integers.
3. Is 3.14 a rational number?
The number 3.14 is a rational number. A rational number can be expressed in the form of a fraction.