Irrational Numbers: Overview, Questions, Preparation

An irrational number is a real number that cannot be stated as a ratio p/q of integers; for instance, √ 2 is an irrational number. Irrational numbers are generally expressed as R\Q, where the retrograde slash sign symbolizes “set minus”. It can also be stated as R – Q, which depicts the difference between the set of real numbers and the set of rational numbers.

About the Topic
Irrational numbers are taught as part of the number system. Also, students are required to learn the value of Pi (π), the Euler’s Number, (e), and the Golden ratio, (φ).

How do you know a number is irrational? 

Real numbers that cannot be expressed as p/q, where p and q are integers and q ≠ 0, are irrational numbers. For instance, √ 2 and √ 3 are irrational. Any number expressed as p/q, where p and q are integers and q ≠ 0, is a rational number.

Properties

Since irrational numbers are a subcategory of real numbers, irrational numbers show the characteristics of real numbers. The characteristics of irrational numbers are:

When irrational numbers and rational numbers are added, the product is an irrational number.  For instance, consider x and y as a rational and an irrational number, respectively. On adding these two numbers, you will get a rational number z.

Upon multiplying an irrational number with a non-zero rational number, the product is an irrational number. Let’s say that xy=z, which is rational, hence, x =z/y is rational. Thus, the number xy has to be irrational.

The LCM of two irrational numbers may or may not exist. 

Upon adding or multiplying two irrational numbers, we may or may not get a rational number. For instance, √2. √2 = 2. We have, √2 which is an irrational number, if it is multiplied twice, then the ultimate product is a rational number. (i.e.) 2.

How to Find an Irrational Number?

Let’s find the irrational numbers between 2 and 3.

The square root of 4 is 2; √4 =2 and the square root of 9 is 3; √9 = 3

Therefore, the irrational numbers between 2 and 3 are √5, √6, √7, and √8. They are not perfect squares and cannot be divided further.

Question 1: Are these numbers rational or irrational: 2, -.45678…, 6.5, √ 3, √ 2

Solution:

Rational numbers– 2, 6.5 as these have terminating decimals.

Irrational numbers– -.45678…, √ 3, √ 2 as these have a non-terminating non-repeating decimal expansion.

Question 2: Check if below numbers are rational or irrational: 2, 5/11, -5.12, 0.31 

Solution:

Since the decimal expansion of a rational number, either terminates or repeats, 2, 5/11, -5.12, 0.31 are all rational numbers.

Question 3: Let x = 0.99999 . . . in the form of p/q.

Solution:

Multiply both sides by 10, we have 

   [⇒ There is only one repeating digit.]

10× x = 10× (0.99999 . . 

or 9x = 9 

Q: What is an irrational number? Give an example.

Q: Are all integers irrational numbers?

Q: Can we say that an irrational number is a real number?

Q: What are the most commonly known irrational numbers?