Irrational Numbers

Irrational numbers can be described as those real numbers which cannot be presented in the ratio form. In simple words, the real numbers that cannot be identified as rational numbers are called irrational numbers. The Pythagorean Philosopher discovered the irrational number during the fifth century BC. Initially, his theory was not taken under consideration, but later it was found that irrational numbers do exist. Throughout this article, the concept of irrational numbers will be analysed and it will be discussed whether irrational numbers can be regarded as real numbers.

What is an irrational number?

An irrational number is a mathematical term that refers to the specific set of real numbers which can never be expressed in the form of a particular fraction. This means that an irrational number can never be expressed in p/q form where q, as well as p, is both integers. Here it should be mentioned that the denominator q can never be equal to 0. Further, if an irrational number is expanded decimally then it can neither be repeated nor terminated.

Irrational number definition

Irrational numbers definition is given by real numbers which cannot be represented in the form of a simple fraction. This means that the numbers can never be expressed in the ratio form that is a/b where ‘b’, as well as a, are integers and b can never be equal to zero. Thus it is a contradiction to rational numbers.

Irrational Numbers Properties

Irrational numbers have a set of different properties which differentiates them from other types of real numbers. These have been outlined in the following.

1. Irrational numbers are always real numbers.
2. Irrational numbers after being multiplied by a rational number except 0 always give an irrational number. This means that if an irrational number b and a rational number c are multiplied then the product ‘bc’ will always be irrational.
3. Irrational numbers involve decimals that do not recur or terminate.
4. Irrational numbers after being added by a rational number always give an irrational number. This means that if an irrational number b and a rational number c are multiplied then the sum b + c will always be irrational.
5. Subtraction, division, addition, and multiplication of 2 or more irrational numbers may be or may not be rational.
6. Considering any two rational numbers, their LCM that is the least common multiple may exist or may not exist.

Irrational Number Set

The Irrational number set can be obtained through writing some numbers that are irrational within brackets. The irrational number set can be achieved through several properties.

1. Each square root that cannot be classified as a perfect square is regarded as an irrational number. For instance, the sets {√5, √83, √71, √91, √11} are irrational numbers.
2. The square root of every prime number can be regarded as an irrational number.
3. Pi Euler’s number and Golden ratio are some famous and well-known irrational numbers. These are denoted as { π, ∅, e}.

Which of the following is an irrational number?

If the above properties are carefully studied, then one can easily identify irrational numbers out of a given set of numbers. In the following section a set of numbers will be provided and which of these numbers are irrational numbers will be identified.

The numbers are 2, 3 , 1.5, 4/5, 1.222222…, √3,  √87, √7, 1/23, 34, 1 + 3i, 2 + 4i, π, -4.

The number given above will be firstly classified under two main parts. Next, which of the numbers are irrational numbers will be identified.

Real Number

• Positive integer: 2, 3, 34
• Negative integer: -4
• Rational numbers: 2, 3, 1.5, 4/5, 1/23, 34, -4
• Irrational number: 1.222222…., √3,  √87, √7, π
• Decimal: 1.5, 1.222222…
• Fractions: 4/5, 1/23
• Complex number: 1 + 3i, 2 + 4i

Conclusion

The main topic on which this article has been written is Irrational numbers which is an important topic in quantitative aptitude. Under this main topic, several subtopics have been discussed. These include what is an irrational number, the Irrational number definition, Irrational number properties, and the Irrational number set. Lastly, a set of different types of numbers have been provided and which of those are irrational numbers have been identified.