What are Irrational Numbers?

What is an irrational number?

Irrational numbers are real numbers that can’t be represented in the form of a ratio. In other words, irrational numbers are real numbers that can’t be expressed in the form of a fraction. P is the symbol often used to represent irrational numbers. Irrational numbers were discovered in the 5th century BC by Hippasus, the Pythagorean philosopher. Irrational numbers are not rational numbers but a logical inconsistency of rational numbers. Irrational numbers are the main concept in mathematics because irrational numbers maintain the continuity of real numbers. Without irrational numbers, geometry, engineering and physics are impossible to do.

Irrational numbers in quantitative aptitude

Irrational numbers were presented because they make everything a ton more straightforward. An irrational number cannot be made by dividing two integers. Decimal irrational numbers continue perpetually without repeating. All square roots which are not an ideal square are irrational numbers. The square root of any indivisible number is an irrational number. An irrational number can be a whole number.

Some of the properties of irrational numbers are the following.

• Irrational numbers consist of non-ending decimals
• Irrational numbers consist of decimals having non-repeating patterns
• Irrational numbers can be represented on a number line
• When an irrational and a rational number are added, the outcome or their total is an irrational number

• The least common multiple (LCM) may or may not exist for any two
• Multiplication, division, addition and subtraction of two irrational numbers may or may not be rational
• Multiplication of irrational numbers by any nonzero rational number provides an irrational number as their product

Irrational numbers are also known as surds. A surd refers to an articulation incorporating a square root, cube root, or other root symbols. Surds are utilised to compose irrational numbers definitively. All surds are viewed as irrational numbers. However, all irrational numbers can’t be viewed as surds. Irrational numbers, which are not the underlying foundations of arithmetical articulations, are not surds.π, and e are irrational numbers but not surds. 

Some of the irrational numbers and their value are given below.

• e – 2.718281
• π – 3.1415926
• √2 – 1.41421356
• √3 – 1.7320508
• √5 – 2.2360679
• √6 – 2.449489
• √7 – 2.6457513
• √11 – 3.3166247
• √13 – 3.60555127
• √17 – 4.12310563
• √19 – 4.35889894
• ∛47 – 3.6088260
• log35 – 1.5440680
• -√3/2 – -0.86602
• √9949 – 99.74467
• √9967 – 99.834863
• √9973 – 99.864908

There are reasons for certain numbers like pi, Euler’s number, Golden ratio to become irrational numbers. Pi is characterised as the proportion of a circle’s boundary to its diameter. The value of Pi is consistent all of the time. Pi (π) = 3.1415926535. Furthermore, it is a non-ending and non-repeating decimal number. Thus ‘pi’ is an irrational number. Euler’s number can be expressed as the quotient of two integers. e = 2.718281….. Furthermore, it is a non-ending and non-repeating decimal number. Thus  Euler’s number is irrational. A golden ratio is an irrational number because the golden ratio cannot be obtained by dividing one integer by another. 

The real-life examples of problems and solutions related to irrational numbers are the following. 

Problem 1: Jack has a bag with five irrational numbers. Jack wants only one irrational number nearest to 1 and should not be less than one how Jack finds the irrational right number. The irrational numbers in the bag are √3, √2, √6, √10, √5.

Solution: We have to find the value of all the irrational numbers in the bag. √3 = 1.732020.., √2 = 1.41421356..,  √6 = 2.449489.., √10 = 3.162277.., √5 = 2.236067… Thus, √2 = 1.41421356… comes closest to 3. Therefore, √2 is the nearest number to 1.

Problem 2: Alex is playing Roll a dice-Number game with Alex. Alex takes a turn and rolls a dice. Alex gets 5. If he gets 5, he is supposed to collect all the irrational numbers from his system. How will Alex collect all the irrational numbers from e, -5, √9,√17, π, -2/8?

Solution: -5, √9 and -2/8 are rational numbers because-5 is an integer, √9 is a perfect square, and -2/8 ends decimals. e, √17 and π are irrational numbers because e, √17 and π have a non-ending and non-repeating decimal. So Alex collects e, √17 and π as irrational numbers.

Conclusion

Irrational numbers are more than rational numbers. There are infinite irrational numbers. All the irrational numbers cannot be enumerated in a list. Some of the properties of irrational numbers can be used to discover them. Prime numbers also help to understand irrational numbers. The decimal value of irrational numbers never stops and repeats. There are certain numbers, such as φ, e, π etc., in irrational numbers compared to rational numbers. Certain properties can get a set of irrational numbers. Knowing the properties of irrational numbers helps to find which of the following is an irrational number.