Irrational Numbers

You must have studied circle in your lower classes. Well does the topic circle make any sense without π? Absolutely not! Also, the Euler’s Number is used extensively in logarithms and algebra. Well, what are this π and e? Yes, they are known as the irrational numbers. Geometrical calculations involve the irrational numbers. So let us study irrational numbers in detail.

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Irrational Numbers

Before studying the irrational numbers, let us define the rational numbers. A rational number is of the form \( \frac{p}{q} \), p = numerator, q= denominator, where p and q are integers and q ≠0.

So irrational number is a number that is not rational that means it is a number that cannot be written in the form \( \frac{p}{q} \). It cannot be written as the ratio of two integers. Representation of irrational numbers on a number line

Here the irrational number is  √2. So if we calculate the value of √2, we get √2 = 1.14121356230951….. the numbers go on into infinity and do not ever repeat, and they do not ever terminate. It cannot be written in p /q form where q is not equal to zero. The value that we get is actually non terminating. Also, there is no pattern in the digits after the decimal. These kinds of numbers are called irrational numbers. 

Browse more Topics under Number Systems

• Rational Numbers
• Irrational Numbers Between two Numbers
• Operations on Irrational Numbers
• Laws for Exponents for Real Numbers

Now consider √3. On calculating we get √3 = 1.732050807. So the pattern that we get is non terminating and non-recurring.  So √3 is also an irrational number. But in the case of √9, √9 = 3 this is a rational number

The square root of any perfect square will always be a rational number and the square root of any number which is not a perfect square will always be an irrational number. Irrational numbers have a decimal expansion that never ends and does not repeat. The most famous irrational number is,

Pi = 3.14…..

Pi is used to calculate the ratio of the circumference of a circle to the diameter of that same circle. It has been calculated to over a quadrillion decimal places, but no pattern has ever been found and so it is an irrational number. The few digits of this pattern look like 3.1415926535897932384626433832795. Few examples of irrational numbers are π, e, φ

• e = This is the Euler’s number and also an irrational number. The first few digits look like 2.7182818284590452353602874713527…
• φ =  This is an irrational number. The first few digits look like 1.61803398874989484820…

Properties of Irrational Numbers

Addition and Subtraction of Irrational numbers

1. The result of an addition of irrational numbers need not be an irrational number

(2 + √3)   +   (4 – √3) = 2 + √3  + 4 – √3 = 6. Here 6 is a rational number. So the result of adding two irrational numbers is not an irrational number.

2. The result of Subtraction of irrational number need not be an irrational number

(5+ √2 ) +  (3 + √2) = 5+ √2 + 3 + √2 = 2. Here 2 is a rational number.

Multiplication and Division of Irrational numbers

1. The product of two irrational numbers can be rational or irrational number.

√2 × √3= 6. Here the result is a rational number.

2. The result of the division of two irrational numbers can be rational or irrational number.

√2 ÷ √3 =\( \frac{√2}{√3} \). Here the result is an irrational number.

Terminating and Non-terminating Decimals

Decimal numbers with the finite number of digits are called as terminating decimals while decimals with the infinite number of digits are called as non-terminating decimals. The number  0.34 is a terminating decimal, while  0.999… a non-terminating decimal. The symbol… means that the 9  extend indefinitely.

Non-Terminating Recurring Decimals

While expressing a fraction in the decimal form, when we perform division we get some remainder. If the division process does not end means we do not get remainder equal to zero then such decimal is known as non-terminating decimal. In some cases, a digit or a block of digits repeats itself in the decimal part, then the decimal is non-terminating recurring decimal. For eg:- 1.666…, 0.141414…

Non-Terminating and Non-recurring Decimals

While expressing a fraction in the decimal form, when we perform division we get some remainder.
If the division process does not end means if we do not get the remainder equal to zero then such decimal is known as non-terminating decimal.

And if a digit or a block of digits does not repeat itself in the decimal part, such decimals are called non-terminating and non-recurring decimals. For eg:- 1.41421356

Conversion of Fractions to Recurring Decimals

Converting fractions to decimals is the same as dividing two whole numbers. For example, convert  \( \frac{2}{11} \) to recurring decimals. \( \frac{2}{11} \) = 0.1818…never ends but repeat the digits 18. Hence, \( \frac{2}{11} \) is expressed in recurring decimal as 0.181818…

Solved Questions for You

Question: State whether the following statements are true or false.

√n is not irrational if n is a perfect square.

1. True
2. False

Solution: The correct option is “A”. √4 = 2 where 2 is a rational number. Here n is perfect square the √n is the rational number. √5 = 2.236..is not rational number. But it is an irrational number. Here n is not a perfect square. √n is an irrational number. So √n is not the irrational number if n is a perfect square.

Question: Number of integers lying between 1 to 102 which are divisible by all √2, √3, √5 is

1. 16
2. 17
3. 15
4. 0

Solution: The correct option is “D”. For a number to be divisible by √2, it must be an irrational number. An integer is not an irrational number. So there are no numbers between 1 to 102 which all √2, √3, √5.

Question. Explain irrational number with example?

Answer. Irrational number refers to a number that is not written as a ratio of two integers. In other words, its expression cannot take place as a fraction. For example, the square root of 2 happens to be an irrational number because one cannot write it as a ratio of two integers.

Question. Can we say that 5 is an irrational number?

Answer. Irrational numbers are numbers that are not rational. Every integer happens to be a rational number. This is because each integer n can be written as n/1. For example 5 = 5/1. Therefore, 5 is a rational number and is not an irrational number.

Question. Can we say that zero is an irrational number?

Answer. Zero can be represented as a ratio of itself as well as a ratio of two integers. People believe that 0 is rational because it happens to be an integer.

Question. Irrational numbers are certainly there between the number 1 and 6, can you tell how many?

Answer. There is infinite number of irrational numbers between the number 1 and number 6. Between any two numbers, we have infinite rational as well as irrational numbers.