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What are irrational numbers?

  • Last Updated : 03 Sep, 2021

The arithmetic value which is used for representing the quantity and used in making calculations are defined as Numbers. A symbol like “456” which represents a number is known as a numeral. Without numbers, we can’t do counting of things, date, time, money, etc. these numbers are also used for measurement and used for labeling. The properties of numbers make them helpful in performing arithmetic operations on them. These numbers can be written in numeric forms and also in words.

For example, 3 is written as three in words, 35 is written as thirty-five in words, etc. Students can write the numbers from 1 to 100 in words to learn more. There are different types of numbers, which we can learn. They are whole and natural numbers, odd and even numbers, rational and irrational numbers, etc

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Number system 

A Number system is a method of showing numbers by writing, which is a mathematical way of representing the numbers of a given set, by using the numbers or symbols in a mathematical manner. The writing system for denoting numbers using digits or symbols in a logical manner is defined as a number system. The numeral system which represents a useful set of numbers also  Reflects the arithmetic and algebraic structure of a number and Provides standard representation. The digits from 0 to 9 can be used to form all the numbers. With these digits, anyone can create infinite numbers. For example, 156,3907, 3456, 1298, 784859 etc.



Irrational Number

Before explaining irrational numbers, let’s have a brief about rational numbers. The numbers which can be expressed as a ratio between two integers are defined as rational numbers. It is the form of  a/b here “a” is the numerator and “b” is the denominator, where a and b are integer and b ≠ 0. Some example, the fractions 1/5 and −2222/ 8 are both rational numbers. All integers are included in the rational numbers and we can write any integer “z” as the ratio of z/1.

The number which is not rational or we cannot write in form of fraction a/b is defined as Irrational numbers. Here √2 is an irrational number, if calculated the value of √2, it will be √2 = 1.14121356230951, and will the numbers go on into infinity and will not ever repeat, and they don’t ever terminate. It can’t be written in a /b form where b is not equal to zero. The value getting is actually non-terminating and there is no pattern in the digits after the decimal. These types of numbers are called irrational numbers.

Consider √3 while calculating, √3 = 1.732050807. The pattern received is non-recurring and non-terminating. So √3 here is also an irrational number. But the case of √9 here √9 = 3 this is a rational number. Square root of perfect square will always be a rational number. The square root of any number which is not a perfect square will always be an irrational number. Irrational numbers can have a decimal expansion that never ends and does not repeat. The very famous irrational number is

Pi(π) = 3.14

Pi(π) is used to calculate the ratio of the circumference of a circle to the diameter of that same circle. Lots of calculation has been done to over a quadrillion decimal places but still no pattern has ever been found and so it is an irrational number. The pattern looks like 3.1415926535897932384626433832795. Here are some examples of irrational numbers are π, e, φ

e = Euler’s number, and also an irrational number. The first few digits look like 2.7182818284590452353602874713527654…

φ = It’s an irrational number. The first digits look like 1.61803398874989484820…

Some properties of Irrational numbers

Addition and Subtraction of Irrational numbers



According to this, the result of an addition of irrational numbers need not be an irrational number

(4 + √3)   +   (6 – √3) = 4 + √3  + 6 – √3 = 10. Here 10 is a rational number.

By this, the result of adding two irrational numbers is not an irrational number.

As per this, the result of Subtraction of irrational number need not be an irrational number

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

Multiplication and Division of Irrational numbers 

According to this, the product of two irrational numbers can be a rational or irrational number.

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

As per this, the result of the division of two irrational numbers can be a rational or irrational number.

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



Sample Problems

Question 1: Which of the numbers are Rational Numbers or Irrational Numbers?

5, -2, -.45678…, 6.5, √ 3, √ 2, √5

Solution:

Here, 5 , -2 , 6.5 are all rational number as its can be expressed as a fraction and have terminating decimal. As 5 can be written as 5/1 and -2 can be written as -2/1 , and 6.5 as 65/10.

Whereas, √3 , √2 , √5 , -.45678…. are all irrational numbers as its cannot be expressed in fraction or having non-terminating, non repeating decimal, here if √5 is equal to non terminating decimal.. √5 = 2.2360679, and same for the √3 = 1.732.. here these represents the Irrational number.

Question 2: Which of these are Irrational numbers?

0.5, π, 1/3, 0.857857

Solution:

The numbers that cannot be expressed as fraction are irrational numbers. So here 0.5 can be written as 1/2 And 1/3 itself a fraction n 0.857857 can be written as 8578/1000 .so these are rational numbers. π is the only irrational here which can’t be expressed as fraction. 

Question 3: When you multiply two irrationals, it gives you different results. How? 



Answer:

Multiply √3 × √3, then it will give result as a rational number

√9 = 3 , Here 3 is rational number.

In second case, If we multiply, √3 × √5 , then it will give result as an irrational number.

=√15 , Here √15 Is an irrational number

So multiplication of two irrational numbers can give you both the result as rational or irrational.

Question 4:  Identify whether the following numbers are Rational or Irrational?

√2, 84, 8.432432432…, 3.14159265358979…, √11, 33/3.

Solution:

84, 8.432432432…, and 33/3  are Rational numbers as either they are Integers or their decimal expansions are terminating, repeating.

√2, 3.14159265358979…, and √11 are Irrational numbers as their decimal Expansions are Non-terminating, Non-repeating.

Question 5: Identify is 6.5 a rational or irrational number?

Solution: 

The number 4.5 is a rational number. Since rational numbers can also be expressed as decimals with repeating digits after the decimal point. Here we can write 6.5 as 65/10 and further write it as 15/2 = 6.5 so its a rational number.




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