Rational and Irrational Numbers

Mathematics is a game of numbers. So, number is a kind of arithmetical element which can be an item, a word, or a symbol that represents an amount and has various applications in counting, measuring and other similar areas. Integers, a whole numbers, natural number, real number, and complex numbers are all examples of numbers. 

So, the real no. are divided into two categories: rational and irrational. So, the Rational no. are integers that can be stated in the form a/b with both the numerator and denominator being integers, while the irrational no. are all those that cannot be represented in a fraction. So, in this item, we will explore irrational and rational numbers, instances of the rational and irrational numbers, the difference between rational and irrational numbers, and more about it.

Rational Number and Irrational Number: Definition

The numbers which can be represented as a fraction, and as a positive, negative no., and 0, are as rational numbers. It can be expressed as x/y, where y is not zero.

The term “rational” comes from the word “ratio,” which refers to a comparison of the two or more than two values or integer numbers, which are often known as a fraction. Basically, it is the ratio of 2 integers in the easy term.

5/3, for example, is a rational number. It denotes the division of an integer 5 by another integer 3. More examples of a rational number are as followed-

• 1/5 – The denominator and numerator are both integers
• 5 – Can be written as 5/1, where 5 is quotient of the integers 5 and 1
• 25 –square root of it can be also written as 5, which is quotient of fraction 5/1 
• 0.5 – All terminating decimals are rational no., and it can be represented as 5 divided by 10 or as 1 divided by 2
• All recurring decimals are rational at 0.55555

Irrational numbers are those numbers that are not rational numbers. For example, the irrational no. can be written in the form of decimals and not in the form of fractions, which means that they cannot be written as per a ratio of 2 integers. Moreover, subsequently the decimal point, the irrational number has an immeasurable amount of non-repeating numerals. 

An example of an irrational number is as follows: √5 = 2.2360. Other examples of irrational numbers are as followed-

• √2 – √2 is irrational since it cannot be simplified
• √7/5 – Although the provided number is a fraction, it does not meet all of the criteria for being classified as a rational number
• Hence, both the numerator and the denominator must be the integers, but √7 is not one
• As a result, the given value is an irrational number
• 5/0 is the irrational fraction since the denominator is zero.
• Π is also an irrational number as decimal values are infinite, never repeat, and never follow a pattern
• As a result, pi’s value is indeed not entirely equivalent to any of the fractions
• Subsequently, the number 22 divided by 7 is only a rough estimate
• The decimals here are neither recurring nor termination at 0.31311311113
• As a result, it can’t be stated as the quotient of the fraction

Difference between Rational and Irrational Numbers 

1. The term rational numbervrefers to the number that may be represented as the ratio of two integers. A number that cannot be stated as the ratio of two integers is called an irrational number.
2. A rational number is made up of numbers that are finite or recurring in nature, whereas an irrational number is made up of non-terminating and non-repeating numbers.
3. Perfect squares, such as 9, 4, 25, 49, etc, are included in the category of rational numbers whereas   surds like 2, 3, 5, 7, etc, are included in Irrational Numbers.
4. In rational numbers, both the denominator and numerator are whole numbers with a non-zero denominator, whereas irrational numbers cannot be expressed in fractional form.

Rules of Rational and Irrational Number

• Rule 1: The sum of two rational no. is also a rational number, for instance, the sum of 1/2 and 1/3 equal to (3+2)/6 which is 5/6
• Rule 2: When two rational numbers are multiplied, the outcome is rational, for example, 1/2 multiplied by 1/3 equal to 1/6
• Rule 3: The sum of two irrational numbers is not always irrational, for example, √2  added with √2 is equal to 2√2, which is an irrational number but then 2 added with 2√5 and (2√5) equal to 2, are a rational number
• Rule 4: The product of two irrational numbers is not always irrational, for instance, √2 multiplied by √3 is √6, which are irrational numbers, then√2 multiplied by √2 equal to √4 = 2, which are Rational numbers

Some other Examples

Rational Number

• Taxes can also be articulated as a fraction
• When the entire pizza is divided into two or more two sections
• When you have done half of your risk, such as 50 percent of the work
• Hockey players symbolise their goals with rational numbers

Irrational Number

These numbers are mostly used in mathematics, though they can also be used in real-life circumstances. These numbers allow us to build models that include important ideas like derivatives, integrals, diverse analytical geometry conclusions, and trigonometric rules, among other things. These concepts are also used in real-world situations.

Conclusion

It is evident that rational numbers can be expressed both in fraction form and decimals. An irrational number, on the other hand, can only be expressed in decimals and not in a fraction form. Moreover, all the integers are rational numbers, but all the non-integers are not irrational numbers.