Introduction to Rational and Irrational Numbers
All the numbers used in Mathematical computations are broadly classified into two kinds. They are real numbers and imaginary numbers. Real numbers are those numbers that exist in reality and are used in most of the Mathematical computations. Imaginary numbers are the numbers that do not exist in reality. However, they are assumed to exist to ease a few Mathematical and Scientific computations. Real numbers comprise the entire list of rational and irrational numbers. The chart below describes the difference between rational and irrational numbers.
Rational Numbers Definition:
Rational numbers are the numbers that can be written in the form of a fraction where numerator and denominator are integers. In the case of rational numbers, numerator and denominator should be coprime and denominator should not be equal to zero. Mathematically, rational numbers definition is given as the number a/b if a and b are coprimes, and b is not equal to zero.
Mathematics is a subject where you can score well if you are clear with the concepts and formulas. You should maintain a separate revision note to write down the formulas, theories, and procedures so that you can easily revise them before exams. This chapter covers everything about rational and irrational numbers.
Vedantu provides all problems along with solutions covering the syllabus. You can always refer to the website if you would like to work out some problems to check your level of understanding. By practicing in this way, you could actually learn the concepts by heart and save time during the exam. Also learn the marking schemes of any problem so you will get an idea of how step marks are given.
While sitting in online classes, parents should always keep an eye on their children as this is a new environment and they will find it difficult to cope with the new situation. They will be having their mobile phones and net connection all the time so they might easily get distracted with other things. They might waste lots of time without our knowledge. But apart from all that we should make students concentrate and put in their efforts in preparing for the exams. So enrolling your students in the Vedantu website will keep them engaged with academics and make the learning process fun as well. They should find the learning process interactive and interesting other than seem boring and difficult.
Definition
Rational number, in mathematics, means a number can be represented as the quotient p/q of two integers such that q ≠ 0. In addition to all the fractions available, the set of rational numbers includes all the integers that are formed, each of which can be written as a quotient with the given integer as the numerator and 1 as the denominator.
Irrational numbers are numbers that generally cannot be expressed as the ratio of two possible whole numbers. This is just the opposite to rational numbers, like 2, 7, one-fifth, and -13/9, which can be and are, expressed as the ratio of two whole numbers.
FAQs on Rational and Irrational Numbers
1. Do irrational numbers obey closure property?
Irrational numbers do not obey closure property.
When two irrational numbers are added, the sum need not be irrational. The sum of 2 + √3 and 4 - √3 is equal to 6 which is not irrational.
When two irrational numbers are subtracted, the difference may not be irrational. The difference between 5√2 and 5√2 is 0 which is a rational number.
When two irrational numbers are multiplied, the product need not be irrational. The product of √7 and √7 is equal to √49 = 7. This is not irrational.
If two irrational numbers are divided, the quotient need not be irrational. If √6 and 7√6 is divided, the quotient is 1/7 which is a rational number.
2. What happens when basic Mathematical operations are performed between a rational number and an irrational number?
If any of the mathematical operations are performed between a rational and an irrational number, the result obtained is irrational.
When an irrational number is added to a rational number, the sum obtained is an irrational number.
The difference between rational and irrational numbers is an irrational number.
The product of two numbers is irrational if one and only one of the numbers is irrational.
The quotient obtained by dividing a rational number by an irrational number or vice versa is an irrational number.
3. How do you know if any particular number is irrational?
Simply all the numbers that are not rational are considered irrational. Accordingly, an irrational number can be written as a decimal, but not as a fraction. They will have endless non-repeating digits to the right of their decimal point. Examples of irrational numbers are √2, √3, √5, √11, √21, π(Pi). Students should be able to find the difference between a rational and irrational number at the end of this chapter.
4. What is the major difference that is found between rational and irrational numbers?
Numbers that can be expressed as a ratio of two given numbers (p/q form) are termed rational numbers. Numbers that cannot be expressed as a ratio of two numbers are termed irrational numbers. In the rational form of numbers, the decimal expansion is terminating or recurring and repeating but it is non-recurring and non-terminating in the case of irrational numbers. Example of a rational number is 0.33333, 0.656565, 1.75, and an example of an irrational number is π, √13, e
5. Where is a rational number used in real life?
There are many areas where we use rational numbers in our lives. For instance, Statisticians use integers and rational numbers to present their results in the forum. For example, in any outdoor or field game, you use rational numbers to simplify how many yards are gained or lost during the match. We also use taxes in the form of fractions and they are used by banks greatly for the calculation of interest and loans percentage. Simply when you share a pizza or anything from 8 pieces depending on the size. When you have completed your homework half portion we say that we have completed 50%, i. e 1/2.
6. Where is an irrational number used in real life?
Even irrational numbers are found really useful in many ways. One of the most practical and effective applications of irrational numbers is to find the circumference of a given circle. C = 2πr uses the irrational number π ≈ 3.14159... 5. pi=3.141592654 generally people use it to deal with any type of circle, sphere, and check computer accuracy. It is also used in solving algebra and algorithms too. There are many more effective uses of both rational and irrational numbers in our daily life in many fields.
7. Is studying this chapter necessary when compared to other topics?
Understanding and studying rational and irrational numbers in-depth requires not only the importance of mathematical ideas but also helps in recognizing how these ideas actually relate to one another. Your understanding continues to grow with experience and as a result of opportunities around you to embrace new ideas and find new connections among the already familiar ones.