by Tiwari Academy
The basics of Irrational Numbers starts from very beginning of schooling. The use of it is so vast that it is an important topic for almost all the classes in primary, secondary and senior secondary. Get here a brief overview of Irrational Numbers and its application with video explanation.
About – Irrational Numbers
In the topic Revisiting Irrational Numbers, students of 10th Standard will study proving a rational or irrational number algebraically. To prove that a number is irrational, you can use the following steps:
First of all, assume that the number is rational. This means that it can be written in the form of a ratio of two integers, such as p/q, where p and q are integers and q is not equal to zero or we can say that p and q are co-prime integers.
Video Representation of Irrational Numbers
Contradiction Method for proving Irrational
Now show that this assumption leads to a contradiction that our assumed rational number is irrational. For this, we should use the fact that p and q have a common factor. Earlier we have supposed that p and q are co-prime, but if p and q have common factor then these are not co-prime. So, this contradict our assumption.
Conditions for an Irrational
Use this contradiction to conclude that the number must be irrational. Since the assumption that the number is rational leads to a contradiction, it must be false, and therefore the number must be irrational.
The Square Root of 5 is irrational Number
Here is an example of how to prove that the square root of 5 is irrational using these steps:
Assume that the square root of 5 is rational, which means that it can be written in the form of a ratio of two integers, such as p/q, where p and q are co-prime integers and q is not equal to zero.
Conclusion – The square root of 5
Show that this assumption leads to a contradiction. The square root of 5 can be written in fraction form as p/q that is no common factor between p and q. However, if the square root of 5 is rational, it can be written in decimal form as a finite or repeating decimal or p should be divisible by q. Later on, the assumption that the square root of 5 is rational leads to a contradiction.
Conclude that the square root of 5 is irrational. Since the assumption that the square root of 5 is rational leads to a contradiction, it must be false, and therefore the square root of 5 is irrational.
