REAL NUMBERS (LECTURE 3)

LESSON 3

Dear Students,

Yesterday we covered the following learning outcomes:

· summarize the Fundamental theorem of Arithmetic

· Generalise the relationship between HCF and LCM.

·  

Let us go through the guidelines for  the  blog once again:

·         The text in Red, is to be written in your register

·         The text in blue is to be viewed by clicking on it

·         The text in green is to be practiced for home work

·         I will be with you on this blog from _9:30AM___ till ___10:20AM__so that I can address to any queries that you may have when you go through the lesson. However, if you post any query after this, I will address to that only on the next day during the same slot.

1.      take your SET-A Mathematics Register

2.      Our handwriting reflects a lot about us. It will be awesome if you use good presentation and cursive hand writing

3.      Make a column on the right hand side, if you need to do any rough work

4.      Leave two lines where you finished yesterday’s work and draw a horizontal line

5.      Write today's date on the line after that.

6.      Pending Queries from yesterday (if any).......

7.      Please write the learning outcomes as mentioned below 

I will be able to  :

· recall what is an irrational number.

· prove     √p is irrational,where p is prime number

 So, let's start with a quick revision of irrational numbers.

Click on this  link 

If you all have gone through the link, then please attempt the  questions given in the image below. 

Now,let's assume   p = 3 (prime no.)  and   'a' be some positive integer = 6

so, a² =36

we all know, 3 divides 36 means  p divides a²

also, 3 divides 6  means p divides a

If you have understood this example, then think  if it happens only in the case when  'p' is prime

If you replace 'p with some composite like 4 or 9,then will this be true??

Based on the above example, We will generalise one result:

Thm 1.3

Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.

In the beginning of the blog, you  have seen that √ 2 is an irrational number and it has non terminating and non repeating type of decimal expansion.

Now, you will learn how to prove √ 2 as  an irrational number

Please click on the link for the proof

https://youtu.be/mX91_3GQqLY

Make a note of theorem 1.4 (with proof)   given on Page 12 ,in your register. 

After watching the above video, you can easily prove √ 3 as  an irrational.

Just follow the steps given in video and solve.

Example:  Prove that√ 3 is irrational.

Just follow the steps given in theorem 1.4 and solve

( Simply replace 2 with 3 and 2²(4) with 3²(9) 

Let us assume, to the contrary, √3 is rational.

That is, we can find integers a and b (≠ 0) such that √3 = a/ b

Suppose a and b have a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime. 

So, √3 b  =   a 

Squaring on both sides, and rearranging, we get 3b² = a². 

Therefore, a² is divisible by 3, 

and by Theorem 1.3, it follows that a is also divisible by 3.

 So, we can write a = 3c for some integer c.

Substituting for a, we get 3b² = 9c², that is, b² = 3c². 

This means that b² is divisible by 3, and so b is also divisible by 3 (using Theorem 1.3 with p = 3). 

Therefore, a and b have at least 3 as a common factor. 

But this contradicts the fact that a and b are coprime.

This contradiction has arisen because of our incorrect assumption that √3 is rational. 

So, we conclude that √3 is irrational. 

Now , you can attempt Q1 of Ex 1.3 as H.W

That is  all for today !

Good morning and thank you boys !

Revise the work done in class today! 

We are sure you will take all precautions  to take care of yourself and your surroundings.It is of utmost importance right now.Do write the SDG -3, Good health and well being.

See you tomorrow for the next class !