REAL NUMBERS( LECTURE 4)

Lesson 4 Real Numbers

Dear Students, 

Good Morning Everyone!!

Yesterday we covered the following learning outcomes:

1.recall what is an irrational number.

2.prove  √p is irrational,where p is prime number

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  :

•                     Prove that sum or difference of a rational and an irrational number is irrational.

•                     Prove that the product and quotient of a non  zero rational and an irrational number is irrational.

Students,  we will be   often using the words rational and irrational in the lesson.

So, I would like to take  your attention towards  the rational and judicious use of our natural resources. It is  definitely the need of the hour !!

Make a note ,SDG #17 Partnership for the Goals

In Class IX you have learnt about   “Operations on Real Numbers”

1.      If you  add, subtract, multiply or divide (except by zero) two rational numbers, you  will get a rational number.

2.       However, the sum, difference, quotients and products of irrational numbers are not always irrational.

For eg. √2 + (-√2) =0 (rational)

For eg.√2 + (-√2) =0 (rational)

√3 x √3 =3      (rational)

3.      But what happens when we add and multiply a rational number with an

irrational number?

 For example, √3 is irrational. What about 2√3  and 2 +√3 ?

We know, √3 = 1.73205……….(non terminating and non repeating)

So,   2√3    =    3.4641…….

And 2 +√ 3 =    3.73205…..

Since, both of these   have non terminating and   non repeating  type of decimal expansion

Therefore , both are irrational.

Similarly, we can show that difference and quotient of rational and irrational no. is an irrational.(** for product and quotient we have to include non zero rational number)

In case you are asked to prove this, then  how will you do??

Look at the  image given below 

The image shows  conversation between students

                                                        

 What did you understand  from this conversation??

Why did the contradiction arise at the end ??

Actually, their assumption that √2 + ½ is rational ,was incorrect .This  led to contradiction.

Therefore, √2 + ½ is not rational but is irrational.

 Please view the example given in the link below for better understanding

https://youtu.be/OlWccczOdjY

 Now, look at the solution of  Ex 1.3 ,Q3 ( part 3) and note it down  in your register

Ques : Prove  that 6 + √2  is irrational.

** You can only use contradiction method as given below to prove such questions. Don’t use the property that sum of rational and irrational  is irrational.**

Solution : Let us assume, to the contrary, that 6 + √2  is rational.

That is, we can find co-prime a and b (b ≠ 0) such that   6 + √2 = a/b

Therefore,   ( a/b )  - 6 = √2  Rearranging this equation, we get

√2 =  (a - 6b)/b

Since a and b are integers, we get     (a - 6b)/b  is rational, and so √2 is rational.

But this contradicts the fact that  √2  is irrational.

This contradiction has arisen because of our incorrect assumption .

So, we conclude that   6 + √2  is irrational.

Please follow the above solution to complete Q 2 of Ex 1.3  as H. W

Please click on the  link ,it will help you to understand the proof of

” Product of (non zero)  rational and irrational  is irrational “

https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:irrational-numbers/x2f8bb11595b61c86:sums-and-products-of-rational-and-irrational-numbers/v/proof-that-rational-times-irrational-is-irrational

Look at the image below and  note down the solution of  EX 1.3 Q3 part (ii)

Now,  you will be able to attempt Q 3 (i) as H.W

Students, we have already discussed that sum and difference of 2 irrationals can be rational or irrational.

So, answer the question: Is √3 + √5  an irrational or rational?

Yes, the answer is Irrational

Look at the image below for the proof  (H.W : Make a note of the proof in your notebook)

Today’s  lesson ends here!!

Complete your H.W and Revise the work done in class today!!

Good morning and thank you boys !

See you tomorrow for the next class !