NUMBER SYSTEM(LECTURE 4)

LESSON 4  NUMBER SYSTEM

Dear Students,

Good morning ! 

We have covered the following learning outcomes so far:

• Recall what are rational and  irational numbers
• Locating Irrational numbers on a number  line.  
• Find decimal expansion of  given number
• Identify the type of number on basis of decimal expansion

Any queries from Lesson 2 or 3? Then please post as comment below !!

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

·         The text in Red, is to be written in your note book

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

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

        Follow the instructions (such as write / view / recall....) as and when they appear in my blog

·         I will be with you on this blog in your maths slot 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 Maths slot.

·         Feel free to clarify your doubts by dropping a comment before going ahead in the lesson. PLEASE DO WRITE YOUR NAME along with your query in the comment box. 🙏

1. Take your SET-A Mathematics Note Book.
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.  : Responsible Consumption and Production. Remember we contemplated on SDG-1 in yesterday’s class ? Let us be responsible global citizens.
6. Please write the learning outcomes which is given below :

 I will be able to  

Recall what are rational numbers / irrational numbers  

Find decimal expansion of  given number

 Find the type of number on basis of decimal expansion
 To Expess  Numbers (Non terminating repeating   )  in  p/q form

  click on  the link 

Have you  been through the video?

So, Now answer the following

1,  A rational number  This rational number can also be known as

      (a)   a natural number

(b)   a whole number

(c)   a fraction

(d)   a real number

5.  The rational number  can also be written as

      (a)   0.3

(b)   

(c)   0.33

(d)   

Answer these in the comments  column

Lets attempt Ex1.3   Q4'  to Q9

4.  Let x = 0.99999 . . . in the form  Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Ans.  x   =  0.9999......  --------eq (1)

       Multiply both sides by 10, we have

     

       10× x = 10× (0.99999 . . .)
         10 x   = 9.9999........----------(2)
subtracting  eq (1) from eq(2)

       or  9 x = 9
         x   = 1 

5.  What can the maximum number of digits be in the repeating block of digits in the decimal expansion of  Perform the division to check your answer.

Ans.

           Since, the number of entries in the repeating block of digits is less than the divisor. In  the divisor is 17.

           ⇒ The maximum number of digits in the repeating block is 16. To perform the long division, we have

           

           The remainder 1 is the same digit from which we started the division.

           

           Thus, there are 16 digits in the repeating block in the decimal expansion of  Hence, our answer is verified.

6.  Look at several examples of rational numbers in the form  where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

Ans.

           Let us look at decimal expansion of the following terminating rational numbers:

           

           We observe that the prime factorisation of q (i.e. denominator) has only powers of 2 or powers of 5 or powers of both.

           Note:

           If the denominator of a rational number

           (in its standard form) has prime factors either 2 or 5 or both, then and only then it can be represented as a terminating decimal.

7.  Write three numbers whose decimal expansions are non-terminating

           non-recurring.

Ans.

           

8.  Find three different irrational numbers between the rational numbers 

Ans.

           To express decimal expansion of  we have:

           

           As there are an infinite number of irrational numbers between  any three of them can be:

           (i)      0.750750075000750. . .

           (ii)    0.767076700767000767. . .

           (iii)   0.78080078008000780. . .

9.  Classify the following numbers as rational or irrational:

       

Ans.

       (i)   ⇒23 is not a perfect square.

              ⇒  is an irrational number.

       (ii)   ⇒225 = 15 × 15 = 152

              ⇒225 is a perfect square.

              Thus,  is a rational number.

       (iii)   ⇒0.3796 is a terminating decimal,

              ⇒It is a rational number.

       (iv)   

              Since,  is a non-terminating and recurring (repeating) decimal.

              ⇒It is a rational number.

       (v)   Since, 1.101001000100001... is a non-terminating and non-repeating decimal number.

              ⇒It is an irrational number.

Do it in your class register

We have double period today, so some more questions for you to do as class work
1. Express in decimal form: (i) 3 1            (ii)2/11
                                                    8

2. Express each of the following as a fraction in simplest form.
(i) 0.324324324......   (ii)0.544444444444.....     (iii)0.163636363.......

Lets practise  (HW)

Q. 1  State whether the following rational numbers will have a terminating decimal expansion or non-terminating repeating decimal expansion: 

   
   (a)          ( b)  

(c)   
(b)   
            Justify your answer.
Q2 .Express   as a fraction in teh simplest form.

  Q3.  Simplify 

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Good morning and thank you boys !