Number system class IX

Type of numbers, rational and irrational numbers, decimal representation of rational and irrational numbers, venn diagram, solved examples from NCERT

1. NUMBER systems CHAPTER 1

2. TYPE OF NUMBERS Natural numbers (N): the counting numbers 1,2,3……. Whole numbers (W): Zero along with all natural numbers 0,1,2,3…. Integers (Z): Negative and positive numbers along with zero …..-3,-2,-1,0,1,2,3….. Rational numbers (Q): All integers, fractions and decimal numbers 0.25, -17, , ,  

3. RATIONAL NUMBERS Any number that can be written in the form , integers , q ≠ 0. p and q have no common factors other than 1 (that is, p and q are co-prime)   Do you think a natural number is a rational ? Yes, 3 can be written as   Do you think a 2.5 is a rational ? Yes, 2.5 can be written as =  

4. NUMBER SYSTEM

5. Decimal representation of rational numbers = 0.7   = 2.5   = 1.75   All these are the examples of terminating decimal representation Note: remainder is zero

6. Decimal representation of rational numbers = 0.666…. = 0.   = 0.142857142857…. = 0.   = 0.166666… =0.1   Note: remainder is never zero Recurring decimal numbers

7. Am I a terminating or recurring decimal ???   If denominator has factors 2 and 5 only then is terminating decimal Otherwise recurring decimal or non terminating decimal = 0.24666… =0.24 This is recurring decimal.  

8. REPRESENTATION ON NUMBER LINE    

9. FINDING RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBERS

10. CONVERTING DECIMALS TO RATIONAL NUMBERS 0.3 = 0.75 = = BUT WHAT IF WE NEED TO CONVERT 0.3333…. 1.2727…. 0.2353535….. into form.  

11. CONVERTING DECIMALS TO RATIONAL NUMBERS let x = 0.3333... ( i ) Now here is where the trick comes in. Multiply ( i ) by 10 10 x = 10 × (0.333...) = 3.333... Now, 3.3333... = 3 + x, since x = 0.3333... Therefore, 10 x = 3 + x Solving for x, we get 9x = 3, i.e., x =  

12. CONVERTING DECIMALS TO RATIONAL NUMBERS Let x = 1.272727... Since two digits are repeating, we multiply x by 100 100 x = 127.2727... So, 100 x = 126 + 1.272727... 100x = 126 + x Therefore, 100 x – x = 126, i.e., 99 x = 126 ., x = =  

13. CONVERTING DECIMALS TO RATIONAL NUMBERS Let x = 0. 23535…. Over here, note that 2 does not repeat, but the block 35 repeats. Since two digits are repeating, we multiply x by 100 to get 100 x = 23.53535... So, 100 x = 23.3 + 0.23535... 100x = 23.3 + x Therefore, 99 x = 23.3 i.e., 99 x = , which gives x =  

14. NCERT QUESTION EX.1.3 (Q.2)