Number System questions Lesson#1 मे प्राकृतिक संख्याए Natural Number, Whole Number, Integers, डेसीमल और rational नंबर के बारे मे बताते हुए recurring number को p/q फोरम मे बदला गया है जिन्हे उदाहरण की सहायता से समझाया गया है –
Number System questions Basic :-
Natural Number (N) : 1, 2, 3, 4, 5, 6, 7, …………………. ∞
Whole Number (W) : 0, 1, 2, 3, 4, 5, 6, …………………….. ∞
Integers (Z) : -∞ ………… -3, -2, -1, 0, 1, 2, 3, ………… ∞
Decimals 0.1, 0.33, 2.4, ……………
(i) terminating Decimals : 1/4 = 0.25
(ii) & non-terminating :
a) repeating non-terminating ⇒ 1/3 = 0.3333…., 1/6 = 0.16666……., 1/11 = 0.090909…..
b) non-repeating non-terminating ⇒ π = 3.141…., √2 = 1.414……, √3 = 1.732…… = irrational number
Rational Number :
p/q, q ≠ 0 & p,q ∈ Z
Irrational Numbers : cannot be written in p/q form
3.14 < π < 22/7 [Note this]
Here 1/7 = 0.142857142857 non-terminating and repeating
Convert a recurring number to p/q form in Number System questions
1/3 = 0.3333.... = 0.\bar{3},
1/6 = 0.1666.... = 0.1\bar{6} ,
1/11 = 0.090909.... = 0.\bar{09}For Example :- 0.1666….
x = 0.1666…. —— 1
10x = 1.666…. (10 से गुणा करने पर)
10x = 1 + 0.6666…. ——- 2
9x = 1 + 0.5
9x = 3/2
x = 1/6
Q1. 0.4444 …. = 0.\bar{4}
x = 0.4444…… ———– 1
10x = 4.444…..
10x = 4 + 0.444….. ———- 2
9x = 4
x = 4/9
Q2. 0.5555 …. = 0.\bar{5}
x = 5/9
Q3. 0.\bar{1}\bar{2}
= 12/99
Q4. x = 0.121212….
x = 0.121212…. ———– 1
100x = 12.121212…. ———– 2
99x = 12 [समीकरण 2 मे से 1 घटाने पर]
x = 12/99
Q5. 0.\bar{2}\bar{3}\bar{4}
= 234/999
Q6. 0.\bar{5}\bar{6}\bar{7}\bar{8}
= 5678/9999
Q7. 0.2555 …..
x = 0.2555…. ——- 1
10x = 2.555….
10x = 2 + 0.555…. ——– 2
9x = 2 + 0.3 [समीकरण 2 मे से 1 घटाने पर]
x = 23/90
Or [By short trick]
= 0.2 \bar{5}
= (25-2)/90
Q8. 0.3 \bar{6}
= (36-3)/90
Q9. 0.21555…
= 0.21 \bar{5} = (215-21)/900 = 194/900
Q10. 0.2151515…
= 0.2 \bar{1} \bar{5} = (215-2)/990 = 213/990
Q11. 0.2343434…..
= 0.2 \bar{3}\bar{4} = (234-2)/990 = 232/990
Q12. 0.21 \bar{3}\bar{4}
= (2134-21)/9900 = 2113/9900
Q13. 0.001 \bar{2}\bar{3}
= (123-1)/99000 = 122/99000
