9thClassMathUnit 2:RealandComplexNumbers
PunjabTextbookBoard2026-FULLEXERCISESWITH
QUESTIONS
CompleteStepbyStepSolutions
EXERCISE2.1-REALNUMBERS
Q1.Classifyeachofthefollowingnumbersasnatural,
whole,integer,rational, irrationalandrealnumber:
i) -5 i ) 0 i i) √9 iv)2/3 v)√2 vi) 0.25 vii) π
vii ) -8/1
SOLUTION:
,i) -5→Integer,Rational, Real
i ) 0→Whole,Integer,Rational, Real
i i) √9=3→Natural, Whole,Integer,Rational, Real
iv)2/3→Rational, Real
v)√2→Irrational, Real
vi) 0.25=1/4→Rational, Real
vii) π=3.14159...→Irrational, Real
vii ) -8/1=-8→Integer,Rational, Real
Q2.Findfourrationalnumbersbetween1/3and2/3.
SOLUTION:
1/3=4/12,2/3=8/12
Numbersbetween4/12and8/12are:
, 5/12,6/12=1/2,7/12
Fourth:3/8=9/24
Ans:5/12,1/2,7/12,3/8
Q3.Provethat√2is anirrationalnumber.
SOLUTION:Proofbycontradiction
Let√2=p/qwherep,qarecoprimeintegers,q≠0
Squaring:2=p²/q²→p²=2q²
Sop²is even→pis even.Letp=2k
4k²=2q²→q²=2k²→qis even
Butpandqbothevencontradictscoprimeassumption
Hence√2is irrational.
PunjabTextbookBoard2026-FULLEXERCISESWITH
QUESTIONS
CompleteStepbyStepSolutions
EXERCISE2.1-REALNUMBERS
Q1.Classifyeachofthefollowingnumbersasnatural,
whole,integer,rational, irrationalandrealnumber:
i) -5 i ) 0 i i) √9 iv)2/3 v)√2 vi) 0.25 vii) π
vii ) -8/1
SOLUTION:
,i) -5→Integer,Rational, Real
i ) 0→Whole,Integer,Rational, Real
i i) √9=3→Natural, Whole,Integer,Rational, Real
iv)2/3→Rational, Real
v)√2→Irrational, Real
vi) 0.25=1/4→Rational, Real
vii) π=3.14159...→Irrational, Real
vii ) -8/1=-8→Integer,Rational, Real
Q2.Findfourrationalnumbersbetween1/3and2/3.
SOLUTION:
1/3=4/12,2/3=8/12
Numbersbetween4/12and8/12are:
, 5/12,6/12=1/2,7/12
Fourth:3/8=9/24
Ans:5/12,1/2,7/12,3/8
Q3.Provethat√2is anirrationalnumber.
SOLUTION:Proofbycontradiction
Let√2=p/qwherep,qarecoprimeintegers,q≠0
Squaring:2=p²/q²→p²=2q²
Sop²is even→pis even.Letp=2k
4k²=2q²→q²=2k²→qis even
Butpandqbothevencontradictscoprimeassumption
Hence√2is irrational.
