Complex Numbers

Board: JEE Main |Exam:JEEMain

JEEMainstudynotesforComplexNumbersbyAjayYadav.ThischaptercoversallconceptsrequiredforJEEMainwithproblem-solvingtechniques.

KeyConcepts

AlgebraofComplexNumbers

z=a+ib.Addition,multiplication,divisionusingconjugate.i²=-1.Powersoficycleevery4.

GeometricalRepresentation

Argandplane.Modulus|z|,argumentarg(z).Polarform:z=r(cosθ+isinθ).Euler:eθ=cosθ+isinθ.

DeMoivre'sTheorem

(cosθ+isinθ)ⁿ=cos(nθ)+isin(nθ).nthrootsofunity:1,ω,ω²whereω=e²π/ⁿ.Cuberoots:1+ω+ω²=0.

GeometryofComplexNumbers

Rotation:z₂-z₀=|z₂-z₀|eθ.Equationofcircle:|z-z₀|=r.Line:arg((z-z&#8321)/(z&#8322-z&#8321))=α.

ImportantFormulas

|z|	|a+ib|=√(a²+b²)
arg(z)	tan⁻¹(b/a)
Polar	z=r(cosθ+isinθ)
Cuberootsofunity	1+ω+ω²=0,ω³=1

SolvedExamples

Example1:Find|3+4i|.

Solution:√(9+16)=5.

Example2:Findcuberootsofunity.

Solution:1,(-1+i√3)/2,(-1-i√3)/2.Verifysum=0.

Example3:If|z-1|=|z+1|, find locus of z.

Solution: |z-1|²=|z+1|²⇒(x-1)²+y²=(x+1)²+y²⇒-2x=2x⇒x=0(y-axis).

PracticeQuestions

1. Simplify(1+i)/(1-i).
2. Findsquarerootof-5+12i.
3. Ifωiscuberootofunity,find(1+ω-ω²)⁶.
4. Findlocusofzif|z-2i|=|z+2i|.
5. Find the argument of (1+i)/(1-i).

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