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Complex Numbers revision notes

Every idea in the Complex Numbers chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 26 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Algebra of complex numbers

Powers of i
2× asked, last 2024
i4k=1,i4k+1=i,i4k+2=−1,i4k+3=−i,in+in+1+in+2+in+3=0i^{4k}=1,\quad i^{4k+1}=i,\quad i^{4k+2}=-1,\quad i^{4k+3}=-i,\qquad i^{n}+i^{n+1}+i^{n+2}+i^{n+3}=0
kk any integer. Reduce the exponent mod 4; for n≥4n\ge4, n!n! is a multiple of 4, so in!=1i^{n!}=1.
Re\text{Re}Im\text{Im}1, i41,\ i^4i, i5i,\ i^5i2=−1i^2=-1i3=−ii^3=-i
Multiplying by ii is a quarter turn: the powers repeat every 4
  • Not 1 for every even power. Only multiples of 4 give 1, and i2=i6=−1i^{2}=i^{6}=-1. In ∑(1−in!)\sum(1-i^{n!}) the terms n=2,3n=2,3 are each 2; every term from n=4n=4 on is 0.
  • Not −4 −9=36=6\sqrt{-4}\,\sqrt{-9}=\sqrt{36}=6. The rule ab=ab\sqrt a\sqrt b=\sqrt{ab} fails when both are negative. Write −a=ia\sqrt{-a}=i\sqrt a first: (2i)(3i)=−6(2i)(3i)=-6.
Equality, real and purely imaginary
15× asked, last 2026
a+ib=c+id  ⟺  a=c, b=d,z∈R  ⟺  z=zˉ,z purely imaginary  ⟺  z=−zˉ (z≠0)a+ib=c+id\iff a=c,\ b=d,\qquad z\in\mathbb R\iff z=\bar z,\qquad z\ \text{purely imaginary}\iff z=-\bar z\ (z\ne0)
a,b,c,da,b,c,d real; one complex equation gives two real equations. If one side is a modulus, the other's imaginary part is 0.
  • Quick method. For (a+ib)/(c+id)(a+ib)/(c+id) multiply by c−idc-id: the quotient is real iff bc−ad=0bc-ad=0, purely imaginary iff ac+bd=0ac+bd=0 (and bc≠adbc\ne ad). Then list every θ\theta in the stated interval, not just [0,2π)[0,2\pi).
Quadratics with complex roots
12× asked, last 2026
z=−b±b2−4ac2a,z1+z2=−ba,z1z2=caz=\frac{-b\pm\sqrt{b^2-4ac}}{2a},\qquad z_1+z_2=-\frac ba,\quad z_1z_2=\frac ca
Vieta holds for complex coefficients too. Real a,b,ca,b,c make non-real roots a conjugate pair; otherwise they need not be.
  • Not a conjugate pair when a coefficient is complex. x2−(3−2i)x−(2i−2)=0x^2-(3-2i)x-(2i-2)=0 has roots 2−2i2-2i and 11. Use "zˉ\bar z is also a root" only for real coefficients.
  • Quick method. Reduce symmetric expressions to s=z1+z2s=z_1+z_2, p=z1z2p=z_1z_2: z14+z24=(s2−2p)2−2p2z_1^4+z_2^4=(s^2-2p)^2-2p^2, β3−α3=(β−α)(s2−p)\beta^3-\alpha^3=(\beta-\alpha)(s^2-p). A polynomial at a root of q(z)q(z): divide by qq, keep the remainder.
Square root of a complex number
4× asked, last 2026
a+ib=±(x+iy):x2−y2=a,2xy=b,x2+y2=a2+b2\sqrt{a+ib}=\pm(x+iy):\quad x^2-y^2=a,\quad 2xy=b,\quad x^2+y^2=\sqrt{a^2+b^2}
b>0b>0: x,yx,y have the same sign; b<0b<0: opposite signs. E.g. 5+12i=±(3+2i)\sqrt{5+12i}=\pm(3+2i).
Re\text{Re}Im\text{Im}5+12i5+12i3+2i3+2i−(3+2i)-(3+2i)θ2\frac\theta2θ\theta
5+12i=±(3+2i)\sqrt{5+12i}=\pm(3+2i): half the argument, 13\sqrt{13} the modulus, two opposite answers
Equations in z, z̄ and |z|
11× asked, last 2026
z2=kzˉ (k≠0) ⇒ z=0 or ∣z∣=∣k∣,  z3=k∣z∣2z^2=k\bar z\ (k\ne0)\ \Rightarrow\ z=0\ \text{or}\ |z|=|k|,\ \ z^3=k|z|^2
Take modulus first to get ∣z∣|z|, then multiply by zz, or put z=x+iyz=x+iy. Count z=0z=0 separately.
  • Not one ∣z∣2|z|^2 per equation. All nonzero roots of z2=kzˉz^2=k\bar z share ∣z∣=∣k∣|z|=|k|, and there are three of them. 4z2+zˉ=04z^2+\bar z=0: ∣z∣=1/4|z|=1/4, so ∑∣z∣2=3/16\sum|z|^2=3/16.
  • Quick method. Put z=x+iyz=x+iy and split: one part usually factors, e.g. x(2y+1)=0x(2y+1)=0, giving cases to test in the other. A real parameter gives a circle and a line: a root exists iff their distance ≤\le radius.

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Practise Complex Numbers

21 more concepts in 4 topics

Conjugate, modulus and argument; Polar form, De Moivre and roots of unity; Loci in the Argand plane; and Distances, rotation and regions. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Conjugate, modulus and argument

  • Conjugate5×
  • Modulus9×
  • Sum, difference and parallelogram law6×
  • Triangle inequality3×
  • Principal argument9×
  • Argument of products and quotients5×

Polar form, De Moivre and roots of unity

  • Polar and Euler form6×
  • Points on the unit circle6×
  • De Moivre's theorem18×
  • Cube roots of unity15×

Loci in the Argand plane

  • Circle, disc and ring7×
  • General equation of a circle7×
  • Perpendicular bisector and lines14×
  • Apollonius circle5×
  • Arc of fixed argument11×
  • Ellipse, square and parabola9×

Distances, rotation and regions

  • Nearest and farthest points18×
  • Rotation7×
  • Collinear points, equilateral triangles4×
  • Areas of regions4×
  • Intersections of loci12×

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