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Complex Numbers formula sheet

Every formula in the Complex Numbers chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 27 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 formula, and "last" is the latest year one appeared. ★ marks formulas 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=−iin+in+1+in+2+in+3=0\begin{aligned}&i^{4k}=1,\quad i^{4k+1}=i,\quad i^{4k+2}=-1,\quad i^{4k+3}=-i\\&i^n+i^{n+1}+i^{n+2}+i^{n+3}=0\end{aligned}
Reduce the exponent mod 4; in!=1i^{n!}=1 for n≥4n\ge4. −4−9=(2i)(3i)=−6\sqrt{-4}\sqrt{-9}=(2i)(3i)=-6, not 6.
Equality, real, imaginary
15× asked, last 2026
a+ib=c+id  ⟺  a=c, b=dz∈R  ⟺  z=zˉz purely imaginary  ⟺  z=−zˉ (z≠0)\begin{aligned}&a+ib=c+id\iff a=c,\ b=d\\&z\in\mathbb R\iff z=\bar z\\&z\text{ purely imaginary}\iff z=-\bar z\ (z\ne0)\end{aligned}
(a+ib)/(c+id)(a+ib)/(c+id) real iff bc=adbc=ad; purely imaginary iff ac+bd=0ac+bd=0.
Quadratics
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
Conjugate pair only for real coefficients. Use s=z1+z2s=z_1+z_2, p=z1z2p=z_1z_2 for symmetric expressions.
Square root
4× asked, last 2026
a+ib=±(x+iy): x2−y2=a, 2xy=bx2+y2=a2+b2\begin{aligned}&\sqrt{a+ib}=\pm(x+iy):\ x^2-y^2=a,\ 2xy=b\\&x^2+y^2=\sqrt{a^2+b^2}\end{aligned}
b>0b>0: same signs; b<0b<0: opposite. 5+12i=±(3+2i)\sqrt{5+12i}=\pm(3+2i).
Equations in z, z̄, |z|
11× asked, last 2026
z2=kzˉ (k≠0)⇒z=0 or ∣z∣=∣k∣z^2=k\bar z\ (k\ne0)\Rightarrow z=0\text{ or }|z|=|k|
Take modulus first, or put z=x+iyz=x+iy and split. Count z=0z=0 separately; three nonzero roots here.

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Practise Complex Numbers

22 more formulas 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 and parallelogram law6×
  • Triangle inequality3×
  • Principal argument9×
  • Argument rules5×

Polar form, De Moivre and roots of unity

  • Polar and Euler form6×
  • Unit circle6×
  • De Moivre's theorem18×
  • nth roots of unity
  • Cube roots of unity15×

Loci in the Argand plane

  • Circle, disc, ring7×
  • General circle7×
  • Perpendicular bisector and lines14×
  • Apollonius circle5×
  • Arc of fixed argument11×
  • Ellipse and square9×

Distances, rotation and regions

  • Nearest and farthest18×
  • Rotation7×
  • Collinear and equilateral4×
  • Areas of regions4×
  • Intersections of loci12×

Practise Complex Numbers next

About 160 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

Practise Complex Numbers

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