Complex Numbers — Complete JEE Mains 2026 Revision Capsule
Compact, exam-focused capsule for complex numbers. Learn the algebra, geometry, identities, De Moivre’s rules, roots of unity, loci in Argand plane, and typical JEE question patterns — all formulas in MathJax for crisp rendering.
$z=a+ib$, $\Re(z)=a$, $\Im(z)=b$, $|z|=\sqrt{a^2+b^2}$
$z=r(\cos\theta+i\sin\theta)=re^{i\theta}$
$z^n = r^n e^{in\theta}$, $n$th roots given by $r^{1/n}e^{i(\theta+2k\pi)/n}$
1. Basics & Notation
Complex number: $z=a+ib$ where $a,b\in\mathbb{R}$ and $i^2=-1$. Conjugate: $\bar z = a-ib$. Modulus: $|z|=\sqrt{a^2+b^2}$. Argument: $\arg z = \theta$ where $z$ corresponds to point $(a,b)$ with $\theta=\tan^{-1}(b/a)$ (careful with quadrant).
2. Algebraic Operations & Properties
Add / Subtract: componentwise: $(a+ib)\pm(c+id)=(a\pm c)+i(b\pm d)$.
Multiply: \[ (a+ib)(c+id) = (ac-bd) + i(ad+bc) \]
Divide: rationalize by conjugate: \[ \frac{a+ib}{c+id} = \frac{(a+ib)(c-id)}{c^2+d^2}. \]
Conjugate rules: \[ \overline{z+w}=\bar z + \bar w,\quad \overline{zw}=\bar z\bar w,\quad z\bar z=|z|^2. \]
3. Polar & Euler Form; De Moivre
Polar: $z=r(\cos\theta+i\sin\theta)$ where $r=|z|$, $\theta=\arg z$. Euler: $z=re^{i\theta}$.
De Moivre: \[ [r(\cos\theta+i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta). \]
Roots (n-th roots): if $z=re^{i\theta}$, solutions of $w^n=z$ are \[ w_k = r^{1/n}\,e^{i(\theta+2k\pi)/n},\quad k=0,1,\dots,n-1. \]
Roots of unity: $1^{1/n}$ are $e^{2\pi i k/n}$; they form regular n-gon on unit circle. Sum of all n-th roots of unity = 0.
4. Geometry & Loci on Argand Plane
Interpret $z=x+iy$ as point $(x,y)$. Common loci:
- Circle: $|z-a|=R$ is circle center $a$ radius $R$.
- Perpendicular bisector: $|z-z_1|=|z-z_2|$ (locus of points equidistant → line).
- Line (Cartesian): $\Re(\bar a z) = c$ represents straight line.
- Arg condition: $\arg\left(\dfrac{z-z_1}{z-z_2}\right)=\alpha$ → locus is arc/angle subtended by segment $z_1z_2$. For $\alpha=\pm\pi/2$ get circle with diameter.
Distance: $|z_1-z_2|$ gives Euclidean distance between points.
5. Transformations (Affine): $w=az+b$
Mapping $w=az+b$ (with $a=re^{i\phi}$):
- Scale by $r$ (dilation) if $|a|=r\neq1$.
- Rotate by $\phi$ (argument adds $\phi$).
- Translate by $b$ (adds vector $b$).
- These combined give a similarity transform: rotation + scaling + translation.
Special: $w=\bar z$ is reflection across real axis; $w=-z$ rotation by $\pi$ about origin.
6. Useful Identities & Inequalities
- Triangle inequality: $|z_1+z_2|\le |z_1|+|z_2|$.
- Reverse triangle: $||z_1|-|z_2||\le |z_1-z_2|$.
- Polar multiplication: $|z_1 z_2|=|z_1||z_2|$, $\arg(z_1 z_2)=\arg z_1+\arg z_2$ (mod $2\pi$).
- Arg relation: $\arg\left(\dfrac{z_1}{z_2}\right)=\arg z_1 - \arg z_2$.
7. Complex Equations — Common Patterns
Typical equations and solution techniques:
- Real/Imag parts: Write $z=x+iy$, equate real and imaginary parts.
- Conjugate tricks: Use $\bar z$ to remove imaginary parts, or multiply by conjugate.
- Modulus equations: $|z-a|=c$ → circle; $|z-a|=|z-b|$ → perpendicular bisector.
- Arg equations: Convert to tangent or use ratio $\frac{z-a}{z-b}$ and equate argument.
- Polynomial equations: Use factorization, roots of unity, symmetry.
8. Roots of Unity & Regular Polygons
n-th roots of unity: solutions of $z^n=1$ are: \[ \omega_k = e^{2\pi i k/n},\quad k=0,1,\dots,n-1. \]
Properties:
- They lie on unit circle, vertices of regular n-gon.
- Sum $\sum_{k=0}^{n-1} \omega_k = 0$.
- Primitive root: $\omega=e^{2\pi i /n}$ and powers give others.
9. Geometry Problems — Area / Perimeter
Area of triangle with vertices $z_1,z_2,z_3$: \[ \text{Area} = \frac12 \left| \Im\left( (z_2-z_1)\overline{(z_3-z_1)} \right) \right|. \]
Perpendicularity / parallelism: vectors $z_2-z_1$ and $z_3-z_1$:
- Perpiff: $(z_2-z_1)\overline{(z_3-z_1)}$ is purely imaginary.
- Parallel: quotient is real and positive.
10. Worked Examples (Step-by-step)
Example 1: If $z=3+4i$, find $|z|$, $\arg z$, and $\bar z$.
Solution: $|z|=\sqrt{3^2+4^2}=5$. $\arg z=\tan^{-1}(4/3)$. $\bar z=3-4i$.
Example 2: Solve $z + \bar z = 6$ and $z\bar z = 25$.
Solution: Let $z=x+iy$. Then $2x=6\Rightarrow x=3$. And $x^2+y^2=25\Rightarrow 9+y^2=25\Rightarrow y^2=16\Rightarrow y=\pm4$. So $z=3\pm4i$.
Example 3: Find the cube roots of 8 (i.e., solve $w^3=8$).
Solution: $8=8e^{i0}$, so roots: $w_k=8^{1/3} e^{i(0+2k\pi)/3}=2e^{2\pi i k/3}$ for $k=0,1,2$. Explicit: $2,\ 2e^{2\pi i/3},\ 2e^{4\pi i/3}$.
11. JEE Traps & Quick Tips
- Quadrant care: $\arg z = \tan^{-1}(b/a)$ needs quadrant correction.
- Multi-valued roots: $z^{1/n}$ provides n distinct roots — include all k values.
- Don't treat $\sqrt{z}$ like real case: complex square roots have two values; choose principal if defined.
- Dot vs complex product: $(\sqrt{z_1 z_2})$ is not generally equal to $\sqrt{z_1}\sqrt{z_2}$ in a branch; use polar to handle consistency.
- Use conjugate: equations mixing $z$ and $\bar z$ usually give real/imag conditions; convert to x,y.
12. Practice Problems (Try first)
- Find all complex z such that $|z-1|=2$ and $\arg(z)=\pi/4$.
- Solve $z^2 + (1-2i)z + (2-2i)=0$.
- Find points z such that $\arg\left(\dfrac{z-1}{z+1}\right)=\pi/2$ (geometric locus).
- Show that the centroid of triangle with vertices $z_1,z_2,z_3$ is $\dfrac{z_1+z_2+z_3}{3}$.
- Find sum of all 6th roots of unity multiplied by their conjugates.
Show Answers & Hints
- Point on circle of radius 2 centered at 1 with argument $\pi/4$ ⇒ $z=1+2e^{i\pi/4}=1+2\left(\dfrac{\sqrt2}{2}+i\dfrac{\sqrt2}{2}\right)=1+\sqrt2+i\sqrt2$.
- Solve quadratic using formula or convert to x+iy; roots are complex — use quadratic formula directly in complex arithmetic.
- $\arg\left(\dfrac{z-1}{z+1}\right)=\pi/2$ means $\dfrac{z-1}{z+1}$ is purely imaginary ⇒ equate real part to 0 and find circle (Apollonius-type locus) — result is circle passing through ±1.
- Centroid: average of vertices follows from vector addition; trivial from coordinates.
- 6th roots of unity are $\omega^k$. Each multiplied by its conjugate gives $1$ (since $|\omega^k|=1$) so sum = 6.
13. One-line Takeaways
- Switch freely between Cartesian and polar forms — each simplifies different problems.
- Use conjugates to handle real/imag constraints; use polar for multiplication, powers, and roots.
- Roots of unity and symmetry solve many polynomial problems elegantly.
- Argand geometry often converts tricky algebra into clean geometry — visualize!
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