Summary and Quick Revision

The Chapter in One Idea

Everything here flows from one invention:

Declare $i^2 = -1$, and every number becomes $z = a + ib$. Add, subtract and multiply exactly as with binomials (replacing $i^2$ by $-1$); divide by multiplying with the conjugate. The pair $(a, b)$ is a point in the plane, $|z|$ is its distance from the origin, and $\bar{z}$ is its mirror image in the real axis.

Hold that picture — algebra on the left, geometry on the right — and every formula below is a one-line consequence.

Powers of $i$ and Arithmetic — Formula Card

• The $i$-cycle: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, then it repeats. For $i^n$, divide $n$ by 4 and keep the remainder: $i^n = i^{n \bmod 4}$.
• Four in a row vanish: $i^n + i^{n+1} + i^{n+2} + i^{n+3} = 0$ for every integer $n$.
• Negative powers: $i^{-1} = -i$; in general $i^{-n} = \frac{1}{i^n}$, then rationalise.
• Square roots of negatives: for $a > 0$, $\sqrt{-a} = i\sqrt{a}$. Convert to $i$-form before multiplying — $\sqrt{-2}\,\sqrt{-3} = i\sqrt{2} \cdot i\sqrt{3} = -\sqrt{6}$, not $+\sqrt{6}$.
• Equality: $a + ib = c + id \Leftrightarrow a = c$ and $b = d$ — one complex equation is two real equations.
• Operations: add and subtract part by part; multiply like binomials with $i^2 = -1$; $(a+ib)(a-ib) = a^2 + b^2$.
• Inverse and division: $z^{-1} = \frac{\bar{z}}{|z|^2}$; to compute $\frac{z_1}{z_2}$, multiply top and bottom by $\bar{z_2}$.
• Speed shortcut: $(1+i)^2 = 2i$, $(1-i)^2 = -2i$, $\frac{1+i}{1-i} = i$, $\frac{1-i}{1+i} = -i$.

Modulus and Conjugate — Formula Card

For $z = a + ib$: $\ |z| = \sqrt{a^2 + b^2}$, $\ \bar{z} = a - ib$.

• The central identity: $z\bar{z} = |z|^2$ — it powers division, inverses and half the JEE problems on this chapter.
• Products and quotients: $|z_1 z_2| = |z_1|\,|z_2|$; $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$; $\overline{z_1 z_2} = \bar{z_1}\,\bar{z_2}$; $\overline{z_1 \pm z_2} = \bar{z_1} \pm \bar{z_2}$.
• Extracting parts: $z + \bar{z} = 2\,\text{Re}(z)$ and $z - \bar{z} = 2i\,\text{Im}(z)$.
• Reality tests: $z = \bar{z} \Leftrightarrow z$ is real; $z = -\bar{z} \Leftrightarrow z$ is purely imaginary (or 0).
• Never compare complex numbers with $<$ or $>$ — order exists only for their moduli (real numbers).
• Modulus of a power: $|z^n| = |z|^n$ — take moduli before expanding any product or power.

Argand Plane and Polar Form — Formula Card

• The picture: $z = a + ib$ is the point $(a, b)$; real numbers sit on the $x$-axis, purely imaginary numbers on the $y$-axis.
• Modulus = distance from the origin; more generally $|z_1 - z_2|$ is the distance between the two points.
• Conjugate = mirror image in the real axis.
• Standard loci: $|z| = r$ is a circle of radius $r$ about the origin; $|z - z_0| = r$ is a circle about $z_0$; $|z - z_1| = |z - z_2|$ is the perpendicular bisector of the segment joining $z_1, z_2$.
• Polar form: $z = r(\cos\theta + i\sin\theta)$ with $r = |z|$ and $\theta = \arg z$; principal argument lies in $(-\pi, \pi]$.
• Quadrant-first method for $\arg z$: plot the point, find the acute reference angle $\alpha = \tan^{-1}\left|\frac{b}{a}\right|$, then: QI $\to \alpha$, QII $\to \pi - \alpha$, QIII $\to -(\pi - \alpha)$, QIV $\to -\alpha$.
• Arguments add: $\arg(z_1 z_2) = \arg z_1 + \arg z_2$ (adjusted into the principal range).

JEE Quick Card — Quadratics, Roots and $\omega$

Quadratics with real coefficients ($ax^2 + bx + c = 0$):

• $D = b^2 - 4ac < 0 \Rightarrow$ roots $x = \frac{-b \pm i\sqrt{4ac - b^2}}{2a}$ — a conjugate pair; complex roots of real quadratics always come in conjugate pairs.
• Root relations: $\alpha + \beta = -\frac{b}{a}$, $\ \alpha\beta = \frac{c}{a}$ — often enough to answer without finding the roots.

Square roots of $a + ib$: set $(x + iy)^2 = a + ib$ and solve the three equations $x^2 - y^2 = a$, $2xy = b$, $x^2 + y^2 = \sqrt{a^2 + b^2}$; the sign of $b$ fixes the relative signs of $x, y$.

Cube roots of unity: $1, \omega, \omega^2$ where $\omega = \frac{-1 + i\sqrt{3}}{2}$:

• $\omega^3 = 1$, $\quad 1 + \omega + \omega^2 = 0$, $\quad \omega^2 = \bar{\omega} = \frac{1}{\omega}$.
• Reduce $\omega^n$ by $n \bmod 3$; substitute $1 + \omega = -\omega^2$ and $1 + \omega^2 = -\omega$ on sight.

Triangle inequality and max-min: $|z_1 + z_2| \le |z_1| + |z_2|$ and $\big||z_1| - |z_2|\big| \le |z_1 - z_2|$. If $|z - z_0| = r$, then $|z|$ ranges over $\left[\,\big||z_0| - r\big|,\ |z_0| + r\,\right]$.

Parallelogram law: $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2\left(|z_1|^2 + |z_2|^2\right)$.

Last-Minute Mistake Checklist

Before the exam, scan this list — each item is a real mark lost by real students:

1. $\sqrt{-a}\,\sqrt{-b} = -\sqrt{ab}$ for $a, b > 0$ — convert to $i$-form first; the rule $\sqrt{a}\sqrt{b} = \sqrt{ab}$ fails for two negatives.
2. Reducing $i^n$: use $n \bmod 4$, not $n \bmod 2$ — and $i^0 = 1$ when the remainder is 0.
3. $(1 - i)^2 = -2i$, not $2i$ — track the middle-term sign.
4. Dividing: multiply by the conjugate of the denominator, not the numerator.
5. $|z|^2 = z\bar{z}$, not $z^2$ — writing $|z|^2 = z^2$ silently assumes $z$ is real.
6. Principal argument lies in $(-\pi, \pi]$: for the third quadrant it is negative (e.g. $\arg(-1-i) = -\frac{3\pi}{4}$, not $\frac{5\pi}{4}$).
7. The argument formula $\tan^{-1}\frac{b}{a}$ alone is not enough — plot the quadrant first, then place the reference angle.
8. "One root is $2 - 3i$, coefficients real" $\Rightarrow$ the other root is $2 + 3i$ — use sum and product; do not solve from scratch.
9. $\omega^2 + \omega = -1$ (not $+1$), and $\omega^{2026} = \omega^{2026 \bmod 3} = \omega$ — reduce mod 3, not mod 4.
10. Never write $z_1 < z_2$ for non-real numbers — inequalities apply to $|z|$, $\text{Re}(z)$, $\text{Im}(z)$ only.