Complex Numbers and the Argand Plane
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Direct answer
A complex number z = a + ib has modulus |z| = sqrt(a^2 + b^2) and argument arg z = tan^-1(b/a) placed in the correct quadrant by the signs of a and b. Writing z = r(cos theta + i sin theta) = r e^(i theta) turns multiplication into addition of arguments and multiplication of moduli, which is the key idea of the chapter: rotations, roots of unity and Argand-plane loci all flow from it.
What you must remember
- Powers of i cycle in steps of four (i, -1, -i, 1); the conjugate z-bar satisfies z × z-bar = |z|^2, which is the clean way to handle division.
- Modulus and argument algebra: |z1 z2| = |z1| × |z2|, arg(z1 z2) = arg z1 + arg z2; division subtracts arguments and divides moduli; the principal argument lies in (-pi, pi].
- De Moivre's theorem: (cos theta + i sin theta)^n = cos(n theta) + i sin(n theta); the n nth roots sit equally spaced on a circle forming a regular polygon, and their sum for roots of unity is 0.
- Cube roots of unity 1, omega, omega^2 with 1 + omega + omega^2 = 0 and omega^3 = 1; every integer power of omega reduces to 1, omega or omega^2.
- Triangle inequality: |z1 + z2| <= |z1| + |z2|, with equality when z1 and z2 point the same way; also |z1 + z2|^2 + |z1 - z2|^2 = 2(|z1|^2 + |z2|^2).
- Rotation: multiplying by i rotates by 90 degrees, by e^(i theta) by theta; (z - z1)/(z2 - z1) purely imaginary means a right angle at z1.
- Loci on the Argand plane: |z - z0| = r is a circle; |z - z1| = |z - z2| is the perpendicular bisector of the join; |z - z1| + |z - z2| = constant is an ellipse; arg((z - z1)/(z - z2)) = theta is an arc of a circle through z1 and z2.
Common confusion
The argument quadrant is the standard bleed. tan^-1(b/a) returns only a reference angle; the principal argument must be fixed using the signs of both a and b — for z = -1 + i, arg z = 3pi/4, not pi/4. A close second: real-coefficient equations pair roots with conjugates, quietly handing you the second root when one is revealed.
Exam-focused takeaway
JEE Main tests polar conversions, modulus-argument algebra, cube roots of unity and straightforward loci as quick numericals. JEE Advanced builds geometry on top: rotations about a vertex, roots of unity as regular-polygon vertices, and maximum or minimum of |z| on a given locus, often fused with other chapters. Statement questions probe principal-argument precision — state the quadrant before quoting tan^-1.
Frequently asked questions
What is the range of the principal argument?
(-pi, pi]; every non-zero complex number has exactly one principal argument in this interval.
How do I rotate a complex number about a point?
Translate the point to the origin, multiply by e^(i theta) to rotate by theta (by i for 90 degrees), then translate back.
What is the sum of all n nth roots of unity?
Zero; the roots satisfy x^n - 1 = 0, whose x^(n - 1) term has coefficient 0 — geometrically, their vector sum is the centre.
How do I find the square root of a complex number?
Assume the root x + iy, square, equate real and imaginary parts, and solve with the sign condition.
Which locus does |z - z1| - |z - z2| = constant describe?
A hyperbola with foci at z1 and z2, provided the constant is smaller than the distance between the foci.