Algebra (Complex Numbers, Quadratic Equations, Binomial)

Syllabus Unit: Mathematics

Why it matters

Chapter overview

NDA's algebra unit moves from complex numbers (extending the real number system so every quadratic has a solution) through the quadratic equations that dominate numerical questions, into the counting techniques of permutations, combinations and the binomial theorem, and finally sequences, series and matrices — together the most formula-dense and fastest-to-score portion of the Mathematics paper.

Mathematics & General Ability Mathematics 40% weightage

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Complex Number z = a + bi

1Real Part (a)The horizontal coordinate on the Argand plane

2Imaginary Part (b)The vertical coordinate, coefficient of i

3Modulus |z|√(a²+b²), distance from the origin

4Argument (θ)tan⁻¹(b/a), angle from the positive real axis

1Complex Numbers & the Argand Plane

A complex number packs two real numbers into one, letting √(-1) exist as 'i' and giving every quadratic a solution.

Learning Objectives

Express a complex number in standard form a+bi and identify its real and imaginary parts.
Plot a complex number on the Argand plane and compute its modulus and argument.
Perform addition, subtraction, multiplication and division of complex numbers.
State and apply De Moivre's theorem for powers and roots of complex numbers.

Key Terms

Imaginary Unit (i)Defined by i²=-1; powers cycle as i, -1, -i, 1, repeating every 4th power.

Modulus |z|The distance of z=a+bi from the origin on the Argand plane, √(a²+b²).

Argument (amp z)The angle θ that the line to z makes with the positive real axis, θ=tan⁻¹(b/a).

Conjugate (z̄)a−bi for z=a+bi; the product z·z̄ = |z|² is always a real number.

A **complex number** z = a + bi extends the real numbers by introducing i, defined so that i² = -1. Here a is the real part and b is the imaginary part — together they let every polynomial equation, including x²+1=0, finally have a solution.

The Argand plane

Plot z=a+bi as the point (a,b): the horizontal axis is the real axis, the vertical axis is the imaginary axis. This turns algebra into geometry.

Modulus |z| = √(a²+b²) — the distance of the point from the origin.
Argument θ = tan⁻¹(b/a) — the angle from the positive real axis (adjusted for the correct quadrant).
Polar form: z = r(cosθ + i sinθ), where r = |z|.

Common MistakeAlways fix the quadrant by the signs of a and b before quoting θ — tan⁻¹(b/a) alone can give the wrong quadrant.

Algebra of complex numbers

(a+bi) ± (c+di) = (a±c) + (b±d)i; (a+bi)(c+di) = (ac−bd) + (ad+bc)i

Division is done by multiplying numerator and denominator by the conjugate of the denominator, which converts the denominator into a real number.

De Moivre's theorem

(cosθ + i sinθ)^n = cos(nθ) + i sin(nθ)

This single result powers both raising a complex number to a power and finding its nth roots — a recurring NDA shortcut.