Complex Numbers and Quadratic Equations

In this chapter: the imaginary unit i, complex numbers and their equality, addition, multiplication, division and inverse, powers of i, square roots of negative numbers, modulus and conjugate, the Argand plane, polar form and argument, quadratic equations with negative discriminant.

Complex numbers

Define i = √−1, so that i² = −1. A number of the form z = a + ib, where a and b are real, is a complex number. We call a the real part, Re z, and b the imaginary part, Im z. Note that Im z is the real number b, not ib. Every real number a is the complex number a + i0; a number 0 + ib with b ≠ 0 is called purely imaginary.

Two complex numbers are equal only if their real parts are equal and their imaginary parts are equal: a + ib = c + id ⇔ a = c and b = d. This one line turns a single complex equation into two real equations, and most problems are solved this way.

Algebra of complex numbers

Let z₁ = a + ib and z₂ = c + id.

Addition and multiplication are closed, commutative and associative, and multiplication distributes over addition. The additive identity is 0 + i0 and the multiplicative identity is 1 + i0. Division is multiplication by the inverse: z₁/z₂ = z₁ · (1/z₂). In practice, multiply the numerator and denominator by the conjugate of the denominator.

Worked example: Express (5 + √2 i)/(1 − √2 i) in the form a + ib.
Solution: Multiply top and bottom by 1 + √2 i. Numerator: (5 + √2 i)(1 + √2 i) = 5 + 5√2 i + √2 i + 2i² = 3 + 6√2 i. Denominator: 1² + (√2)² = 3. So the quotient is 1 + 2√2 i.

Powers of i

The powers of i repeat in a cycle of four: i, −1, −i, 1. For any integer k:

So divide the power by 4 and look at the remainder: i³⁵ = i³² · i³ = −i. Negative powers work too, since 1/i = −i; for example i⁻³⁹ = 1/i³⁹ = 1/(−i) = i. The sum of four consecutive powers of i is always 0.

Square roots of negative numbers

For a positive real a, √(−a) = i√a, so √(−16) = 4i. The rule √a × √b = √(ab) holds when at least one of a, b is non-negative, but it fails when both are negative: √(−1) × √(−1) = i × i = −1, whereas √((−1)(−1)) = √1 = 1. Convert to i first and then multiply. The usual algebraic identities, such as (z₁ + z₂)² = z₁² + 2z₁z₂ + z₂², remain true for complex numbers.

Modulus and conjugate

For z = a + ib, the modulus is |z| = √a² + b² and the conjugate is z̄ = a − ib. For instance, |3 − 4i| = 5 and the conjugate of 3 − 4i is 3 + 4i.

The product rule for modulus saves a lot of work: |(1 + i)(2 + 3i)| = √2 × √13 = √26, with no need to multiply out.

The Argand plane and polar form

The complex number x + iy corresponds to the point (x, y) in a plane whose horizontal axis is the real axis and vertical axis the imaginary axis. This is the Argand plane. The modulus |z| is then the distance of the point from the origin, and the conjugate is the reflection of the point in the real axis.

With r = |z| and θ the angle OP makes with the positive real axis, x = r cos θ and y = r sin θ, so

θ is the argument, arg z. It is determined only up to multiples of 2π, so we use the principal argument, the value with −π < θ ≤ π. To find it, first find the acute angle α with tan α = |y/x|, then place it by quadrant:

Worked example: Write z = −1 − i√3 in polar form.
Solution: r = √(1 + 3) = 2. The acute angle α satisfies tan α = √3/1, so α = π/3. The point (−1, −√3) is in the third quadrant, so the principal argument is −(π − π/3) = −2π/3. Hence z = 2[cos(−2π/3) + i sin(−2π/3)]. Check: 2 cos(−2π/3) = −1 and 2 sin(−2π/3) = −√3.

Quadratic equations

For ax² + bx + c = 0 with real a, b, c and a ≠ 0, the discriminant is D = b² − 4ac. When D < 0 there are no real roots, but there are two complex roots:

These two roots are conjugates of each other. In general, non-real roots of a polynomial equation with real coefficients always come in conjugate pairs.

For JEE you also need the relations with the coefficients, familiar from Class 10: if α and β are the roots, α + β = −b/a and αβ = c/a, and the equation with roots α, β is x² − (α + β)x + αβ = 0.

Worked example: Solve x² + x + 1 = 0.
Solution: D = 1 − 4 = −3 < 0, so x = (−1 ± i√3)/2. Check with the relations: the sum of the roots is −1 = −b/a, and the product is (1 + 3)/4 = 1 = c/a. These two roots are the non-real cube roots of unity, usually written ω and ω², with ω³ = 1 and 1 + ω + ω² = 0.

Common mistakes: (1) Writing Im(3 + 4i) = 4i; the imaginary part is 4. (2) Multiplying √(−4) × √(−9) as √36 = 6; the correct value is 2i × 3i = −6. (3) Reading the argument straight off a calculator as tan⁻¹(y/x) without checking the quadrant. (4) Giving 4π/3 as a principal argument; it must lie in (−π, π], so the answer is −2π/3. (5) Forgetting that conjugate pairs are guaranteed only when the coefficients are real. (6) Dividing complex numbers term by term instead of multiplying by the conjugate of the denominator.

Exam focus

• Reducing expressions to a + ib, including quotients and high powers of i.
• Equating real and imaginary parts to find unknowns.
• Modulus and conjugate properties, especially z z̄ = |z|² and |z₁z₂| = |z₁||z₂|.
• Polar form and the principal argument in all four quadrants.
• Nature of roots from the discriminant, sum and product of roots, and solving quadratics with D < 0.

Practice questions