Complex Numbers and Quadratic Equations — Study Notes

1. Introduction

The real number system is incomplete in a fundamental way: the equation x^2 + 1 = 0 has no real solution because the square of a real number is never negative. To overcome this limitation, mathematicians introduced a new number, denoted by i, defined so that i^2 = -1. The symbol i is called the imaginary unit, and this chapter develops the entire system of complex numbers built upon it.

The chapter begins by defining complex numbers and the operations of addition, multiplication, and division on them. It then introduces the conjugate and modulus of a complex number, the Argand plane representation, and the polar form, including the famous formula of De Moivre. The chapter also explores the quadratic formula when the discriminant is negative, producing complex roots, and proves the fundamental theorem that every quadratic equation has exactly two roots in the complex number system.

Complex numbers are not just a theoretical curiosity. They are indispensable in electrical engineering, signal processing, quantum mechanics, and fluid dynamics. In mathematics, they connect algebra and geometry through the Argand plane and provide a natural setting for the Fundamental Theorem of Algebra. This chapter is the gateway to all of that.

2. Complex Numbers: Definition and Operations

A number of the form a + bi, where a and b are real numbers and i^2 = -1, is called a complex number. It is usually denoted by z = a + bi.

• The real part of z is a, written Re(z) = a.
• The imaginary part of z is b, written Im(z) = b.

Two complex numbers z1 = a + bi and z2 = c + di are equal if and only if a = c and b = d. A complex number with b = 0 is purely real; with a = 0, it is called purely imaginary.

Algebraic Operations

• Addition: (a + bi) + (c + di) = (a + c) + (b + d)i.
• Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i.
• Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i.
• Division: (a + bi)/(c + di) = [(a + bi)(c - di)]/(c^2 + d^2), valid when c + di is not zero.

The operations of addition and multiplication satisfy the commutative, associative, and distributive laws, just as in the real number system. The additive identity is 0 + 0i and the multiplicative identity is 1 + 0i.

3. Integral Powers of i

The powers of i follow a cyclic pattern:

i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, and then the cycle repeats.

In general, i^(4k) = 1, i^(4k+1) = i, i^(4k+2) = -1, i^(4k+3) = -i. To evaluate i^n, divide n by 4 and use the remainder.

Example: i^57 = i^(56+1) = i^56 x i = (i^4)^14 x i = 1 x i = i.

Also note i + i^2 + i^3 + i^4 = 0, and in general the sum of four consecutive powers of i is zero.

4. Conjugate and Modulus of a Complex Number

Conjugate

The conjugate of z = a + bi is z bar = a - bi. The conjugate is obtained by changing the sign of the imaginary part.

Modulus

The modulus of z = a + bi is |z| = sqrt(a^2 + b^2), which is always a non-negative real number.

Properties

• z . z bar = |z|^2.
• (z1 + z2) bar = z1 bar + z2 bar.
• (z1 z2) bar = z1 bar . z2 bar.
• z + z bar = 2 Re(z).
• z - z bar = 2i Im(z).
• |z1 z2| = |z1| |z2|.
• |z1/z2| = |z1|/|z2| for z2 not equal to 0.
• |z1 + z2| <= |z1| + |z2| (triangle inequality).

5. The Argand Plane

A complex number z = a + bi can be represented as the point (a, b) in the coordinate plane, called the complex plane or Argand plane. The x-axis is the real axis and the y-axis is the imaginary axis.

• The modulus |z| is the distance of the point from the origin.
• The distance between z1 = a1 + b1 i and z2 = a2 + b2 i is |z1 - z2| = sqrt((a1 - a2)^2 + (b1 - b2)^2).

The locus of points satisfying |z - z0| = r is a circle of radius r centred at z0.

6. Polar Form of a Complex Number

Let z = a + bi be a complex number represented by point P(a, b) in the Argand plane. Let r = |z| and theta be the angle made by OP with the positive real axis. Then:

a = r cos theta, b = r sin theta

so that z = r(cos theta + i sin theta). This is the polar form of z. Here r is the modulus and theta is called the argument of z, written arg(z).

The argument is not unique; adding any multiple of 2pi gives the same complex number. The value of the argument in (-pi, pi] is called the principal argument.

Conversion Between Forms

From Cartesian to polar: r = sqrt(a^2 + b^2), tan theta = b/a (with the correct quadrant taken into account). From polar to Cartesian: a = r cos theta, b = r sin theta.

Multiplication and Division in Polar Form

If z1 = r1 (cos theta1 + i sin theta1) and z2 = r2 (cos theta2 + i sin theta2), then:

z1 z2 = r1 r2 (cos (theta1 + theta2) + i sin (theta1 + theta2)) z1/z2 = (r1/r2)(cos (theta1 - theta2) + i sin (theta1 - theta2))

De Moivre's Theorem

(cos theta + i sin theta)^n = cos (n theta) + i sin (n theta) for any integer n.

7. Quadratic Equations with Real Coefficients

For the quadratic equation ax^2 + bx + c = 0 with a not equal to 0 and a, b, c real, the roots are:

x = (-b +/- sqrt(b^2 - 4ac))/(2a)

The quantity D = b^2 - 4ac is called the discriminant.

• If D > 0, the roots are real and distinct.
• If D = 0, the roots are real and equal.
• If D < 0, the roots are complex and are conjugates of each other.

When D < 0, the square root of the negative discriminant is handled using i. For example, the roots of x^2 + 1 = 0 are x = -i and x = i.

Relation Between Roots and Coefficients

If alpha and beta are the roots of ax^2 + bx + c = 0, then:

alpha + beta = -b/a and alpha . beta = c/a

These relations remain valid even when the roots are complex.

8. Quadratic Equations with Complex Coefficients

For the equation a z^2 + b z + c = 0 with complex coefficients, the same quadratic formula applies:

z = (-b +/- sqrt(b^2 - 4ac))/(2a)

In the complex number system, every quadratic equation has exactly two roots, and they may be complex. The sum and product relations alpha + beta = -b/a and alpha . beta = c/a hold here as well.

This result is a special case of the Fundamental Theorem of Algebra, which states that every polynomial equation of degree n has exactly n roots counted with multiplicity in the complex number system.

Quick Revision Tables

Table 1: Powers of i and Identities

Table 2: Roots of a Quadratic Equation by Discriminant

Mind Map

Important Diagrams (SVG)

Diagram 1: Argand Plane Representation

Diagram 2: Polar Form and the Unit Circle of Arguments

Common Mistakes

1. Writing i = -1 instead of i^2 = -1. The square of i is -1; i itself is not a real number and has no simple real value.
2. Confusing the conjugate with the negative of the complex number. The conjugate of a + bi is a - bi, while the negative is -a - bi.
3. Dividing by a complex number without multiplying the numerator and denominator by the conjugate of the denominator.
4. Evaluating powers of i incorrectly. Divide the exponent by 4 and use the remainder; i^57 = i, not -i.
5. Forgetting the minus sign when expanding (a + bi)(c + di); the i^2 = -1 term makes the real part ac - bd, not ac + bd.
6. Stating that the modulus of a complex number can be negative. |z| = sqrt(a^2 + b^2) is always non-negative.
7. In the quadratic formula, forgetting the +/- sign, or mishandling the negative discriminant instead of introducing i.
8. Finding the argument in the wrong quadrant. tan theta = b/a is ambiguous; the quadrant of the point (a, b) determines the correct angle.
9. Adding 2pi to arguments incorrectly; the principal argument must lie in (-pi, pi], and the angle must be chosen consistently.

Exam Tips

1. To simplify a quotient, always multiply numerator and denominator by the conjugate of the denominator and use the identity (c + di)(c - di) = c^2 + d^2.
2. For powers of i, remember the remainder pattern: remainders 0, 1, 2, 3 give 1, i, -1, -i respectively.
3. When using the quadratic formula with a negative discriminant, write sqrt(-D) as i sqrt(D) and simplify fully.
4. To convert a complex number to polar form, sketch or imagine the point in the Argand plane first to determine the correct quadrant of the argument.
5. Use the relations alpha + beta = -b/a and alpha . beta = c/a to build a quadratic equation whose roots are given, including when the roots are complex conjugates.
6. Practise the cyclic sum: i + i^2 + i^3 + i^4 = 0, which often simplifies lengthy expressions quickly.
7. In objective exams, test the answer by substituting the given root into the equation; if the discriminant is negative the roots must be conjugates of each other.

Conclusion

Complex numbers complete the number system by providing solutions to every polynomial equation, and they unify algebra with geometry through the Argand plane. The operations of conjugation, the modulus, and the polar form give us powerful tools for computation and visualisation, while De Moivre's theorem links complex numbers to trigonometry. The quadratic formula, extended to complex roots, ensures that every quadratic equation has exactly two roots, a fact that underlies the Fundamental Theorem of Algebra. These ideas are essential for advanced mathematics, physics, and engineering, and they form the basis for the study of complex analysis in higher education.