Chapter 4: Complex Numbers and Quadratic Equations

4.1 Introduction

The extension of the real number system to the complex number system arises because there is no real number \( x \) that satisfies the equation \( x^2 + 1 = 0 \). We introduce a symbol \( i \) defined as \( i = \sqrt{-1} \), such that \( i^2 = -1 \). This allows us to define solutions for all polynomial equations.

4.2 Complex Numbers

A number of the form \( a + ib \), where \( a \) and \( b \) are real numbers, is defined to be a complex number. For the complex number \( z = a + ib \):

• \( a \) is called the real part, denoted by \( \text{Re}(z) \).
• \( b \) is called the imaginary part, denoted by \( \text{Im}(z) \).

Two complex numbers \( z_1 = a + ib \) and \( z_2 = c + id \) are equal if \( a = c \) and \( b = d \).

Algebraic Properties

1. Addition: \( (a + ib) + (c + id) = (a+c) + i(b+d) \)
2. Difference: \( (a + ib) - (c + id) = (a-c) + i(b-d) \)
3. Multiplication: \( (a + ib)(c + id) = (ac - bd) + i(ad + bc) \)
4. Division: \( \frac{z_1}{z_2} = \frac{z_1 \bar{z_2}}{z_2 \bar{z_2}} \)

Modulus and Conjugate

Let \( z = a + ib \).

• The conjugate of \( z \) is denoted by \( \bar{z} = a - ib \).
• The modulus of \( z \) is denoted by \( |z| = \sqrt{a^2 + b^2} \).

Important properties:

• \( z\bar{z} = |z|^2 \)
• \( |z_1 z_2| = |z_1| |z_2| \)
• \( \overline{z_1 \pm z_2} = \bar{z}_1 \pm \bar{z}_2 \)
• \( \overline{z_1 z_2} = \bar{z}_1 \bar{z}_2 \)

4.3 Argand Plane

A complex number \( z = x + iy \) can be represented geometrically by the unique point \( P(x, y) \) in the Cartesian plane, known as the Argand Plane or Complex Plane. The x-axis is called the Real Axis, and the y-axis is called the Imaginary Axis.

4.4 Quadratic Equations

For the quadratic equation \( ax^2 + bx + c = 0 \) with real coefficients, where \( D = b^2 - 4ac < 0 \), the roots are complex conjugate pairs: \[ x = \frac{-b \pm i\sqrt{4ac - b^2}}{2a} \]

Competency-Based Questions

Question 1:

Evaluate the following expression if \( z = 2 - \sqrt{-3} \).

Find the value of \( z^2 - 4z + 13 \).

Answer 1:

First, express \( z \) in the standard form.

\[ z = 2 - \sqrt{3}i \]

We need to find the value of \( z^2 - 4z + 13 \).

Let’s first compute \( z^2 \):

\[ z^2 = (2 - \sqrt{3}i)^2 = 2^2 - 2(2)(\sqrt{3}i) + (\sqrt{3}i)^2 \]

\[ z^2 = 4 - 4\sqrt{3}i + 3i^2 = 4 - 4\sqrt{3}i - 3 = 1 - 4\sqrt{3}i \]

Now calculate \( -4z \):

\[ -4z = -4(2 - \sqrt{3}i) = -8 + 4\sqrt{3}i \]

Substitute everything into the expression:

\[ z^2 - 4z + 13 = (1 - 4\sqrt{3}i) + (-8 + 4\sqrt{3}i) + 13 \]

The imaginary parts cancel out: \( -4\sqrt{3}i + 4\sqrt{3}i = 0 \).

Add the real parts:

\[ 1 - 8 + 13 = 6 \]

Conclusion: The value of the expression is 6.

Question 2:

An electrical circuit has an impedance \( Z_{1} = 4 + 3i \) ohms in series with another impedance \( Z_{2} = 2 - i \) ohms. Find the total impedance, and then find its modulus.

Answer 2:

For series impedances, the total impedance \( Z_{total} \) is the sum:

\[ Z_{total} = Z_{1} + Z_{2} = (4 + 3i) + (2 - i) \]

\[ Z_{total} = (4 + 2) + (3i - i) = 6 + 2i \]

Now compute the modulus:

\[ |Z_{total}| = |6 + 2i| = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \]

Conclusion: The total impedance is \( 6 + 2i \) ohms, and its magnitude is \( 2\sqrt{10} \) ohms.