Complex Numbers and Quadratic Equations

In the real number system $\mathbb{R}$, square roots of negative real numbers do not exist because the square of any real number is always non-negative ($x^2 \ge 0$ for all $x \in \mathbb{R}$). Simple quadratic equations such as $x^2 + 1 = 0 \implies x^2 = -1$ possess no real solutions. To overcome this limitation, the Swiss mathematician Leonhard Euler introduced the symbol $\mathbf{i}$ (iota) representing the imaginary unit.

The imaginary unit is defined by:

Higher positive integral powers of $i$ exhibit a strict 4-step cyclic pattern:

• $i^1 = i$
• $i^2 = -1$
• $i^3 = i^2 \cdot i = (-1) \cdot i = -i$
• $i^4 = (i^2)^2 = (-1)^2 = 1$

For any integer $n \in \mathbb{Z}$, dividing $n$ by $4$ using Euclid's division lemma gives $n = 4k + r$, where $k \in \mathbb{Z}$ and remainder $r \in \{0, 1, 2, 3\}$. Hence:

Negative powers are evaluated similarly: $i^{-1} = \frac{1}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i$, $i^{-2} = -1$, $i^{-3} = i$, $i^{-4} = 1$.

A number of the form:

is defined as a complex number. The set of all complex numbers is denoted by $\mathbb{C}$.

• $\mathbf{a = \text{Re}(z)}$ is called the Real Part of $z$.
• $\mathbf{b = \text{Im}(z)}$ is called the Imaginary Part of $z$ (Note: $\text{Im}(z)$ is the real coefficient $b$, not $ib$).
• If $b = 0$, $z = a$ is called purely real. Thus, every real number is a complex number ($\mathbb{R} \subset \mathbb{C}$).
• If $a = 0$ and $b \neq 0$, $z = ib$ is called purely imaginary.
• The number $0 = 0 + 0i$ is simultaneously purely real and purely imaginary.

Two complex numbers $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$ are said to be equal ($z_1 = z_2$) if and only if their real parts are equal and their imaginary parts are equal:

Order Property Note: There is no inequality or order relation in $\mathbb{C}$. Expressions like $z_1 > z_2$ or $z_1 < z_2$ are meaningless unless both numbers are purely real.

Arithmetic operations on complex numbers follow standard polynomial algebraic laws combined with the substitution $i^2 = -1$. The algebraic structure $(\mathbb{C}, +, \cdot)$ forms a mathematical field.

Let $z_1 = a + ib$ and $z_2 = c + id$ where $a, b, c, d \in \mathbb{R}$:

For any non-zero complex number $z = a + ib \neq 0$, its multiplicative inverse $z^{-1} = \frac{1}{z}$ satisfies $z \cdot z^{-1} = 1$:

If $z = a + ib$, its complex conjugate, denoted by $\bar{z}$, is obtained by reversing the algebraic sign of its imaginary part:

Geometrically, $\bar{z}$ represents the reflection (mirror image) of point $z$ across the real axis.

For any complex numbers $z, z_1, z_2 \in \mathbb{C}$:

1. $\overline{(\bar{z})} = z$ (Involution law)
2. $z + \bar{z} = (a + ib) + (a - ib) = 2a = \mathbf{2\,\text{Re}(z)}$ (Purely real)
3. $z - \bar{z} = (a + ib) - (a - ib) = 2ib = \mathbf{2i\,\text{Im}(z)}$ (Purely imaginary or zero)
4. $z = \bar{z} \iff \text{Im}(z) = 0 \iff z \text{ is purely real}$
5. $z + \bar{z} = 0 \iff \text{Re}(z) = 0 \iff z \text{ is purely imaginary}$
6. $\mathbf{z \cdot \bar{z} = (a + ib)(a - ib) = a^2 - (ib)^2 = a^2 + b^2 = |z|^2}$
7. $\overline{z_1 \pm z_2} = \bar{z}_1 \pm \bar{z}_2$
8. $\overline{z_1 z_2} = \bar{z}_1 \cdot \bar{z}_2$
9. $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z}_1}{\bar{z}_2} \quad (z_2 \neq 0)$
10. $\overline{z^n} = (\bar{z})^n$ for any $n \in \mathbb{N}$

Just as real numbers correspond to points on a 1D real number line, complex numbers correspond in a one-to-one fashion with points in a 2D Cartesian plane known as the Argand Plane or Gaussian Complex Plane.

Every complex number $z = a + ib$ is uniquely associated with the ordered pair of real numbers $(a, b)$ and plotted as a point $P(a, b)$:

• The horizontal $X$-axis is called the Real Axis (points $(a, 0)$ represent real numbers $a$).
• The vertical $Y$-axis is called the Imaginary Axis (points $(0, b)$ represent imaginary numbers $ib$).
• The origin $O(0, 0)$ represents the complex number $0 = 0 + 0i$.

The modulus (or absolute value) of $z = a + ib$, denoted by $|z|$, is the geometric Euclidean distance of the point $P(a, b)$ from the origin $O(0, 0)$:

Core Properties of Modulus:

• $|z| \ge 0$, and $|z| = 0 \iff z = 0$.
• $|z| = |-z| = |\bar{z}| = |-\bar{z}| = \sqrt{a^2 + b^2}$.
• $-|z| \le \text{Re}(z) \le |z|$ and $-|z| \le \text{Im}(z) \le |z|$.
• $z \cdot \bar{z} = |z|^2 \implies \mathbf{z^{-1} = \frac{\bar{z}}{|z|^2}}$.
• Multiplicative Property: $|z_1 z_2| = |z_1| \cdot |z_2|$ and $|z_1 z_2 \cdots z_n| = |z_1| |z_2| \cdots |z_n|$.
• Division Property: $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} \quad (z_2 \neq 0)$.
• Triangle Inequality: $$\mathbf{|z_1 + z_2| \le |z_1| + |z_2|} \quad \text{and} \quad \mathbf{|z_1 - z_2| \ge ||z_1| - |z_2||}$$ Equality $|z_1 + z_2| = |z_1| + |z_2|$ holds if and only if the origin $O$, $z_1$, and $z_2$ are collinear and $z_1, z_2$ lie on the same side of the origin ($\text{Arg}(z_1) = \text{Arg}(z_2)$).
• Parallelogram Law of Vectors: $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2(|z_1|^2 + |z_2|^2)$.

The directed angle $\theta$ made by the position vector $\vec{OP}$ with the positive direction of the real axis is called the argument or amplitude of $z$, denoted by $\arg(z)$ or $\text{amp}(z)$. Since $\theta$ is periodic with period $2\pi$, if $\theta$ is an argument, then $\theta + 2k\pi$ ($k \in \mathbb{Z}$) is also an argument.

To ensure a single unique value, the Principal Argument $\text{Arg}(z) = \theta$ is strictly restricted to the semi-open interval:

Algorithm for Finding Principal Argument:

1. Calculate the acute reference angle: $\mathbf{\alpha = \tan^{-1}\left|\frac{b}{a}\right|}$, where $0 < \alpha < \frac{\pi}{2}$.
2. Determine the quadrant of $P(a, b)$ and calculate $\theta = \text{Arg}(z)$ as follows:

• Positive Real Axis ($z = a > 0$): $\theta = 0$
• Negative Real Axis ($z = -a < 0$): $\theta = \pi$
• Positive Imaginary Axis ($z = ib, b > 0$): $\theta = \frac{\pi}{2}$
• Negative Imaginary Axis ($z = -ib, b > 0$): $\theta = -\frac{\pi}{2}$
• For $z = 0$, the modulus $|0| = 0$, but the argument is undefined.

While Cartesian coordinates $z = a + ib$ make addition and subtraction straightforward, multiplication, division, and raising to powers become dramatically simpler when expressed in polar trigonometric coordinates.

Let $P(a, b)$ represent $z = a + ib \neq 0$ with modulus $r = |z| > 0$ and principal argument $\theta = \text{Arg}(z)$. From right-angled triangle projection:

Substituting into $z = a + ib$ yields the polar form:

According to Euler's profound mathematical formula, $e^{i\theta} = \cos\theta + i\sin\theta$. Hence, any complex number can be expressed compactly in exponential form:

Let $z_1 = r_1 e^{i\theta_1} = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2 e^{i\theta_2} = r_2(\cos\theta_2 + i\sin\theta_2)$:

1. Multiplication: $$z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)} = r_1 r_2 [\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$$ $$\mathbf{|z_1 z_2| = |z_1| |z_2| = r_1 r_2} \quad \text{and} \quad \mathbf{\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)}$$ Geometric meaning: Multiplying $z_1$ by $z_2$ scales its distance from origin by $r_2$ and rotates its position vector anti-clockwise by angle $\theta_2$.
2. Division: $$\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)} = \frac{r_1}{r_2} [\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]$$ $$\mathbf{\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} = \frac{r_1}{r_2}} \quad \text{and} \quad \mathbf{\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)}$$

For any integer $n \in \mathbb{Z}$ and any real angle $\theta$:

Consequently, for $z = r(\cos\theta + i\sin\theta)$:

Extracting the square root of a complex number algebraically and solving the polynomial equation $x^3 = 1$ to uncover the geometric symmetry of the cube roots of unity are two signature topics in the WBCHSE Class 11 curriculum.

Let $\sqrt{a + ib} = x + iy$, where $x, y \in \mathbb{R}$. Squaring both sides:

Equating real and imaginary parts:

1. $x^2 - y^2 = a$
2. $2xy = b$

Using the algebraic identity $(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 = a^2 + b^2$:

Adding and subtracting the equations for $(x^2 + y^2)$ and $(x^2 - y^2)$:

The signs of $x$ and $y$ are dictated by $2xy = b$:

Standard Square Roots to Memorize:

• $\sqrt{i} = \pm \frac{1 + i}{\sqrt{2}}$
• $\sqrt{-i} = \pm \frac{1 - i}{\sqrt{2}}$

Consider the cubic equation $x^3 = 1 \implies x^3 - 1 = 0$:

This yields three roots:

1. Real root: $x = 1$
2. Complex roots from $x^2 + x + 1 = 0$: $$x = \frac{-1 \pm \sqrt{1^2 - 4(1)(1)}}{2} = \frac{-1 \pm \sqrt{-3}}{2} = \mathbf{\frac{-1 \pm i\sqrt{3}}{2}}$$

Designating one complex root as $\mathbf{\omega}$ (omega):

Thus, each imaginary cube root of unity is the square of the other!

In secondary school mathematics, when the discriminant $\Delta = b^2 - 4ac < 0$, students were told that the quadratic equation has "no real roots". In Class 11, with the complex number system $\mathbb{C}$, every quadratic equation possesses precisely two roots.

For $a, b, c \in \mathbb{R}$ with $a \neq 0$, the roots of $ax^2 + bx + c = 0$ are given by the quadratic formula:

When the discriminant $\Delta = b^2 - 4ac < 0$, we factor out $-1$ from the radical:

Thus, the roots are:

Crucial Condition: The coefficients $a, b, c$ must be real. If the coefficients are complex (e.g. $x^2 - ix + 2 = 0$), roots do not necessarily occur in conjugate pairs!

Let $\alpha, \beta \in \mathbb{C}$ be the roots of $ax^2 + bx + c = 0$:

• Sum of Roots: $\mathbf{\alpha + \beta = -\frac{b}{a}}$
• Product of Roots: $\mathbf{\alpha \beta = \frac{c}{a}}$
• Formation of Equation: A quadratic equation having roots $\alpha$ and $\beta$ is given by: $$\mathbf{x^2 - (\alpha + \beta)x + (\alpha \beta) = 0}$$