Complex Numbers and Quadratic Equations - Complex Numbers

The Imaginary Unit $i$: The concept of complex numbers begins with the imaginary unit $i$, defined as $i = \sqrt{-1}$, such that $i^2 = -1$. This allows us to solve equations like $x^2 + 1 = 0$, which have no real solutions. Visually, while real numbers exist on a single horizontal line, the introduction of $i$ adds a second dimension (the vertical axis) to our number system.

Complex Number Definition: A complex number $z$ is expressed in the form $a + ib$, where $a$ and $b$ are real numbers. Here, $a$ is the real part, denoted by $Re(z)$, and $b$ is the imaginary part, denoted by $Im(z)$. If $a = 0$, the number is 'purely imaginary'; if $b = 0$, the number is 'purely real'.

The Argand Plane: A complex number $z = a + ib$ is represented visually as a point $P(a, b)$ on a 2D coordinate plane called the Argand Plane (or Complex Plane). The horizontal x-axis is known as the 'Real Axis', and the vertical y-axis is the 'Imaginary Axis'. Every complex number corresponds to a unique point in this plane, and vice-versa.

Modulus of a Complex Number: The modulus of $z = a + ib$, denoted by $|z|$, is the non-negative real number $\sqrt{a^2 + b^2}$. Geometrically, $|z|$ represents the distance of the point $z$ from the origin $O(0, 0)$ in the Argand plane. It can be visualized as the length of the vector connecting the origin to the point $(a, b)$.

Conjugate of a Complex Number: The conjugate of $z = a + ib$ is denoted by $\bar{z}$ and is defined as $a - ib$. Visually, $\bar{z}$ is the mirror image of the point $z$ with respect to the Real Axis. For example, if $z$ is in the first quadrant at $(a, b)$, its conjugate $\bar{z}$ will be directly below it in the fourth quadrant at $(a, -b)$.

Equality and Algebra: Two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ are equal if and only if $a = c$ and $b = d$. For addition and subtraction, we combine the real parts and imaginary parts separately. For multiplication, we use the distributive property (FOIL) and replace $i^2$ with $-1$. Division is performed by multiplying both the numerator and denominator by the conjugate of the denominator to make the denominator a real number.

Powers of $i$: The powers of $i$ follow a cyclic pattern of four: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$. For any integer $n$, $i^{4n} = 1$, $i^{4n+1} = i$, $i^{4n+2} = -1$, and $i^{4n+3} = -i$. This cycle allows us to simplify any high power of $i$ by finding the remainder when the exponent is divided by 4.

Square Roots of Negative Real Numbers: For any positive real number $a$, the square root of $-a$ is expressed as $\sqrt{-a} = \sqrt{-1} \cdot \sqrt{a} = i\sqrt{a}$. In quadratic equations $ax^2 + bx + c = 0$, if the discriminant $D = b^2 - 4ac < 0$, the roots are complex and given by $x = \frac{-b \pm i\sqrt{4ac - b^2}}{2a}$.