Complex numbers extend the real number system, adding a new dimension to mathematical problem-solving. They facilitate the exploration of numbers beyond real ones, notably including the square roots of negative numbers.

Understanding Complex Numbers

• Definition: A complex number is expressed as $z = a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined as $\sqrt{-1}$.
• Imaginary Unit $( i )$: The core of the imaginary part of complex numbers, defined as $\sqrt{-1}$.
• Example: The complex number $2 + 3i$ has a real part of 2 and an imaginary part of 3.

Image courtesy of Cuemath

Real and Imaginary Parts

• Real Part $( \text{Re } z )$: In $z = a + bi$, $a$ is the real component.
• Imaginary Part $( \text{Im } z )$: In $z = a + bi$, $bi$ is the imaginary component.
• Example: For $4 + 5i$, $\text{Re } z = 4$ and $\text{Im } z = 5$.

Modulus of a Complex Number

• Formula: The modulus of $z$, denoted as $|z|$, is $|z| = \sqrt{a^2 + b^2}$.
• Physical Interpretation: Represents the distance from the origin in the complex plane.
• Example: For $1 + 1i$, the modulus is $\sqrt{1^2 + 1^2} = \sqrt{2}$.

Argument of a Complex Number

• Definition: The argument of a complex number, arg $( z )$, is the angle in radians from the positive real axis to the line from the origin to $z$.
• Calculation: Typically $\tan^{-1}(b/a)$.
• Geometric Interpretation: The direction of ( z ) in the complex plane.
• Example: For $1 + \sqrt{3}i$, arg$( z )$ is $\tan^{-1}(\sqrt{3}/1) = \pi/3$ radians.

Complex Conjugate

• Definition: The conjugate of $z = a + bi$ is $z^* = a - bi$.
• Usage: Useful in division, finding modulus, and argument.
• Example: The conjugate of $3 + 4i$ is $3 - 4i$.