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.

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## 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 $3 - 4i$.