This problem aims to find the **real** and the **imaginary part** of a **complex number.** The concept required to solve this problem includes **complex numbers,** **conjugates, rectangular forms, polar forms,** and** magnitude of a complex number.** Now, **complex numbers** are the numerical values that are represented in the form of:

\[ z = x + y\iota\]

Where, $x$, $y$ are **real numerals,** and $\iota$ is an **imaginary numeral** and its value is$(\sqrt{-1})$. This form is called the **rectangular coordinate** form of a **complex number.**

The **magnitude** of a **complex number** can be obtained by taking the **square root** of the sum of **squares** of **coefficients** of the **complex number,** let’s say $z = x + \iota y$, the **magnitude** $|z|$, can be taken as:

\[ |z| = \sqrt{x^2 + y^2} \]

One other way to think of **magnitude** is the **distance** of $(z)$ from the **source** of the **complex number** **plane.**

## Expert Answer

To find the **polar form** of the given **complex number,** we will first calculate their **sum** to construct a **binomial form.** Two **complex numbers** can be summed using the **formula:**

\[ = (a_1 + b_1\iota) + (a_2 + b_2\iota) \]

\[ = (a_1 + a_2) + (b_1 + b_2)\iota \]

\[ = (a + b\iota) \]

The given **complex numbers** are $(1 + 2\iota) + (1 + 3\iota)$, substituting it gives us:

\[ = (1 + 2\iota) + (1 + 3 \iota) \]

\[ = (1+ 1) + (2+ 3)\iota \]

\[ = 2 + 5\iota \]

The next step is to find the **polar form,** which is another way to express the **rectangular coordinate** form of a **complex number.** It is given as:

\[ z = r( \cos \theta +\iota\sin\theta) \]

Where $(r)$ is the **length** of the **vector,** yielded as $r^2 = a^2+b^2$,

and $\theta$ is the **angle** created with the **real axis.**

Let’s calculate the **value** of $r$ by **plugging** in $a=2$ and $b=5$:

\[ r = \sqrt{a^2 + b^2} \]

\[ r = \sqrt{2^2 + 5^2} \]

\[ r = \sqrt{29} \]

\[ r \approx 5.39 \]

Now **finding** the $\theta$:

\[ \theta = \tan^{-1}(\dfrac{b}{a}) \]

\[ \theta = \tan^{-1}(\dfrac{5}{2}) \]

\[ \theta = 68.2^{\circ} \]

Plugging in these values in the above **formula** gives us:

\[ z = r( \cos\theta + \iota\sin\theta) \]

\[ z = \sqrt{29}(\cos(68.2) +\iota \sin(68.2)) \]

## Numerical Result

The **polar form** of the **rectangular coordinate complex** number is $z = \sqrt{29}(\cos(68.2) + \iota\sin(68.2))$.

## Example

Express the **rectangular form** of $5 + 2\iota$ in **polar form.**

It is **given** as:

\[ z = r(\cos\theta + \iota\sin\theta) \]

**Calculating** the value of $r$:

\[ r = \sqrt{a^2+b^2} \]

\[ r = \sqrt{5^2+2^2} \]

\[ r = \sqrt{29} \]

Now **finding** the $\theta$:

\[ \tan\theta = (\dfrac{b}{a}) \]

\[ \theta = \tan^{-1}(\dfrac{b}{a}) \]

\[ \theta = \tan^{-1}(\dfrac{2}{5}) \]

\[ \theta = 0.38^{\circ} \]

**Plugging** in these values in the above **formula** gives us:

\[ z = r(\cos\theta + \iota\sin\theta) \]

\[ z = \sqrt{29}(\cos(0.38) +\iota\sin(0.38)) \]

\[ z = 5.39(\cos(0.38) + \iota\sin(0.38)) \]